Working with WKB waves far from the semiclassical limit

Physics Reports 397 (2004) 359 – 449 www.elsevier.com/locate/physrep

Working with WKB waves far from the semiclassical limit Harald Friedricha; b;∗ , Johannes Trosta b

a Physik-Department, Technische Universitat Munchen, 85747 Garching, Germany Department of Theoretical Physics and Atomic and Molecular Physics Laboratories, RSPhysSE, Australian National University, Canberra, A.C.T. 0200, Australia

Accepted 1 April 2004 editor: J. Eichler

Abstract WKB wave functions are expected to be accurate approximations of exact quantum mechanical wave functions mainly near the semiclassical limit of the quantum mechanical Schr4odinger equation. The accuracy of WKB wave functions is, however, a local property of the Schr4odinger equation, and the failure of the WKB approximation may be restricted to a small “quantal region” of coordinate space, even under conditions which are far from the semiclassical limit and close to the anticlassical or extreme quantum limit of the Schr4odinger equation. In many physically important situations, exact or highly accurate approximate wave functions are available for the quantal region where the WKB approximation breaks down, and together with WKB wave functions in residual space provide highly accurate solutions of the full problem. WKB wave functions can thus be used to derive exact or highly accurate quantum mechanical results, even far from the semiclassical limit. We present a wide range of applications, including the derivation of properties of bound and continuum states near the threshold of a potential, which are important for understanding many results observed in experiments with cold atoms. c 2004 Elsevier B.V. All rights reserved.
PACS: 03.65.Sq; 03.75.Be; 34.20.Cf; 03.65.Xp Keywords: Modi>ed WKB theory; Semiclassical and anticlassical limits; Threshold e?ects; Quantum reAection

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Semiclassical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 ∗

Corresponding author. Physics T30, Technical University Munich, James-Franck Str., 85747 Garching, Germany. Tel.: +49(89)28912729; fax: +49(89)28914656. E-mail addresses: [email protected] (H. Friedrich), [email protected] (J. Trost). c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2004.04.001

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2.1. The semiclassical and the anticlassical limit of the Schr4odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The WKB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Accuracy of WKB wave functions as a local property of the Schr4odinger equation . . . . . . . . . . . . . . . . . . . . . 2.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Beyond the semiclassical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Connection formulas at classical turning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The reAection phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Scattering by a repulsive singular potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Quantization in a potential well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Example: Circle billard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Example: Potential wells with long-ranged attractive tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Near the threshold of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Scattering lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Example: Sharp-step potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Example: Attractive homogeneous potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Near-threshold quantization and level densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Example: The Lennard-Jones potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Nonhomogeneous potential tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. The transition from a >nite number to in>nitely many bound states, inverse-square tails . . . . . . . . . . . . . . . . . 4.5. Tunnelling through a centrifugal barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Quantum reAection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Homogeneous potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Quantum reAection of atoms by surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Coupled channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

363 367 369 371 373 373 374 375 381 382 383 389 395 396 399 399 401 404 406 410 417 421 422 425 429 439 443 446 446

1. Introduction The relation between classical mechanics and quantum mechanics has been a central theme of physics ever since quantum mechanics was discovered at the beginning of the last century. Schr4odinger’s wave equation of quantum mechanics was published in 1926 [1], and in the same year Wentzel [2], Kramers [3] and Brillouin [4] developed the semiclassical approximation now known as the WKB approximation. It is largely equivalent to a formulation given earlier by Je?reys [5] in analogy to an approximation in wave optics which is expected to work well when the wavelengths are small compared to typical lengths in the physical system under investigation. The formulation of the WKB approximation is straightforward for conservative systems with just one degree of freedom and easily generalized [6] to systems with several degrees of freedom as long as they are integrable, meaning essentially, that their multidimensional classical dynamics can be decomposed into the dynamics of a corresponding number of conservative one-dimensional systems. It has been known since 1917 [7], that such generalizations are not straightforward for nonintegrable systems. Seemingly simple systems such as the three-body problem in celestial mechanics are nonintegrable, as was already pointed out by PoincarPe [8–10] who also observed that a possible

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consequence of nonintegrability is an extreme sensitivity of the dynamical evolution on small variations of the initial conditions, a phenomenon nowadays called “chaos”. The subtle complexities contained in the evolution of deterministic classical systems became a subject of general interest in the last quarter of the twentieth century, as continuous advances in computer capabilities enabled detailed numerical studies of nonintegrable classical systems on a large scale. This development rekindled interest in the fundamental relation between classical mechanics and quantum mechanics, in particular for nonintegrable systems with largely chaotic classical dynamics [11–13], but also for integrable [14,15] and almost integrable systems [16,17]. New semiclassical theories were developed, in which properties of a quantum spectrum such as the energy level density are expressed in terms of a sum over all periodic orbits of the corresponding classical system. An overwhelmingly large share of the immense body of research based on such periodic orbit theories has been devoted to “simple” model systems, mostly with one or two and rarely with more than two degrees of freedom. There have, however, also been more realistic applications in nuclear physics [13], solid state physics [18] and, in particular, atomic physics [19–23] which have brought forward many intriguing new insights on the interplay between classical dynamics and quantum mechanics. These developments have reinstated classical mechanics as a useful and relevant theory for atomic (and molecular) systems, even though quantum mechanics is the more general and undisputedly the valid theory in this domain. Semiclassical theories are expected to work well near the semiclassical limit of the Schr4odinger equation, which can be de>ned formally as the limit ˝ → 0. However, physically interesting situations do not always ful>ll the conditions of the semiclassical limit. Furthermore, semiclassical approximations may not be appropriate for all quantum states, even near the semiclassical limit. Consider, for example, an integrable system such as the circle billard [24–26]—a free particle moving in a two-dimensional space bounded by a circle. The bound states can be labelled by two good quantum numbers, one for the angular momentum around the centre of the circle and one for the radial motion; the semiclassical limit is the high-energy limit in this case. At high energies, there are many states for which both quantum numbers are large, but there are always some states for which one of the two quantum numbers is small. There is no energy regime where all states can be considered to be semiclassical. This is just one example showing the necessity to consider improvements of the conventional semiclassical approximations. Improvements of >rst-order semiclassical approximations are often formulated by including corrections of higher order in ˝ in the approximation of the Schr4odinger equation [27]. This works quite well in many cases, but it is not a general solution, because the resulting series is asymptotic and not convergent, and the dependence of physical quantities on the small parameter ˝ is in general nonanalytic near the semiclassical limit, ˝ → 0. “Soft” quantum e?ects due to interferences of contributions related to di?erent classical paths connecting given initial and >nal states may be well described via a series expansion in the small parameter ˝, but this approach is not appropriate for the description of “hard” quantum e?ects related to classically forbidden processes [28]. For example, probabilities for tunnelling through a smooth, analytic potential barrier are generally exponentially suppressed towards the semiclassical limit, so they do not contribute in any >nite order of ˝. Alternative methods of improving the conventional WKB approximation are proposed frequently, and we shall just mention a few recent examples: Bronzan [29] proposed a modi>ed WKB method which accurately reproduces the nodeless ground state of a system with one or more degrees of freedom; Gomez and Adhikari [30] introduced an algebraic reformulation of the WKB quantization condition which is useful in strongly con>ning potentials; a supersymmetric version of WKB

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quantization [31] is known [32,33] to yield exact energy eigenvalues for a large class of special (“shape invariant”) potentials. Further treatments of the one-dimensional Schr4odinger equation are based on the transformation of the second-order linear di?erential equation to a nonlinear >rst-order equation of the Ricatti type, which can be approximated via the quasilinearization method as described by Mandelzweig and Krivec [34,35]. The work reviewed in this article is based on a di?erent, less formal approach to improving the conventional WKB method. It exploits the fact, that the accuracy of WKB wave functions is a local property of the Schr4odinger equation. Away from the semiclassical limit, WKB wave functions may not be useful approximations in the whole of coordinate space, but there may be—and often are—large regions of coordinate space where the WKB wave functions are highly accurate, even if the conditions of the semiclassical limit are far from being ful>lled for the Schr4odinger equation as a whole. In many physically important situations, exact or highly accurate wave functions can be obtained for the “quantal regions” of coordinate space where the WKB approximation breaks down, e.g. when the potential in the Schr4odinger equation has a simple structure allowing analytical solutions in this region. By matching the exact or highly accurate wave functions from the quantal region to the WKB wave functions which are accurate in the “WKB region” outside the quantal region, we can construct wave functions which are accurate solutions of the Schr4odinger equation in the whole of coordinate space. These globally accurate wave functions can be used to derive quantum mechanical properties of the physical system which are highly accurate or asymptotically exact, even under conditions which are far from the semiclassical limit, all the way to the anticlassical or extreme quantum limit. This is demonstrated in a variety of applications in the following sections. Section 2 contains a brief review of some elements of semiclassical theory, including a discussion of the semiclassical and anticlassical limits of the Schr4odinger equation and of the (local) condition for the accuracy of WKB wave functions. In Section 3 we discuss improvements of the WKB method based on a generalization of the conventional connection formulas relating the WKB waves on either side of a classical turning point. This greatly increases the range of applicability and the performance of the WKB approximation as demonstrated in three di?erent situations: scattering by a repulsive potential, quantization in a potential well and classically forbidden tunnelling through a potential barrier. Section 4 is devoted to the narrow range of energies immediately above or below the threshold of a potential. In atomic and molecular physics, the near-threshold regime is of substantial current interest because it is relevant for the description of experiments involving ultra-cold atoms and molecules. Recent progress in the cooling of atoms to temperatures below the microkelvin regime and the successful construction of atomic degenerate quantum gases was rewarded with the Nobel Prize for physics in the years 1997 and 2001. There is intense and growing interest and activity in the >eld of ultra-cold atoms, e.g. in the preparation and understanding of quantum degenerate gases of bosonic and fermionic atoms [36,37], and also in the construction of atom-optical elements such as atom waveguides [38,39]. For potentials with tails falling o? faster than 1=r 2 the threshold represents the anticlassical or extreme quantum limit of the Schr4odinger equation where conventional WKB expansions break down. It turns out that several important near-threshold properties of the potential such as the quantization rule and the level density below threshold, and the mean scattering length and the reAectivity of the potential tail above threshold, are determined by three independent tail parameters which can be derived from the zero-energy solutions of the Schr4odinger equation. This information is of considerable practical use, not only in relation to cold atoms, but also for

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the accurate description of molecular reactions in near-threshold situations, e.g. for understanding the inAuence of quantum e?ects in the vibrational dissociation of molecules close to the dissociation threshold [40]. One example of “hard” quantum e?ects in the near-threshold region is quantum reAection by attractive potential tails as occur in the interaction of atoms and molecules with each other and with surfaces. In the quantum reAection of atoms by surfaces, the incoming atoms are reAected at distances of many hundreds or thousands of atomic units where the atom–surface interaction is accurately described by a local potential of the van der Waals type with retardation e?ects. The quantum reAected atoms do not reach the moderate distances of a few atomic units, where the atom–surface interaction becomes more intricate and leads to inelastic reactions or capture (sticking). For potentials falling o? faster than −1=r 2 , the probability for quantum reAection approaches unity at threshold, so it is always an important e?ect at suSciently low energies. The rapid progress in the >eld of cold atoms has made the quantum reAection regime increasingly accessible to laboratory experiments [41,42] and a number of investigations of this fundamental quantum process can be expected in the near future. The theory of quantum reAection is described in some detail in Section 5.

2. Semiclassical theory In this section we give a brief and necessarily sketchy review of some elements of semiclassical theory which will be needed for the subsequent sections. For more elaborate discussions, the reader is referred to textbooks and reviews [13,21,43–47]. 2.1. The semiclassical and the anticlassical limit of the Schrodinger equation Our starting point is the time-independent Schr4odinger equation for a particle of mass M moving under the inAuence of a potential V (r),



(r) +

2M [E − V (r)] (r) = 0 : ˝2

(1)

Most of the considerations of this section apply also to higher-dimensional systems, but subsequent sections are based essentially on the one-dimensional Schr4odinger equation (1). Planck’s constant ˝ de>nes a system-independent reference for measuring classical actions in the corresponding classical system. These depend on the local classical momentum
p(r) = 2M[E − V (r)] (2) and are given by  
Sc = p(r) dr = 2M[E − V (r)] dr :

(3)

In the more general case of D dimensions, r and p are replaced by D-dimensional vectors ˜r, p ˜ with components ri , pi , and the action that a particle accumulates on a trajectory C in 2D-dimensional

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phase space is  Sc =

D 

C i=1

pi dri :

(4)

When the potential is radially symmetric, V (˜r) = V (|˜r|), only the motion along the radial coordinate r = |˜r| is nontrivial; this is again described by a one-dimensional Schr4odinger equation (1), but the energy stored in the angular degrees of freedom shows up as a centrifugal potential after transformation of the equations of motion to spherical coordinates. It is generally accepted that quantum mechanical e?ects are important when typical classical actions are of the same order as or smaller than Planck’s constant ˝, and that the semiclassical limit, where quantum mechanical e?ects become less important, is the regime where typical classical actions are large compared to ˝. This leads to the formulation of a de>nition of the semiclassical limit of the Schr4odinger equation as “˝ → 0”, which is a little unfortunate, because ˝ is a well de>ned constant roughly equal to 10−34 J s. A more satisfactory de>nition of the semiclassical limit is to require Sc
˝

(5)

for typical actions Sc , and the “anticlassical” or “extreme quantum” limit is appropriately de>ned as Sc ˝ :

(6)

From Eq. (3) we see that increasing the mass M in the Schr4odinger equation leads towards the semiclassical limit, whereas for small M one can approach the anticlassical limit. In an atomic or molecular system, however, mass is generally a >xed and not a tunable parameter. Eq. (3) also suggests that the semiclassical limit might be reached for high energies, but this depends on the nature of the potential, as shown below. When the potential consists of a sum of homogeneous terms, the semiclassical and anticlassical limits can be reached for various combinations of large and/or small values of the energy and the strengths of the homogeneous terms, as discussed in Ref. [48], see also Section 5.3.4 in Ref. [49]. Here we give a brief derivation for one homogeneous term, i.e. for a potential with the property Vd (r) = d Vd (r) ;

(7)

the real number d denotes the degree of homogeneity, e.g., d = −1 for Coulomb potentials and d = +2 for the harmonic oscillator. Classical motion in homogeneous potentials has the property of mechanical similarity [50], i.e. if r(t) is a valid solution of the equations of motion at energy E, then r(1−d=2 t) is a solution at energy E
= d E. This rescaling of energy with a factor j ≡ d and of the coordinates according to s = r has the following e?ect on the classical action (3): 
√ Sc (jE) = 2M ds jE − Vd (s) 1 = j2

√

 2M

ds




E − j−1 Vd (s)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 1

= j2

√

1 1 = j2+d

 2M

ds

√

 2M




365

E − Vd (r)

dr




E − Vd (r) = j1=2+1=d Sc (E) :

(8)

This means that an increase in the absolute value |E| of the energy, j ¿ 1, results in an increase of the action Sc if and only if 1 1 + ¿0 ; 2 d

(9)

or equivalently d¿0

or

d¡ − 2 :

(10)

For homogeneous potentials of positive degree, such as all homogeneous oscillator potentials V ˙ |r|d , d ¿ 0, the semiclassical limit Sc =˝ → ∞ is reached for |E| → ∞, and the limit of vanishing energy, E → 0, corresponds to the anticlassical or extreme quantum limit. This intuitively reasonable behaviour also holds for negative degrees of homogeneity as long as d ¡ − 2. In contrast, for negative degrees of homogeneity in the range −2 ¡ d ¡ 0, the opposite and perhaps counterintuitive situation occurs: the limit of vanishing energy E → 0 de>nes the semiclassical limit, whereas |E| → ∞ is the anticlassical, extreme quantum limit. All attractive or repulsive Coulomb-type potentials, for which d = −1, fall into this category. This is consistent with the observation that the energy eigenvalues of the hydrogen atom scale the inverse square of the principal quantum number n and that the semiclassical limit corresponds to n → ∞, E → 0. For positive energies, the di?erential cross section for scattering by a 1=r potential in three spatial dimensions, the Rutherford cross section, is known to be identical in classical and quantum mechanics, but in two spatial dimensions this is not the case—here the quantum mechanical di?erential cross section approaches the classical result for E → 0, but for E → ∞ it deviates from the classical result and the prediction of the quantum mechanical Born approximation becomes more accurate [51–54]. The >ndings above may still be applicable for weakly perturbed homogeneous potentials, as long as the disturbance of the homogeneity (7) becomes negligible in the semiclassical or anticlassical limit. In some cases, the potential is a superposition of homogeneous terms with compatible semiclassical limits, e.g. a superposition of a quartic and an harmonic oscillator, V (r) = c2 r 2 + c4 r 4 . In this case, the high-energy limit E → ∞ corresponds to the semiclassical limit for both terms and for the potential as a whole. When the potential contains terms with incompatible limits, then the semiclassical limit may be reached neither for |E| → ∞ nor for E → 0. A hydrogen atom in a magnetic >eld is an example for such a situation. The electron feels the Coulomb attraction of the proton, and the magnetic >eld contributes a harmonic oscillator potential perpendicular to the >eld direction [19]. The semiclassical limit is not reached for high energies because of the Coulombic term, and it is not reached for low energies because of the harmonic contribution. For a given magnetic >eld strength, the system has no semiclassical limit. Note, however, that an appropriate simultaneous variation of energy and >eld strength can lead to the semiclassical or the anticlassical limit for this system [48].

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In order to see how the variation of the strength of a homogeneous potential a?ects the proximity to the semiclassical or anticlassical limits, consider the potential Vd (r) = V0 Ud (r) ;

(11)

where Ud (r) is a given homogeneous potential with degree d and V0 is a variable strength parameter. If r(t) is a solution of the classical equations of motion for the potential Ud (i.e. V0 = 1) at energy E, then r(t) is a solution for the potential strength V0 = −d at the same energy. For the action (3) this implies, 
Sc (V0 ) = ds 2M[E − V0 Ud (s)]  =

dr




2M[E − Ud (r)] = (V0 )−1=d Sc (1) :

(12)

For constant energy, the semiclassical limit Sc =˝ → ∞ is reached for V0 → 0 when the degree d of homogeneity is positive, and for V0 → ∞ when d is negative. The case d = −2, i.e. potentials proportional to the inverse square of the coordinates, is special and important. Such potentials occur in the interaction of a monopole charge with a dipole and as centrifugal potential in the radial part of the Schr4odinger equation in two or more dimensions. The classical action (8) becomes independent of energy for d = −2, but there is still a dependence on the potential strength according to Eq. (12). Since d is negative, the semiclassical limit is for in>nite potential strength while the anticlassical limit corresponds to the limit of vanishing strength of the inverse-square potential. For nonhomogeneous potentials with two or more energy or length scales, the de>nition of the semiclassical limit is not always straightforward and unambiguous. Consider, for example the Woods-Saxon potential step, VWS (r) = −

V0 ; 1 + exp(r=)

(13)

which is characterized by its depth V0 = ˝2 (K0 )2 =(2M) and the di?useness parameter . A quantum particle approaching such a step with a total positive energy E = ˝2 k 2 =(2M) is partially reAected (see Section 5), and the reAection probability PR is known [43,46] to be   sinh[(q − k)] 2 PR = ; (14) sinh[(q + k)]
where q = k 2 + (K0 )2 is the asymptotic wave number on the down side of the step, r → −∞. In the Schr4odinger equation (1) we can consider the formal limit ˝ → 0 while keeping the energy E and potential strength V0 >xed. This corresponds to taking the limit k → ∞ and K0 → ∞, and the reAection probability (14) becomes, PR

k;K0 →∞

∼

exp(−4k) ;

(15)

which is now independent of q. If on the other hand, we respect the fact that ˝ is >xed and study the high-energy (k → ∞) limit of Eq. (1), then the parameter K0 de>ning the height of the step

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

remains constant and the formula for the reAection probability becomes, 2  (K0 )2 k →∞ PR ∼  exp(−4k) : k

367

(16)

We see that the high-energy limit for PR di?ers from the “formal” semiclassical limit (15) by a factor inversely proportional to the energy. This is another subtle example demonstrating that the high energy limit need not coincide with the semiclassical limit Sc =˝ → ∞. 2.2. The WKB approximation For vanishing or constant potential, the local classical momentum (2) is constant, and the Schr4odinger equation (1) has plane wave solutions,   i (r) ˙ exp ± pr : (17) ˝ When the potential is not constant, p is a function of r, and looking at Eq. (17) suggests a more general ansatz for the wave function,   i S(r) : (18) (r) = exp ˝ Inserting (18) in the Schr4odinger equation (1) gives a di?erential equation for the function S(r), S
(r)2 − i˝S

(r) = p(r)2 : In order to obtain an approximate solution for S(r), and thus for the wave function S(r) in a formal series in ˝, which is regarded as a small parameter,  2 ˝ ˝ S2 (r) + · · · : S(r) = S0 (r) + S1 (r) + i i

(19) (r), we expand (20)

In the spirit of conventional semiclassical theory, the expansion (20) assumes that all other relevant quantities of the dimension of an action are large compared to ˝, as discussed in the preceding section. Inserting Eq. (20) in Eq. (19) gives a di?erential equation for the Si (r), p2 (r) − (S0
)2 + i˝(S0

+ 2S0
S1
) + ˝2 (S1

+ 2S0
S2
+ (S1
)2 ) : : : = 0 :

(21)

Eq. (21) can only be ful>lled if all terms of O(˝n ) vanish independently. Starting with n = 0 we get 
(22) S0 = ±p(r) ⇒ S0 = ± p(r) dr : In the classically allowed regions, the local classical momentum (2) is real, and we shall always assume p(r) to refer to the positive square root of 2M[E − V (r)]; except for a possible sign, S0 is the classical action Sc , Eq. (3). The terms of >rst order in ˝ in Eq. (21) give, S1
= −

S0

p
(r) ; = − 2S0
2p(r)

(23)

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and, after integration, 1 S1 = − ln p(r) : 2

(24)

After inserting these results in the ansatz (18), we obtain the general form of the >rst-order WKB wave function,    r C1 i


(r) = p(r ) dr exp WKB ˝ r0 p(r)    C2 i r

+
; (25) exp − p(r ) dr ˝ r0 p(r) with arbitrary complex coeScients C1 and C2 . In classically allowed regions, Eq. (25) represents a superposition of a rightward travelling (>rst term) and a leftward travelling (second term) wave. The lower integration point r0 is a “point of reference” which determines the phase of each term; this point of reference must always be speci>ed when de>ning a WKB wave function. An alternative choice of WKB waves is given by the sine or cosine of the WKB integrals, e.g.     r   1 1

  : (26) p(r ) dr  − cos  WKB (r) ˙
˝ r0 2 p(r) Two linearly independent WKB wave functions can be obtained by choosing two di?erent phases  in (26), but they must not di?er by an integral multiple of 2. In classically forbidden regions, V (r) ¿ E, the local classical momentum (2) becomes purely imaginary, but WKB wave functions can still provide a feasible approximation of the exact solution of the Schr4odinger equation. Two independent WKB wave functions are now given by    1 r 1


; (27) exp ± (r) ˙ |p(r )| dr WKB ˝ r0 |p(r)| which are exponentially increasing or decreasing functions of r. The terms of higher order in ˝ in Eq. (21) allow a systematic derivation of the terms Sn in the expansion (20). It is convenient [27] to introduce the functions n (r), n = −1; 0; 1; 2; : : :, via  r dr
n−1 (r
); Sn
= n−1 : (28) Sn (r) = r0

The n are then given by −1 = S0
= ±p;

0 = S1
= −

p
; 2p

and the recursion relation   n 1 
 n+1 = − n + j n− j  2−1 j=0

(29)

for n ¿ 0 :

(30)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

In particular, 1 1 = S2
= − (
+ 02 ) = ± 2−1 0



p

3 p
2 − 4p2 8 p3

369

:

(31)

The series de>ned by Eq. (20) is asymptotic and does not converge. The inclusion of higher and higher terms eventually gives less accurate results. There are special potentials where the series terminates. For example, for potentials V (r) ˙

1 ; (ar + b)4

(32)

with real constants a and b, all n (r), n ¿ 1 vanish identically at zero energy, so the WKB wave functions (25) or (27) are exact solutions of the Schr4odinger equation for E = 0. For V (r) = (ar 2 + br +c)−2 and E =0, we >nd 2 ≡ 0, but we are left with the calculation of the remaining integrals in Eq. (28). 2.3. Accuracy of WKB wave functions as a local property of the Schrodinger equation The WKB approximation is, of course, expected to work well near the semiclassical limit ˝ → 0, see Section 2.1. However, since the expansion (20) depends on the coordinate r, the accuracy of a truncated expansion including a given number of terms must be expected to also depend on r. A frequently formulated condition for the accuracy of the WKB approximation is based on the requirement, that the second term in the left-hand side of Eq. (19) should be small compared to the >rst term, |i˝S

(r)||S
(r)2 | ; inserting S
(r) ≈ p(r) according to Eq. (22) gives  
   p (r)   1  d    1 ; ˝ 2  = (r)  p (r) 2  dr

(33)

(34)

which corresponds to the requirement that the local de Broglie wavelength, (r) = 2˝=p(r) should be slowly varying. Note, however, that the leading contribution to S

on the left-hand side of Eq. (33), namely

S0 , is already included via the >rst-order terms (23), (24) in the >rst-order WKB wave functions (25)–(27), so it does not make sense to require this term to be small as a condition for the accuracy of the wave functions. Indeed, the frequently accepted condition (34) is not necessary for the >rst-order WKB wave functions to be accurate. A striking counter-example is the potential (32) at energy E = 0. Although >rst-order WKB wave functions are exact solutions of the Schr4odinger equation, the derivative of the de Broglie wavelength goes to in>nity for r → ∞. As a criterion for the accuracy of the WKB wave functions, it makes more sense to require smallness of the >rst term of the series (20) which is not considered in the de>nition of the wave functions, i.e., the term (31). This idea is supported by considering the modi>ed Schr4odinger equation for which the WKB wave function is an exact solution. By calculating the second derivative of the

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WKB wave function (25) [or (26) or (27)] it is easy to see that Schr4odinger-type equation  p(r)2 3 p
2 p



− WKB (r) − WKB (r) = 0 : WKB (r) + ˝2 4 p2 2p

WKB

is an exact solution of the (35)

It should be a good approximation to the solution of the original Schr4odinger equation if the third term on the left-hand side of (35) is small compared to the second term, i.e. the absolute value of the function   3 (p
)2 p

2 Q(r) = ˝ (36) − 4 p4 2p3 should be much smaller than unity, |Q(r)|1 :

(37)

This corresponds to requiring |1 =−1 | to be small, where 1 is given in Eq. (31); 1 is the derivative of S2 and (−˝2 )S2 is the >rst term in the series (20) not to be considered in the de>nition of the (>rst-order) WKB wave functions. The function Q(r) de>ned in Eq. (36) can be positive or negative, but it is always real, even in a classically forbidden region where the local classical momentum p(r) is purely imaginary. The condition (37) is a local condition, because Q is a function of r. This function has been called “badlands function” [55,56], because it is large in regions of coordinate space where the WKB approximation is bad. On the other hand, large (positive or negative) values of Q(r) mean that quantum e?ects are important in this region of coordinate space. So a more positive name for the function (36) is the “quantality function”. In regions of high quantality, the condition (37) is violated, the WKB approximation is poor and quantum e?ects are important. Now it is also clear that the simple condition Eq. (34) is not in general suScient for the WKB wave function to be an accurate approximation of an exact solution of the Schr4odinger equation. Consider a potential oscillating rapidly with small amplitude and a moderate total energy such that the local de Broglie wavelength 2˝=p(r) behaves as 1 + sin(qr)=q3=2 . In the limit of large q values, the simple condition (34) is ful>lled suggesting good applicability of the WKB approximation. However, √ the term involving the second derivative in Eq. (36) gives a contribution proportional to q sin qr, which results in a diverging quantality for large q. By analogous arguments it follows that small but nonvanishing values of |Q(r)| are not always suScient for the WKB approximation to be accurate; it is in principle possible that p


which contributes to the next-order correction 2 according to Eq. (30) is large even though |Q(r)| is small. Nevertheless, smallness of |Q(r)| is clearly a more reliable indication of the accuracy of the WKB approximation than the simple condition Eq. (34). Outside the quantal region, the WKB wave function is often an excellent approximation to the exact solution of the Schr4odinger equation, even if the global condition for semiclassical approximations, i.e. small ˝, Eq. (5), is not ful>lled. In such situations, it may be possible to >nd accurate solutions of the Schr4odinger equation in the quantal regions—by analytical or numerical means—and then construct globally accurate wave functions by appropriately matching the exact (or highly accurate) wave functions from the quantal region to WKB wave functions in the semiclassical, “WKB regions” where the condition (37) is well ful>lled. This simple philosophy underlies the various applications discussed in the subsequent sections of this review.

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

371

2.4. Examples A simple set of examples is provided by step-wise constant potentials, Vsteps (r) = Vi ;

r ∈ Li ;

(38)

where {Li } represents a covering of coordinate space with a set of intervals, Li = (ri ; ri+1 ). At total energy E, the motion in each interval Li is classically allowed if E ¿ Vi and forbidden if E ¡ Vi . WKB wave functions are exact within each interval. In the classically allowed
intervals there are two independent solutions proportional to exp(±iki r) with wave number ki = 2M(E − Vi )=˝, in the classically forbidden intervals there
are two independent solutions proportional to exp(±%i r) with inverse penetration depth %i = 2M(Vi − E)=˝. The quantality function (36) is zero everywhere, except at the borders ri of the intervals, and a global exact solution of the Schr4odinger equation can be obtained by matching superpositions of the two independent solutions in each interval such that the wave function and its >rst derivative are continuous. This method gives exact results regardless of any consideration of the semiclassical limit. In fact, it is just how one would go about solving the Schr4odinger equation without any knowledge of or reference to WKB wave functions. At each step ri , the >rst derivative of the wave function should be continuous, but the second derivative should not, because in the Schr4odinger equation (1), the discontinuity in the potential must be compensated by a corresponding discontinuity in

. For a single discontinuity separating two regions 1 and 2 where V (r) is constant, we have a sharp-step potential. A quantum particle approaching the step on the upper level is partially reAected, and the reAection probability is √  √ 2  E − V1 − E − V2 q−k 2 √ PR = = √ (39) q+k E − V1 + E − V2 where q and k are the wave numbers on the down side and the up side of the step. Eq. (39) is a standard textbook result; it also follows from the reAection probability for the Woods-Saxon potential (13) in the limit of vanishing di?useness,  → 0. The formula (39) does not contain ˝, i.e. it is not a?ected by taking the formal semiclassical limit, ˝ → 0. One way of understanding this is to realize, that the characteristic length of the potential, de>ned as the distance over which it changes appreciably, is zero for the sharp step. This is always small compared to quantum mechanical lengths such as wavelengths or penetration depths, regardless how close we may be to the semiclassical limit. (See also footnote in Section 25 of Ref. [43]). In this context it is interesting to discuss the case that the potential V is itself continuous, but has a step-like discontinuity in one of its derivatives. Assume that V (i) is continuous for 0 6 i ¡ n and that the nth derivative V (n) (r) is discontinuous at a point r0 . The next derivative, V (n+1) (r), then has a delta-function singularity at r0 , and so do the (n + 1)st derivative of p(r) and the function
(r), cf. Eqs. (29), (31). The function S Sn+1 n+1 (r) has a step-like discontinuity at r0 , and this enters in order ˝n+1 in the expansion (20). The ansatz (18) thus contains a step-like discontinuity of order ˝n at the point r0 , which is incompatible with the requirement, that (r) be a continuous and (at least) twice di?erentiable function. The continuity of the wave function (18) is repaired by adding a second term with an amplitude of order O(˝n ). In the classically allowed regime, this leads to classically forbidden reAection with a reAection amplitude of order O(˝n ). The case n = 0

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

as for the sharp step potential leads to a reAection amplitude of order ˝0 , i.e., independent of ˝. For a continuous potential with a kink, i.e. with a step-like discontinuity in the >rst derivative, the amplitude for classically forbidden reAection vanishes as ˝ in the semiclassical limit. For analytical potentials, which are continuously di?erentiable to all orders, the amplitude for classically forbidden reAection generally vanishes exponentially in the semiclassical limit, e.g. as exp(−C=˝), see Section 5, Eqs. (253), (254). For homogeneous potentials (7), the semiclassical and anticlassical limits can be reached by appropriate variations of the energy and/or the potential strength, see Section 2.1. Negative degrees d corresponding to inverse-power potentials, V& (r) = ±

C& ; r&

&¿0 ;

(40)

occur in the description of various physical phenomena. For example: & = 1 for Coulomb potentials, & = 2 for centrifugal or monopole-dipole potentials, & = 3 for the van der Waals potential between a neutral polarizable particle and a surface, & = 4 for the interaction between a neutral and a charged particle, &=6 for the van der Waals potential between two neutral particles and &=7 for the retarded van der Waals potential between two neutral particles [57]. At zero energy, the local classical momentum (2) in the repulsive or attractive potential (40) is proportional to r −&=2 , and the quantality function (36) has a very simple form, &  & −2 & 1− r : (41) Q(r) ˙ 4 4 As discussed for the potential (32), Q(r) ≡ 0 for the special case & = 4. More importantly, Q(r) is seen to vanish for small r when & ¿ 2 and for large r when & ¡ 2. For Coulombic potentials and near-threshold energies, WKB wave functions become increasingly accurate for r → ∞. For >nite (positive or negative) energies, the inAuence of the potential becomes less important for r → ∞ and the local classical momentum tends to a constant. WKB wave functions become increasingly accurate for r → ∞ for any power & ¿ 0. For positive energies E = ˝2 k 2 =(2M), the error in the WKB wave function decreases asymptotically as 1=(kr)&+1 when its phase is correctly matched [58]. Towards small r values, a >nite energy in the Schr4odinger equation becomes increasingly irrelevant as r → 0, so the result (41) holds in this case as well. For & ¿ 2, WKB wave functions become increasingly accurate for r → 0. For attractive potentials, this singular behaviour cannot continue all the way to the origin, but there may be a region of small but nonvanishing r values where the WKB approximation is highly accurate. Repulsive inverse-power potentials can be meaningfully used all the way down to r = 0, and, for & ¿ 2, the WKB wave functions accurately describe the behaviour of the wave function near the origin. Expressing the strength C& of the potential (40) through a length & , C& = ˝2 (& )&−2 =(2M), we have   (&−2)=2  1 & r →0 &=4 : (42) ; '= (r) ˙ r exp −2' r &−2 For 0 ¡ & ¡ 2, the quantality function diverges for r → 0, and the WKB approximation fails near the origin. For the special case & = 2, the quantality function tends towards a constant as r → 0. For a repulsive inverse-square potential, C2 = ˝2 (=(2M), ( ¿ 0, the WKB wave function WKB and

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

the exact wave function r →0 1 +√( ; WKB (r) ˙ r 2

373

behave a little di?erently near the origin, r →0

1 √

(r) ˙ r 2 +

(+1=4

:

(43)

Inverse-square potentials will be discussed in more detail later on; see in particular Eq. (64) in Section 3.3 and the whole of Section 4.4. 3. Beyond the semiclassical limit As was shown in the preceding section the accuracy of the WKB approximation is a local property of the Schr4odinger equation in coordinate space rather than depending on the global validity of the conditions of the semiclassical limit. By bridging the gaps due to regions of high quantality, appropriate WKB wave functions may be constructed and used to derive accurate quantum results, even under conditions which are globally far from the semiclassical limit. In this section we discuss concrete applications to three di?erent physical situations: scattering by singular potentials, quantization in potential wells and tunnelling through potential barriers. 3.1. Connection formulas at classical turning points At a classical turning point, rt , of a one-dimensional system the local classical momentum (2) vanishes, p(rt ) = 0, and the quantality function Q(r) [Eq. (36)] is singular. The exact wave function may nevertheless be well represented by a superposition of exponentially increasing and decreasing WKB waves (27) on the classically forbidden side of rt , and/or by oscillating WKB waves (25), (26), on the classically allowed side. For a consistent description of both regions, we have to say how the WKB waves on either side of rt are to be connected. In the most general case, the connection formulas can be written as       r     N 2 1  1  r





↔ − (44) cos exp − p(r ) dr p(r ) dr  2  ; ˝  rt ˝  rt p(r) |p(r)|       r U   1 NU  1  r 1 




↔
(45) cos exp p(r ) dr  − p(r ) dr  :   ˝ rt 2 ˝ rt p(r) |p(r)| The two parameters  and N in Eq. (44) can be determined by comparing the exact solution corresponding to an exponentially decreasing wave on the classically forbidden side with the oscillating WKB waves on the allowed side. Since the decaying wave on the forbidden side is unique except for a constant factor, the parameters  and N are well de>ned. In contrast, the asymptotic behaviour of the exponentially increasing solution in Eq. (45) masks any admixtures of the decaying wave; since such an admixture signi>cantly a?ects phase and amplitude of the wave function on the allowed side, the two parameters U and NU are not well de>ned. In conventional semiclassical theory [43–47], the connection formulas are derived assuming that the potential can be approximated by a linear function of r in the vicinity of rt , and that this region extends “suSciently far” to either side of rt . “SuSciently far” means that the exact solutions of the Schr4odinger equation in the linear potential, which can be expressed in terms of Airy functions [59],

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

are valid until the asymptotic forms of the respective Airy functions can be matched to the WKB waves on both sides. For the well de>ned parameters  and N this conventional matching leads to  conv = ; N conv = 1 : (46) 2 The continuity equation relates the four parameters , N , U and NU [60]. To see this consider a superposition of Eqs. (44) and (45) with arbitrary complex coeScients, = A × (44) + B × (45), and calculate the current density, j = (˝=M)I( ∗
). For de>niteness we assume the classically allowed (forbidden) region to lie to the left (right) of rt . Inserting the left-hand sides of Eqs. (44) and (45) in the superposition gives the current density on the classically allowed side,  2  − U ∗ I(A B) sin ; (47) jallowed = M 2 whereas inserting the right-hand sides gives the current density on the classically forbidden side, 2 jforbidden = (48) I(A∗ B)N NU : M The conservation of the current density, jallowed = jforbidden , requires  U  −  : (49) N NU = sin 2 The ill-de>ned parameters U and NU are irrelevant for a totally reAecting potential. Here the wave function must vanish asymptotically in the classically forbidden region, so the coeScient of the exponentially growing solution (45) must vanish. In the more general case, the uncertainty in the de>nition of the barred quantities can be removed by introducing a second condition by convention, e.g. by assuming a >xed phase di?erence of =2 between the oscillating waves in Eqs. (44) and (45). Such a choice is used in the de>nition of the irregular continuum wave function in scattering theory [49,61]. From Eq. (49) it would follow that NU = 1=N . However, other choices are possible as long as the left-hand sides of Eqs. (44) and (45) remain linearly independent. 3.2. The re?ection phase In the case of total reAection at a classical turning point the current density is zero and it is possible to represent the quantum wave by a real function. The expressions (44) represent the exact wave function in the semiclassical regions on either side of the turning point. The amplitude parameter N can be used to calculate the particle density | |2 in the classically forbidden region [62], but the choice of N is not so important in the classically allowed region if the normalization of the wave function is either irrelevant or deducible from other considerations. The more important quantity in many cases is the phase , which inAuences the phase of the WKB wave in the whole of the classically allowed region. Rewriting the left-hand side of Eq. (44) as      r     ei=2 i  i  r ()

 −i


+ e (50) (r) = p(r ) dr exp p(r ) dr exp − WKB   ˝  ˝ p(r) rt

rt

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

375

reveals that  is the phase loss su?ered by the WKB wave due to reAection, the “reAection phase” [63,64]. With the conventional choice (46) of the reAection phase and the amplitude factor, the WKB wave function in the classically allowed region is     r   2 1  conv

 : (51) p(r ) dr  − cos WKB (r) =
˝  rt 4 p(r) For this approximation to be valid, the Airy function must match the exact wave function both in the classically allowed as well as in the classically forbidden region. The potential must be close to linear in a region extending at least a few times the wavelength of the Airy function on the classically allowed side and a few times the penetration depth on the forbidden side. This condition is sometimes hard to meet, even though the general condition (37) for accuracy of the WKB approximation may be well ful>lled on both sides of the classical turning point rt , e.g. when the kinetic energy of the particle is small and thus the wavelength large even far away from rt . Sometimes the slow variation of the potential compared to the wavelength and penetration depth coincides with the linearity requirement. However, there are important examples where these conditions are not ful>lled at the same time. The separation of the linearity requirement and the resulting >xation (46) of the reAection phase from the application of the WKB approximation greatly widens the range of applicability of WKB waves to a variety of new situations. A very simple example is the sharp potential step, V = V0 ,(r), at energies below the step, 0 ¡ E ¡ V0 . On the classically forbidden side of the step, r ¿ 0, the WKB wave function is pro
portional to exp(−%|r|) where % = 2M(V0 − E)=˝ is the inverse penetration depth, and this is an exact solution of the Schr4odinger equation. On the classically allowed side, r ¡ 0, the WKB wave √ function is proportional to cos(k|r| − =2) where k = 2ME=˝ is the wave number, and this too is an exact solution of the Schr4odinger equation. The problem is solved exactly in the whole of coordinate space, if the reAection phase  and amplitude factor N are chosen as  %k : (52)  = 2 arctan(%=k); N = 2 2 % + k2 √ Note that  assumes the semiclassical value =2 only for %=k whereas N =1 only for %=(2± 3)k. The sharp step potential is particularly easy to handle, because the quantal region reduces to a single point; WKB wave functions are exact everywhere, except at the discontinuity in the potential. For a billard system in more than one dimension, i.e. a particle moving in an area (or volume) with a sharp boundary, the generalization from an in>nitely repulsive boundary to a potential of >nite height simply means replacing the Dirichlet boundary condition appropriate for the in>nite step to the conditions (52) for the component normal to the bounding surface. This is one of the few examples where quantum e?ects beyond the semiclassical limit have been successfully included in the semiclassical description of multidimensional systems [24,25,65]. 3.3. Scattering by a repulsive singular potential Repulsive singular potentials have a large application >eld in particle scattering. They are excellent examples to demonstrate the usefulness of the concept of the reAection phase. Consider a repulsive

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homogeneous potential, C& ˝2 (& )&−2 ˝2 v& (r) : V&rep (r) = & = = r 2M r& 2M The quantality function (36) for the potential (53) at energy E = ˝2 k 2 =(2M) is  

&  1 & v& (r) 2 − 1 v Q(r) = (& + 1)k + (r) : & 4 r 2 [k 2 − v& (r)]3 4

(53)

(54)

From Eq. (54) we observe that Q(r) = O(r &−2 ) for small r. As already mentioned in Section 2.4, the WKB approximation becomes increasingly accurate for r → 0 as long as & ¿ 2. This holds in particular for small energies—including vanishing energy. In the special case & = 4 and k = 0 the quantality function vanishes for all r, Q(r) ≡ 0, and WKB wave functions are exact solutions of the Schr4odinger equation, see also Eq. (32). For the large-r limit one has to distinguish the cases E = 0 and E ¿ 0. While the Q(r) diverges as r &−2 (& ¿ 2; & = 4) for zero energy, it approaches zero for large r as r −&−2 for E ¿ 0. The classical turning point is given by rt = & (& k)−2=& :

(55) & −2

as r → 0, so the WKB For 0 ¡ & ¡ 2, the quantality function Q(r) diverges proportional to r approximation deteriorates towards the origin. Decaying WKB waves cannot be expected to be good approximations of the exact wave function on the classically forbidden side of rt for potentials which are less singular than 1=r 2 . On the allowed side of rt however, Q(r) vanishes faster than 1=r 2 for any positive value of & (and E ¿ 0), so the WKB approximation becomes increasingly accurate for r → ∞. Hence an oscillating WKB wave function such as on the left-hand side of Eq. (44),     r   1  2 ()

 ; (56) cos p(r ) dr  − WKB (r) =
˝  rt 2 p(r) is a valid representation of the exact wave function on the classically allowed side of the turning point, regardless of whether the corresponding decaying WKB wave functions are good approximations for r → 0, as is the case for & ¿ 2, or poor approximations, as is the case for 0 ¡ & ¡ 2. In any case, the reAection phase  is to be chosen such that the phase of the WKB wave (56) asymptotically matches the phase of the regular exact wave function which vanishes at r = 0. Inverse-square potentials, corresponding to & = 2, represent a special case, ˝2 ( V( (r) = : (57) 2M r 2 Potentials of this form are of considerable physical importance since they appear in the interaction of a monopole charge with a dipole and as centrifugal potential in radial Schr4odinger equation in two or more dimensions. The centrifugal potential depends on the angular momentum quantum number lD and is of the form (57) with ( = l3 (l3 + 1);

l3 = 0; 1; 2; : : : ;

for three dimensions, and 1 ( = (l2 )2 − ; l2 = 0; ±1; ±2; : : : ; 4 for two dimensions.

(58) (59)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

377

The Schr4odinger equation with an inverse-square potential (57) contains no length or energy scale. Its regular solution is essentially a simple Bessel function of the product kr,   1 r J0 (kr); where 0 = ( + : (60) (r) = 2 4 √ The classical turning point is given by krt = (, and the quantality function is (2 ( 3 5 + : (61) 2 3 2 4 ((kr) − () 2 ((kr) − ()2 Away from the classical turning point, the absolute value of Q(r) is small for large ( corresponding to the semiclassical limit, and large for small ( corresponding to the anticlassical limit, see Section 2.1. For large values of kr, the asymptotic form [59] of the Bessel function implies that the wave function (60) behaves as    1  kr →∞ 1 ; (62) (r) ˙ √ cos kr − (+ − 2 4 4 k Q(r) =

whereas for the WKB wave function (56),   √  kr →∞ 1 () : (63) (− WKB (r) ˙ √ cos kr − 2 2 k In conventional semiclassical theory, the reAection phase in the WKB wave function (63) is assumed to be =2 and the resulting discrepancy of the phases in Eqs. (63) and (62) is repaired with the help of the Langer modi>cation [66]. This consists of modifying the potential used to calculate the WKB wave functions according to 1 ˝2 1 ( → ( + ; V( (r) → V( (r) + : (64) 4 2M 4r 2 This also leads to a correct behaviour of the WKB wave function in the classically forbidden region:
0 ˙ (kr) for kr → 0, 0 = ( + 1=4, cf. Eq. (43) in Section 2.4. However, the classical turning point is shifted. Comparison of Eq. (62) with Eq. (63) suggests an alternative approach. The phases of exact and WKB waves can be made to match asymptotically by choosing the reAection phase  in the following way [63,64],   1 √ = + (+ − ( : (65) 2 4 Examination of higher order asymptotic (kr → ∞) terms shows that the error in the WKB wave function (63) with the reAection phase (65) is proportional to (kr)−3 . This is better by two orders in 1=(kr) than the conventional semiclassical treatment, which is based on a reAection phase =2 and the Langer modi>cation (64); in the conventional treatment, the error in the WKB wave function only falls o? as 1=(kr) asymptotically [63,64]. For the inverse-square potential (57), the semiclassical limit is independent of energy and is reached for ( → ∞, see Section 2.1 and Eq. (61), whereas the anticlassical limit corresponds to ( → 0. As one would expect, the reAection phase (65) approaches =2 in the semiclassical limit;

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in the anticlassical limit it approaches the value , as for a free particle reAected by a hard wall [Eq. (52) in the limit %=k → ∞]. The general repulsive potential (53) proportional to 1=r & with & = 2 is characterized by a strength parameter with the physical dimension of a length, & . The properties of the Schr4odinger equation at energy E = ˝2 k 2 =(2M) depend not on energy and potential strength independently, but only on the product k& . Proximity to the semiclassical limit may be expressed quantitatively by comparing the classical turning point (55) with the quantum mechanical length corresponding to the inverse wave number, k −1 . We call the ratio of these two quantities the reduced classical turning point, a = krt = (k& )1−2=& :

(66)

In the semiclassical limit, the classical turning point is large compared to k −1 , i.e. a → ∞, which corresponds to k → ∞ when & ¿ 2, but to k → 0 when 0 ¡ & ¡ 2, in accordance with Section 2.1. Conversely, the anticlassical limit is given by a → 0. The inverse-square potential (57) >ts into this √ scheme if we identify krt = ( as the reduced classical turning point a. In terms of the classically de>ned quantities, total energy E, particle mass M and strength parameter C& of the potential (53) or (57), the reduced classical turning point (66) is, a=

√ 1 1−1 pas rt E 2 & (C& )1=& 2M = : ˝ ˝

(67)

According to Eq. (67), the reduced classical turning point a is just the classical action obtained by multiplying the asymptotic (r → ∞) classical momentum pas = ˝k and the classical turning point rt , measured in units of Planck’s constant ˝. The reAection phase  in the WKB wave function (56) is directly related to the phase shift 1, which determines the asymptotics of the regular solution of the Schr4odinger equation [43,61], reg (r)

r →∞

˙ sin(kr + 1) :

(68)

Comparing this to the WKB wave function (56) gives an explicit expression for 1 in terms of ,   r  1  

p(r ) dr − kr − 1 = + lim : (69) 2 r →∞ ˝ rt 2 For potentials falling o? faster than 1=r asymptotically, the square bracket above remains >nite as r → ∞ and 1 is a well de>ned constant depending only on the wave number k. For & ¿ 2, the anticlassical limit of the Schr4odinger equation corresponds to the threshold, k → 0, and the phase loss of the WKB wave due to reAection by the singular potential (53) can be derived from the exact solution of the Schr4odinger equation for zero energy,    √ & 1=(2') 1 : (70) ; '= rK±' 2' reg (r) ˙ r &−2 This is the regular solution, reg (0)=0, which is unique to within a constant factor. Since the energy enters the Schr4odinger equation with a term of order E=O(k 2 ), the zero-energy solution (70) remains valid for small energies to order below O(k 2 ). For large values of r, the argument of the Bessel

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379

function K±' in (70) is small, and the leading behaviour of the wave function is, reg (r)

r →∞

˙ '2'

r 2(1 − ') − : 2(1 + ') &

(71)

The asymptotic behaviour of the regular solution of the Schr4odinger equation is given by Eq. (68) and, for potentials falling o? more rapidly than 1=r 3 , the low-energy behaviour of the phase shift is, k →0

1 ∼ n − a0 k ;

(72)

with the scattering length a0 (see Ref. [43] and Section 4.1). So for large r and small k we have, reg (r)

˙ k(r − a0 ) :

(73)

Comparing Eqs. (73) and (71) gives an explicit expression for the scattering lengths of repulsive inverse power-law potentials (53) with & ¿ 3, a0 = '2'

2(1 − ') & ; 2(1 + ')

'=

1 : &−2

(74)

The asymptotic (r → ∞) expression for the action integral in the WKB wave function (56) is,    √  1 r  2(1 − &1 ) a & −1

r →∞ : (75) p(r ) dr ∼ kr − a + O ˝ rt 2 2( 32 − &1 ) kr For & ¿ 3 the phase of the WKB wave function has the correct near-threshold behaviour, see Eqs. (72) and (74), when the reAection phase is given by [58], k →0

 ∼ −

2 √ 2(1 − &1 ) 2(1 − ') k& ;  3 1 (k& )1− & + 2'2' 2(1 + ') 2( 2 − & )

(& ¿ 3) :

(76)

The second term on the right-hand side of Eq. (76) is linear in a and cancels the corresponding contribution from the WKB action integral (75); the third term (proportional to a1+2' ) yields the contribution −ka0 with the scattering length a0 in Eq. (74). Near the semiclassical limit a → ∞, a semiclassical expansion for the phase shifts [67,68] can be used [58] to derive the leading contributions to the reAection phase for the repulsive homogeneous potentials (53),   √  (& + 1)2( &1 ) 1 a→∞  + : (77)  ∼ +O 2 a 12&2( 12 + &1 ) a3 For >nite values of the reduced classical turning point a, between the semiclassical (77) and anticlassical (76) limits, the reAection phases can be obtained by solving the Schr4odinger equation numerically and comparing the phase of the solution with the phase of the WKB wave function (56) as r goes to in>nity. The results are shown in Fig. 1 as functions of a for integer powers & from 2 to 7. In all cases we observe a monotonous decline from the value  in the anticlassical limit a = 0 towards the semiclassical expectation =2 for large a. The smooth dependence of the reAection phases in Fig. 1 on a and & suggests that a simple algebraic formula based on the exact result (65) for & = 2 might be e?ective. Indeed, if we replace

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Fig. 1. Exact reAection phases  for reAection by a repulsive homogeneous potential (53) as function of the reduced classical turning point (66). From [58].

the reduced classical turning point a ≡ aS =

a ; S(&)

√

( in Eq. (65) with a scaled reduced classical turning point,

2 & + 1 2( &1 ) ; S(&) = √ 3  & 2( 12 + &1 )

then the appropriate generalization of Eq. (65), namely  1  (aS )2 + − aS ; = + 2 4

(78)

(79)

reproduces the numerically calculated exact reAection phases with an error never exceeding 0:006 for all powers & shown in Fig. 1 and all energies. The scaling factor S(&) in Eq. (78) was chosen such that the term proportional to 1=a in the large-a expansion (77) is given correctly by the formula a→0 (79). Near the anticlassical limit, the approximate formula Eq. (79) corresponds  ∼  − a=S(&), and the next-to-leading term −a=S(&) does not agree exactly with the result (76). The accurate algebraic approximation (79) of the reAection phases yields a convenient and accurate approximation for the scattering phase shifts via Eq. (69), namely  √   2(1 − &1 )  1 1= −a (80) (aS )2 + − aS : − 4 2 2( 32 − &1 ) 2 4 This is illustrated in Fig. 2, where the phase shifts are plotted as functions of k& . The case & = 1:4 is included in order to show that the formula (79) also gives good results in the regime 1 ¡ & ¡ 2, where the semiclassical limit a → ∞ corresponds to the low-energy limit k → 0. For Coulomb potentials, & = 1, the phase shifts have to be de>ned with respect to appropriately distorted waves rather than plane waves (68), but the concept of the reAection phase and Eq. (79) are still useful [58,69].

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381

Fig. 2. Phase shifts 1 for scattering by a repulsive homogeneous potential (53) proportional to 1=r & , which are functions of k& . The thin solid lines are the exact results and the thin dashed lines are the conventional WKB results, which are obtained by inserting the value =2 for the reAection phase  in the expression (69). The thick dashed lines show the results obtained with Eq. (80). The curves for & = 8; & = 4 and & = 1:4 are shifted by ; 2 and 4 respectively. From [58].

3.4. Quantization in a potential well Consider a binding potential in one dimension with two classical turning points, rl to the left and rr to the right. We assume that there is a region between rl and rr , where WKB wave functions are accurate approximations to the exact solution of the Schr4odinger equation. Then each turning point can be assigned a reAection phase, l and r for the left and right turning point, respectively, and the bound state wave function can be written either as    r 1 l 1

(81) p(r ) dr − cos l (r) ˙
˝ rl 2 p(r) or as

   rr 1 r 1

p(r ) dr − cos r (r) ˙
˝ r 2 p(r)

(82)

at any point r in the WKB region. For the wave function to be continuous around r, the cosines in Eqs. (81) and (82) must agree, at least to within a sign, so the sum of the arguments, which does

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not depend on r, must be an integral multiple of . This yields the rather general quantization rule   l r 1 rr (E) + : (83) p(r) dr = n + ˝ rl (E) 2 2 We retrieve the conventional WKB quantization rule, 

1 rr (E) 0M  ; 0M = 2 ; p(r) dr = n + ˝ rl (E) 4

(84)

by inserting =2 for both l and r in (83) as prescribed by Eq. (46). The parameter 0M is the Maslov index which counts how often a phase loss of =2 occurs due to reAection at a classical turning point during one period of the classical motion. Allowing for reAection phases which are not integral multiples of =2 amounts to allowing nonintegral Maslov indices [64]. The conventional quantization rule (84) works well near the semiclassical limit, but it breaks down near the anticlassical limit of the Schr4odinger equation. The modi>ed quantization rule (83) represents a substantial generalization of the conventional WKB rule, in that it avoids the restrictive assumptions underlying the conventional choice of reAection phases at the turning points; it only requires the WKB approximation to be accurate in some, perhaps quite small region of coordinate space between the turning points. 3.4.1. Example: Circle billard As a simple example consider the circle billard [24–26], a particle moving freely in an area bounded by a circle of radius R. The quantum mechanical wave functions of this √ separable twodimensional problem can be written in polar coordinates as (r; ’) = eil2 ’ l2 (r)= r, and the radial wave function l2 (r) obeys the one-dimensional Schr4odinger equation with the centrifugal potential (57), (59); l2 = 0; ±1; ±2; : : : is the angular momentum quantum number. The √ exact solutions for 2 2 (r) at energy E=˝ k =(2M) are essentially Bessel functions [59], (r) ˙ rJ|l2 | (kr), and bound l2 l2 states for a given angular momentum l2 exist when the wave function vanishes at the boundary: J|l2 | (kn R) = 0. The >rst, second, third, etc. zeros of the Bessel function de>ne bound states with radial quantum number n = 0; n = 1; n = 2, etc., and angular momentum quantum number l2 . The circle billard has served as a popular model system for testing semiclassical periodic orbit theories, and in leading order, standard periodic orbit theories essentially yield the energy eigenvalues obtained via conventional WKB quantization of the separable system. Main [26] has recently used the circle-billard as an example to demonstrate the applicability of an extension of standard periodic orbit theories to include terms of higher order in ˝. In conventional WKB quantization (84) of the radial motion, the Maslov index is taken to be 0M = 3 in order to account for the hard-wall reAection with reAection phase  at r = R; furthermore, the centrifugal potential is taken to be (l2 )2 ˝2 =(2Mr 2 ) in accordance with the Langer modi>cation 2 2 (64). The resulting energy eigenvalues En;WKB |l2 | [in units of ˝ =(2MR )] are tabulated in Table 1 for angular momentum quantum numbers |l2 |=0 and 1 together with the exact eigenvalues En;exact |l2 | , which are just the squares of the zeros of the corresponding Bessel functions. For the modi>ed quantization rule (83) the reAection phase at the outer classical turning point r = R is also taken as , corresponding to hard-wall reAection, but the reAection phase at the inner classical turning point has, for l2 = 0,  the energy independent value =2 + (|l2 | −

(l2 )2 − 14 ) according to Eq. (65); also the centrifugal

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383

Table 1 Energy eigenvalues En; |l2 | [in units of ˝2 =(2MR2 )] for angular momentum quantum numbers l2 = 0 and |l2 | = 1 in the circle billard. The superscript “WKB” refers to conventional WKB quantization of the radial degree of freedom, “mqr” refers to the modi>ed quantization rule (83), “exact” labels the exact results and “(1)” the results obtained by Main [26] in higher-order semiclassical periodic orbit theory n

En;WKB 0

En;mqr 0

En;exact 0

En;(1)0

En;WKB 1

En;mqr 1

En;exact 1

En;(1)1

0 1 2 3 4 5 6 7 8

5.551652 30.22566 74.63888 138.7913 222.6829 326.3138 449.6839 592.7931 755.6416

5.798090 30.47498 74.88861 139.0412 222.9329 326.5637 449.9338 593.0431 755.8916

5.783186 30.47126 74.88701 139.0403 222.9323 326.5634 449.9335 593.0429 755.8914

5.804669 30.47647 74.88960 139.0418 222.9329 326.5656

14.39777 48.95804 103.2445 177.2678 271.0297 384.5306 517.7705 670.7494 843.4676

14.65833 49.21105 103.4959 177.5187 271.2803 384.7809 518.0207 670.9997 843.7178

14.68197 49.21846 103.4995 177.5208 271.2817 384.7819 518.0214 671.0002 843.7182

14.70160 49.22259 103.5015 177.5135 271.2822

potential (57), (59) is left intact—it is not subjected to the Langer modi>cation. A naive application of the modi>ed quantization rule (83) does not work for l2 = 0, i.e. for s-waves in two dimensions, because the centrifugal potential is actually attractive without the Langer modi>cation, and the WKB action integral diverges when taken from r = 0. This can be overcome by shifting the inner integration limit to a small positive value and adjusting the reAection phase accordingly as described in Ref. [70]; see also Section 4.4. The eigenvalues obtained with the modi>ed quantization rule (83) are listed as En;mqr |l2 | in Table 1.

Also included in Table 1 are the energies En;(1)|l2 | obtained via the extension of semiclassical periodic-orbit quantization to higher order in ˝ according to Main [26]. The performance of the various approximations is illustrated in Figs. 3 and 4 showing the errors approx |En; |l2 | − En;exact |l2 | | on a logarithmic scale. The energies obtained in conventional WKB quantization including the Langer modi>cation of the potential are, both for l2 = 0 and |l2 | = 1, too small by an error which is very close to 0:25˝2 =(2MR2 ) and virtually independent of n. Using the modi>ed quantization rule (83)—without the Langer modi>cation—reduces the error by one to three orders of magnitude and yields the same sort of accuracy as the higher-order periodic orbit quantization according to Ref. [26]. In contrast to the higher-order periodic orbit theory, however, the application of the modi>ed quantization rule is very simple indeed—just as simple as applying the conventional WKB quantization rule. For large angular momentum quantum numbers |l2 |, conventional WKB quantization becomes increasingly accurate and the improvement due to the modi>ed quantization rule is less dramatic. For small values of l2 , the modi>ed quantization rule is an extremely simple and powerful tool for obtaining highly accurate energy eigenvalues beyond the conventional WKB approximation. 3.4.2. Example: Potential wells with long-ranged attractive tails In this subsection we consider a deep potential well with a long ranged attractive tail, as occurs, e.g., in the interaction of atoms and molecules with each other and with surfaces. We focus our

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l2=0

0

-1

conv. WKB

log10(error)

mod. quan. rule higher-order p.o. theory

-2

-3

-4 0

1

2

3

4 n

5

6

7

8

Fig. 3. Logarithmic plot of the error |En;approx − En;exact 0 | of various approximations of the energy eigenvalues En; |l2 | of the 0 circle billard for angular momentum number l2 = 0. The >lled triangles are the errors obtained via conventional WKB quantization including the Langer modi>cation of the potential and the empty triangles are the errors of the eigenvalues obtained by Main [26] via higher-order periodic orbit theory. The >lled squares show the errors obtained via the modi>ed quantization rule (83), adapted for the weakly attractive centrifugal potential as described in Ref. [70].

attention on homogeneous tails, C& ˝2 (& )&−2 = − ; &¿2 : (85) r& 2M r& The repulsive homogeneous potentials (53) studied in the previous section can, in principle, be the whole potential in the Schr4odinger equation, but the attractive potentials (85) cannot, otherwise the energy spectrum would be unbounded from below. The full potential must necessarily deviate from the homogeneous form (85) in the vicinity of r = 0, e.g. in the form of a short-ranged repulsive term. For the moment we neglect the inAuence of such a short-ranged repulsive part in the potential and study the properties of the long-ranged attractive tail (85). For negative energies, E = −˝2 %2 =(2M), there is an outer classical turning point rout given by V&att (r) = −

rout = & (%& )−2=& ;

(86)

cf. Eq. (55). In analogy to Eqs. (66), (67) we now de>ne the reduced classical turning point as the ratio of the classical turning point rout and the quantum mechanical penetration depth %−1 , √ 1 1 1 |pas | rout ; (87) a = %rout = (%& )1−2=& = |E| 2 − & (C& )1=& 2M = ˝ ˝ where |pas | = ˝% is the absolute value of the asymptotic (r → ∞) local classical momentum (2), which is now purely imaginary. Again, a is a quantitative measure of the proximity to the semiclassical or the anticlassical limit, a → ∞ being the semiclassical and a → 0 the anticlassical limit.

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

385

|l2|=1

0

-1

conv. WKB

log10(error)

mod. quan. rule higher-order p.o. theory

-2

-3

-4 0

1

2

3

4 n

5

6

7

8

Fig. 4. Logarithmic plot of the error |En;approx − En;exact 1 | of various approximations of the energy eigenvalues En; |l2 | of the 1 circle billard for angular momentum number l2 = ±1. The >lled triangles are the errors obtained via conventional WKB quantization including the Langer modi>cation of the potential and the empty triangles are the errors of the eigenvalues obtained by Main [26] via higher-order periodic orbit theory. The >lled squares show the errors obtained via the modi>ed quantization rule (83).

We assume that the reAection phase r at the outer classical turning point is determined by the homogeneous tail (85) alone; its behaviour near the anticlassical limit can be derived [73] from the zero-energy solutions of the Schr4odinger equation in much the same way as for the repulsive potentials (53) in Section 3.3; for any power & ¿ 2 the result is,

 √ 2(1 − 1 ) 2(1 − ') 1 a→0   ∼ a + 2'2' + ' −  3 &1 & tan sin(') a1+2' ; ' = : (88) 2 & 2(1 + ') &−2 2( 2 − & ) For E = 0, the classical turning point is at +∞ and the reAection phase is  def (89) (0) =0 = + ' : 2 The zero-energy reAection phase 0 is one of three parameters characterizing the near-threshold properties of an attractive potential tail, as discussed in more detail in Section 4. Near the semiclassical limit, the leading behaviour of the reAection phase in the attractive homogeneous potential (85) is [73],

 √ (& + 1)2( &1 ) a→∞   ∼ + a : (90) tan 2 & 12&2( 12 + &1 ) The dependence of  on a in between the anticlassical (88) and semiclassical (90) limits can be derived from numerical solutions of the Schr4odinger equation and the results are shown in Fig. 5. The phases fall monotonically from the threshold value (=2) + ' towards the semiclassical expectation =2 at large a.

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

Fig. 5. Exact reAection phases  for reAection by an attractive homogeneous potential (85) as function of the reduced classical turning point (87). From [73].

An algebraic approximation to the function (a) can now be obtained by generalizing the formula (79) to account for the fact that the value (0) in the anticlassical limit depends on the power & for the attractive potentials,  1  (91) (aR )2 + − aR :  = + 2' 2 4 Here aR stands for a scaled reduced classical turning point, and the scaling factor is chosen such that the formula (91) reproduces the leading contributions (90) to  in the semiclassical limit,

 1 & + 1 2( &1 ) a ; R(&) = √ : (92) aR = (& − 2) tan R(&) & 3  & 2( 12 + &1 ) As a concrete example for the e?ectivity of the modi>ed quantization rule (83) we discuss the Lennard-Jones potential, which is a model for molecular interactions, 

r 6  rmin 12 min : (93) −2 VLJ (r) = j r r The tail of the potential (93) has the homogeneous form (85) with & = 6; the strength parameter 6 is given by  1=4 4M(rmin )2 j = rmin (2BLJ )1=4 : (94) 6 = rmin ˝2 The potential (93) has its minimum value −j at r = rmin , and the energy eigenvalues, measured in units of j, depend only on the reduced strength parameter BLJ =2M(rmin )2 j=˝2 . The energy levels of the potential (93) and WKB approximations thereof were studied in considerable detail by Kirschner and Le Roy [71] and by Paulsson et al. [72]. Following Ref. [72] we choose a reduced strength

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387

Fig. 6. The Lennard-Jones potential (93) with reduced strength parameter BLJ = 104 . Table 2 Selected energy eigenvalues, in units of j, for the Lennard-Jones potential (93) with a strength parameter BLJ = 104 . En represents the exact energy, WEnconv is the error of the conventional WKB approximation (84), and WEnR is the error in the modi>ed WKB quantization (83) with l = =2 and the algebraic approximation (91) for r , & = 6; ' = 14 and R(6) = 2:08287. For a table containing all eigenvalues see Ref. [73] En

109 × WEnconv

109 × WEnR

0 1

−0.941046032 −0.830002083

−85841 −82492

−17508 −16684

11 12

−0.147751411 −0.115225891

−46115 −42250

−7522 −6589

22 23

−0.000198340 −0.000002697

−4493 −1021

−100 +42

n

parameter BLJ = 104 , for which the potential supports 24 bound states corresponding to quantum numbers n = 0; 1; : : : ; 23. The potential is illustrated in Fig. 6. A selection of energy eigenvalues, in units of j, is listed in Table 2; complete lists are contained in Refs. [72,73]. Next to the exact eigenvalues we also show the errors of the conventional WKB eigenvalues, which are obtained via the conventional quantization rule (84), and the last column shows the results obtained with the modi>ed quantization rule (83) when the reAection phase l at the inner classical turning point is kept at =2 while the energy dependence of the outer reAection phase r is accounted for via the approximate algebraic formula (91), with ' = 14 and R(6) = 2:08287 for & = 6.

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5

log10(relative error)

higher order WKB 4

-3

1

2

3

-4 0

5

10

15

20

n

Fig. 7. Relative errors (95) of the energy eigenvalues in the Lennard-Jones potential (93) with strength parameter BLJ =104 . The >lled triangles show the errors of conventional WKB quantization (84) and the >lled squares show the errors of the modi>ed quantization rule (83) when the reAection phase l at the inner classical turning point is kept at =2 while the energy dependence of the outer reAection phase r is accounted for via the approximate algebraic formula (91), with ' = 14 and R(6) = 2:08287 for & = 6. The open triangles numbered 1 to 5 show the relative error for the highest bound state, n = 23, as obtained in Ref. [72] with successive higher-order approximations involving terms up to S2N +1 in the expansion (20); the state becomes unbound for N ¿ 5.

Although the results of conventional WKB quantization seem quite satisfactory at >rst sight, allowing for the energy dependence of the reAection phase at the outer classical turning point improves the accuracy by a factor ranging from >ve for the low-lying states to 45 and 25 for the highest two states. The usefulness of the modi>ed quantization rule becomes clearer when looking at the errors relative to the level spacing, which decreases by a factor of 500 from the bottom to the top of the spectrum,    WEn  rel  :  WEn =  (95) En − E n − 1  Fig. 7 shows the relative errors (95) obtained via conventional WKB quantization (>lled triangles) and via modi>ed quantization including the energy dependence of the outer reAection phase via Eq. (91) (>lled squares). The relative error of conventional WKB quantization grows by an order of magnitude as we approach the anticlassical limit at E = 0. In contrast, accounting for the energy dependence of the outer reAection phase keeps the relative error roughly constant at a comfortably low level. As shown in Ref. [72], higher-order WKB approximations involving terms up to S2N +1 in the expansion (20) substantially reduce the error for all but the last eigenstate, n = 23. The relative errors obtained in higher-order WKB approximations for the n = 23 state are shown as open triangles in Fig. 7. The relative error initially decreases, for N = 1; 2 and 3, but then increases with the order of the approximation. For N ¿ 5 the WKB series no longer yields a bound state with quantum

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389

number n = 23. This is an illustrative demonstration of the asymptotic nature of the expansion (20) which breaks down towards the anticlassical limit. The observation that conventional semiclassical approximations deteriorate towards threshold in a typical molecular potential such as (93) sometimes causes surprise [74], because one might expect such approximations to improve with increasing quantum number. It is, however, not surprising and actually well understood [72,73,75–78], that semiclassical approximations break down near the anticlassical or extreme quantum limit, which is at energy zero for potentials such as (93). A more detailed discussion of quantization near the anticlassical limit is presented in Section 4.2. 3.5. Tunnelling When two classically allowed regions of (in our case one-dimensional) coordinate space are separated by a localized classically forbidden region, a barrier, then a classical particle approaching the barrier from one side is inevitably reAected and cannot be transmitted to the other side. In quantum mechanics there is generally a >nite probability PT for transmission, and the probability PR for reAection is correspondingly less than unity, PR + PT = 1 :

(96)

Although the following discussion focusses on tunnelling through a classically forbidden region, it can also be applied to the transmission through a classically allowed region, for which the probability can be less than unity when quantum e?ects lead to a >nite probability for classically forbidden reAection. Such quantum reAection requires the existence of a region of substantial quantality and is discussed in detail in Section 5. When a particle approaches the barrier (or the quantal region of coordinate space) from the left, the wave function to the left of this region can be expressed by the WKB wave functions       r  1 i r i



(r) ∼
(97) p(r ) dr + Rl exp − p(r ) dr exp ˝ rl ˝ rl p(r) with the reAection amplitude Rl , and the transmitted wave to the right is    r 1 i


p(r ) dr exp (r) ∼ Tl ˝ rr p(r)

(98)

with the transmission amplitude Tl . The points rl and rr are points of reference where the phases of the WKB wave functions vanish. When the particle is incident from the right, Eqs. (97) and (98) are replaced by     r    i 1 i r



; (99) p(r ) dr + Rr exp p(r ) dr (r) ∼
exp − ˝ rr ˝ rr p(r)    1 i r

(100) (r) ∼ Tr
p(r ) dr ; exp − ˝ rl p(r) where Eq. (99) applies for r values to the right and Eq. (100) for r values to the left of the barrier (or quantal region). The reAection and transmission amplitudes in Eqs. (97)–(100) are connected

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through the reciprocity relations (see, e.g. Ref. [79]), T def Tr = Tl =T; Rr = −R∗l ∗ : (101) T If the potential becomes constant on one or both sides of the barrier, then the conventional ansatz with plane waves can be used to de>ne reAection and transmission amplitudes, e.g., 1 (r) ∼ √ {exp[ − ik(r − rr )] + Rr exp[ik(r − rr )]} ˝k

(102)

√ for a particle incident from the right. We have included the factor 1= ˝k to account for the velocity dependence of the particle density in accordance with the continuity equation. When the r dependence of the potential is negligible, the WKB waves in Eq. (99) are identical to the plane waves in (102) except for constant phase factors. The use of WKB waves to de>ne transmission and reAection amplitudes as in Eqs. (97)–(100) does not imply any semiclassical approximation of these amplitudes. The Schr4odinger equation should be solved exactly, and the exact wave functions matched to the incoming and reAected waves or the transmitted wave in the semiclassical regions on either side of the barrier (or quantal region). If there are no semiclassical regions, then the terms “incoming”, “reAected” and “transmitted” cannot be de>ned consistently. The ans4atze (97)–(100) involving WKB wave functions are more general than those using plane waves, because they can also be used when the potential depends strongly on r in the semiclassical region(s). The probabilities PT for transmission and PR for reAection are given by PT = |T |2 ;

PR = |R|2 ;

(103)

and they do not depend on whether the particle is incident from the left or the right, or on whether the amplitudes are de>ned via WKB or plane wave functions. The phase of the reAection amplitude depends on the direction of incidence according to Eq. (101), and also on whether WKB waves or plane waves are used as reference. For example, the reAection amplitude R(p) de>ned with reference to plane waves as in Eq. (102) is related to the r reAection amplitude R(WKB) de>ned via WKB waves according to Eq. (99) by r     1 r (p) (WKB)

R r = Rr : (104) exp lim 2i k(r − rr ) − p(r ) dr r →∞ ˝ rr The exponential on the right-hand side simply accounts for the di?erent phases accumulated by the reference waves on their way to rr and back. The reAection amplitudes do not depend on whether plane waves or WKB waves are used to represent the transmitted wave. If 2M(E − V ) approaches the constant values ˝2 k 2 and ˝2 q2 for r → +∞ and r → −∞ respectively, then the transition amplitude T (p) de>ned via incoming and transmitted plane waves is related to the transition amplitude T (WKB) based on WKB waves by     1 r− (p) (WKB) T =T exp lim i q(r− − rl ) − p(r) dr r± →±∞ ˝ rl    1 r+ p(r) dr : (105) −i k(r+ − rr ) − ˝ rr

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The phases of the reAection and transmission amplitudes also depend on the points of reference rl; r at which the reference waves have vanishing phase. In principle they could be chosen arbitrarily; in the presence of a potential barrier they are conveniently chosen as the left and right classical turning points, respectively. In the presence of a potential barrier, the wave function in the classically forbidden region is approximated by a superposition of exponentially increasing and decreasing WKB waves        1 r 1 r 1



: (106) A exp − |p(r )| dr + B exp + |p(r )| dr forb (r) =
˝ rl ˝ rl |p(r)| We assume that the classical turning points rl and rr are isolated, meaning that there is a region in the classically forbidden domain between rl and rr where the WKB representation (106) is valid. Instead of referring the WKB waves to the left classical turning point rl , we could equally have chosen rr as point of reference. For the derivation of an explicit expression for the transmission amplitude we consider the case that the particle is incident from the left and we make use of Eqs. (97), (98). In order to use the connection formulas (44) and (45) at the right turning point we rewrite Eq. (98) as     r  C 1 CU 1 r r U r

(r) = √ cos + √ cos ; (107) p dr − p dr − p ˝ rr 2 p ˝ rr 2 where 2Tl e−ir =2 C= ; e−ir − e−iUr

U

2Tl e−ir =2 CU = − : e−ir − e−iUr

(108)

The cosines of Eq. (107) can now be matched to the growing and decaying exponentials in the WKB region under the barrier, Eq. (106), according to the connection formulas (44) and (45). The exponentials containing integrals with lower limit rl can be rewritten in terms of exponentials of integrals with upper limit rr by introducing the factor  1 rr ,B (E) = exp[I (E)]; I (E) = |p(r)| dr ; (109) ˝ rl this gives the coeScients A and B of Eq. (106), U A = NU r ,B C;

B=

Nr C : 2,B

(110)

The subscript r indicates application of the connection formulas at the right turning point. By using the connection formulas once again at the left turning point rl we get an expression of the wave function in the classical allowed region to the left of rl . Decomposing the cosines into exponentials and comparison with the WKB wave incident from the left, Eq. (97), gives the coeScient of the incoming WKB wave, which should be unity, as A il =2 B iUl =2 e + e : Nl 2NU l

(111)

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Using Eqs. (110), (108), and (49) yields the general expression for the transmission amplitude   Nl Nr 1 i(Ul +Ur )=2 −1 i(l +r )=2 − e : (112) T = iNl Nr ,B e NU l NU r 4,B This is a very general formula, but it still contains eight parameters, namely the unbarred and barred phases  and amplitudes N at each of the two turning points, where the connection formulas have been applied. For a “dense” barrier, meaning that the exponentiated integral ,B as de>ned by Eq. (109) is large, we might choose to neglect the second term in the large brackets in Eq. (112) as subdominant. This leaves us with an approximate formula for the transmission amplitude T ≈ iNl Nr e−i(l +r )=2 =,B ;

(113)

which contains only the well-de>ned (unbarred) connection parameters. With the standard semiclassical choice (46) for the unbarred connection parameters, Eq. (113) leads to the standard WKB expression for the tunnelling probability, PTWKB (E) = |T |2 = (,B )−2 :

(114)

This formula fails near the top of a barrier, where the two turning points rl and rr coalesce. The expression (114) gives unity at the top of a barrier, whereas the exact result is generally smaller than unity. An improved formula, 1 PTKemble (E) = ; (115) 1 + (,B )2 is due to Kemble [80,81], and it is actually exact at all energies for an inverted quadratic potential, V (r) ˙ −r 2 ; in particular, PTKemble is exactly one half at the top of the barrier. This result is accurate for all barriers which can be approximated by an inverted parabola in a range of r values reaching from the top of the barrier into the semiclassical regions on either side. If we include the subdominant term in Eq. (112) and >x the connection parameters according to conventional WKB theory, Nl; r = NU l; r = 1, l; r = −U l; r = =2, we obtain the result   1 −1 ; (116) T = ,B + 4,B which can be found, e.g., in Ref. [45]. We now look at the behaviour of the tunnelling amplitude and probability near the base of a barrier, where the potential becomes asymptotically constant on at least one side. The limit of small excess energy, E =˝2 k 2 =(2M) → 0, corresponds to the anticlassical limit of the Schr4odinger equation if the potential approaches its asymptotic limit faster than 1=r 2 . For this case it can be shown [82], that both the amplitude N in the connection formula (44) and the amplitude NU in the connection √ formula (45) become proportional to k for k → 0. For potential tails falling o? faster than 1=r 2 the exponentiated √ integral (109) remains >nite at E = 0, so the transmission amplitude (112) is proportional to k near the threshold E → 0, if E = 0 represents the base of the barrier on just one side. The transmission probability through an asymmetric barrier with two di?erent asymptotic levels is thus proportional to the square root of the excess energy above the higher level, in the limit that this excess energy becomes small. If the potential approaches the same asymptotic limit

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on both sides and does so faster than 1=r 2 , then the transmission amplitude (112) is proportional to k and the transmission probability is proportional to E near the base. 2 For a potential barrier √ with tails vanishing faster than 1=r , the quantum mechanical tunnelling probability vanishes as E or as E at the base, E = 0. This behaviour is not reproduced by any of the semiclassical formulas (114), (115) or (116), which all predict a >nite tunnelling probability at the base. For a symmetric barrier, the left and right connection parameters are the same and we can drop the subscripts. Eq. (112) then simpli>es to  −1 N 2 1 iU 2 i T = iN ,B e − 2 e : (117) NU 4,B For the calculation of the transmission probability through a symmetric barrier the phases in Eq. (117) are eliminated via Eq. (49) giving  − 1 2 ,B 1 2 − +1 : (118) PT (E) = |T | = N2 4,B NU 2 If we keep only the dominant term, PT (E) ≈

N4 : (,B )2

As an example consider a symmetric rectangular barrier of length L,  2 ˝ (K0 )2 =(2M) for |r| ¡ L=2 ; V (r) = 0 for |r| ¿ L=2 :

(119)

(120)

This corresponds to a sharp-step potential at both classical turning points; the
phase  and ampli2 2 tude N are given in Eq. (52) as functions of the inverse √ penetration depth % = (K0 ) − k on the classically forbidden side and the wave number k = 2ME=˝ on the classically allowed side of the step. For the sharp-step potential, the WKB waves are exact except at the classical turning point, so the exponentially increasing wave function in the classically forbidden region can be uniquely de>ned; this allows the determination of the barred parameters in the connection formula (45), N NU = ; 2

U = − :

(121)

With Eqs. (52), and (121) for the connection parameters at both turning points and the explicit expression for the exponentiated WKB integral (109), ,B = e%L , the transmission amplitude (117) is T=

e%L (k

+

i%)2

4ik% : − e−%L (k − i%)2

The resulting transmission probability is  −1 2 %2 + k 2 sinh %L + 1 PT = ; 2%k

(122)

(123)

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a standard textbook result. The low-energy (k → 0) behaviour of the expression (123) is, PT ≈

1 16k 2 + O(k 4 ) ; 2 (K0 ) (,B − 1=,B )2

(124)

as could also be calculated via Eq. (118). Now consider symmetric barriers decaying asymptotically as an inverse power of the coordinate, V (r)

|r |→∞

∼

V&rep (|r|) =

˝2 (& )&−2 ; 2M |r|&

(125)

with & ¿ 2 and & positive. The turning points of the homogeneous potential, V& (|r|), are given by rl; r = ∓& (k& )−2=& . In order to calculate the phases and amplitudes in the low energy limit it suSces to consider a potential step which shows the same behaviour as the potential (125) for r → +∞ and stays classically forbidden for energies near zero on the left side, r → −∞. It follows that we can use the results for the reAection by singular repulsive potentials, Section 3.3. After taking into account the proportionality factors in Eqs. (68) and (70) we get the leading contribution to the amplitude factor N , k →0

N ∼

2'&'=2
k& ; 2(1 + ')

'=

1 ; &−2

(126)

which can be inserted into Eq. (119) to give for the tunnelling probability at the base of the barrier k →0

PT ∼

16'2&' (k& )2 : 2(1 + ')4 (,B )2

(127)

The formula (127) is accurate as long as the homogeneous behaviour (125) of the tails of the barrier continues far enough into the barrier, i.e., until the WKB wave functions (106) are accurate. Under these conditions, more accurate values for N going beyond the leading contribution (126) can be obtained by numerically solving the Schr4odinger equation for the homogeneous tail and matching the exact solution to the WKB waves on either side of the turning point. As an example we have calculated N for a homogeneous tail proportional to 1=r 8 and applied the formula (119) to the potential V (r) =

V0 : 1 + (r=)8

(128)

Fig. 8 shows the resulting tunnelling probabilities in a doubly logarithmic plot. The solid line is the exact numerically calculated tunnelling probability and the thick dashed line shows the prediction of the formula (119), with N calculated for the homogeneous tail proportional to 1=r 8 . The straight-line behaviour for small energies demonstrates the proportionality of the tunnelling probability to energy near the base of the barrier, see Eq. (127). The thin dashed line shows the conventional semiclassical result (114), which fails near the base of the barrier, because it remains >nite. For further examples see Refs. [60,82–84]. An accurate description of tunnelling is important for the understanding of energy levels in potentials with two or more wells, e.g. the energy splitting between two almost degenerate levels in a

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√ Fig. 8. Numerically calculated exact tunnelling probabilities (solid line) for the potential (128) with  MV0 =˝=5 together with the prediction (119) based on values of N (k) calculated numerically for a homogeneous potential tail proportional to 1=r 8 (thick dashed line). The thin dashed line shows the prediction of the conventional WKB formula (114). From [60].

symmetric double well is essentially determined by the tunnelling probability through the potential barrier separating the wells [43,85]. Modi>ed WKB quantization techniques utilizing the generalized connection formulas (44), (45) have been applied to the quantization in multiple-well potentials by several authors in the last few years, and substantial improvements over the predictions of the standard WKB procedure have been achieved. For details see Refs. [86–91].

4. Near the threshold of the potential In this section we consider potentials which vanish asymptotically, r → ∞, and we focus on the regime of small positive or negative energies near the threshold E = 0. For potentials falling o? faster than 1=r 2 ; E = 0 corresponds to the anticlassical or extreme quantum limit of the Schr4odinger equation; conventional semiclassical methods are not applicable in this case, but modi>ed methods involving exact wave functions in the quantal regions of coordinate space and WKB wave functions elsewhere can give reliable and accurate results. The Schr4odinger equation (1) contains the energy in order O(E), but the leading near-threshold behaviour of the wave functions is often determined by terms of lower order in E, see e.g. Eqs. (76), (88) for the near-threshold reAection phases in repulsive or attractive homogeneous potentials or Eq. (122) for the near-threshold transmission amplitude through a rectangular potential barrier. If exact solutions of the Schr4odinger equation are known at threshold, E = 0, then they also determine the leading behaviour of the solutions near threshold to order less than E, because the energy can be treated as a perturbation of (higher) order E in the Schr4odinger equation. Examples involving continuum states above threshold and discrete bound states below threshold are presented in this section.

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4.1. Scattering lengths Consider a potential V (r), which vanishes faster than 1=r 2 for r → ∞. For positive energies, E = ˝2 k 2 =(2M) ¿ 0, the regular solution reg (r) of the Schr4odinger equation, which is de>ned by the boundary condition reg (0) = 0, behaves asymptotically as reg (r)

r →∞

˙ sin(kr + 1) ;

(129)

and 1(k) is the phase shift of the exact wave function reg relative to the free wave sin(kr), as already discussed in Section 3.3. The phase shift approaches an integral multiple of  at threshold, as long as the potential falls o? faster than 1=r 2 . For a potential falling o? faster than 1=r 4 , the leading terms of the low-energy behaviour of the phase shift are given by the e?ective range expansion [61,92] 1 1 k →0 k →0 2 + k 2 re? ; 1 ∼ n − ka0 + O(k 3 ) : (130) k cot 1 ∼ − a0 2 The parameter a0 in Eq. (130) is the scattering length and re? the e?ective range of the potential V (r). For a potential behaving asymptotically as C4 ˝2 (4 )2 = − ; (131) r4 2M r 4 the leading near-threshold behaviour of the phase shift is given [93] by  k →0 1(k) ∼ n − ka0 − (k4 )2 ; (132) 3 so the e?ective range expansion (130) begins with the same constant term −1=a0 on the right-hand side, but there is a term linear in k preceding the quadratic one. For all potentials falling o? faster than 1=r 3 , the scattering length a0 dominates the low-energy properties of the scattering system. It is, e.g., a crucial parameter for determining the properties of Bose-Einstein condensates of atomic gases, see [36,94]. For a potential behaving asymptotically as r →∞

V (r) ∼

−

C3 ˝ 2 3 = − ; r3 2M r 3 the leading near-threshold behaviour of the phase shift is given [92] by r →∞

(133)

k →0

(134)

V (r) ∼ −

1(k) ∼ n − k3 ln(k3 ) ;

and it is not possible to de>ne a >nite scattering length. Scattering lengths depend sensitively on the positions of near-threshold bound states, so they are determined by the potential in the whole range of r values, not only by the potential tail. The mean value of the scattering length—averaged, e.g. over a range of well depths—is, however, essentially a property of the potential tail. If there is a WKB region of moderate r-values, where the exact solutions of the Schr4odinger equation (at near-threshold energies) are accurately approximated by WKB wave functions, explicit expressions for the scattering length can be derived [95–97] as brieAy reviewed below. The Schr4odinger equation at threshold (E = 0) has two linearly independent solutions, 0 and 1 , whose asymptotic (r → ∞) behaviour is given by 0 (r)

r →∞

∼ 1 + o(r −1 );

1 (r)

r →∞

∼ r + o(r 0 ) :

(135)

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In the WKB region, the exact threshold solutions 0 ; 1 can be written in WKB form,    ∞ 1 0; 1 1

; p0 (r ) dr − cos 0; 1 (r) = D0; 1
˝ r 2 p0 (r)

(136)

with well de>ned amplitudes D0 and D1 and the phases 0 and 1 . Here p0 (r) is the local classical momentum at threshold,
(137) p0 (r) = −2MV (r) : Since 0 (r) is the solution which remains bounded for r → ∞, the phase in the WKB form of this wave function in the WKB region is just the threshold value of the reAection phase at the outer classical turning point, which lies at in>nity for E = 0, i.e. the zero-energy reAection phase 0 , cf. Eq. (89) in Section 3.4. The asymptotic behaviour (129) of the regular solution of the Schr4odinger equation becomes reg (r)

k →0

˙ sin[k(r − a0 )] ∼ k(r − a0 )

√

(138)

when we insert the near-threshold behaviour (130) of the phase shift 1(k). To order k ˙ E, the regular solution of the Schr4odinger equation for small k thus corresponds to the following linear superposition of the zero-energy solutions 0 and 1 : reg (r)

˙ k[

1 (r)

− a0

0 (r)]

:

(139)

For values of r in the WKB region, 0 and 1 are WKB wave functions (136), so     ∞   ∞  k 1 1 1 0



− a0 D0 cos p0 (r ) dr − p0 (r ) dr − D1 cos reg (r) ˙
˝ r 2 ˝ r 2 p0 (r)   ∞  1 + 1 ˙
p0 (r
) dr
− cos −8 ; (140) ˝ r 4 p0 (r) where 8 is an angle de>ned by   − a0 + D1 =D0 ; tan tan 8 = a0 − D1 =D0 4

(141)

and + ; − stand for the sum and di?erence of the phases in Eq. (136), + = 0 + 1 ;

− = 0 − 1 :

Coming from the inner turning point rin , the WKB wave function    r 1 1 in (E)

cos p(r ) dr − WKB (r) =
˝ rin (E) 2 p(r)

(142)

(143)

is expected to be an accurate approximation of reg (r) for r values in the WKB region. The reAection phase in at the inner turning point will be near =2 if the conditions for conventional matching are ful>lled in the neighbourhood, see Section 3.2, but, even if this is not the case, in can be expected to be a smooth analytical function of the energy E near threshold, in (E) = in (0) + O(E) :

(144)

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The inner turning point rin also depends weakly and smoothly on E, and for r
values between rin and a value r in the WKB region, the local classical momentum p(r
) in Eq. (143) di?ers from its threshold value (137) only in order E near threshold. So to order less than E we can assume E = 0 in Eq. (143),    r 1 in (0) 1


: (145) (r) ≈ (r) ˙ p (r ) dr − cos reg WKB 0 ˝ rin (0) 2 p0 (r) Eqs. (140) and (145) are compatible if and only if the cosines agree at least to within a sign. This leads to an explicit expression for the angle 8 in terms of the threshold value  ∞ S(0) = p0 (r) dr (146) rin (0)

of the action integral, namely − S(0) in (0) + − − − n = (nth − n) + : (147) ˝ 2 4 4 Here we have introduced the threshold quantum number nth which ful>lls the modi>ed WKB quantization rule (83) at E = 0, 8=

S(0) in (0) 0 − = nth  : − (148) 2 2 ˝ nth is an upper bound to the quantum numbers n = 0; 1; 2 : : : of the negative energy bound states and is usually not an integer. An integer value of nth indicates a zero-energy bound state. The number of negative energy bound states is [nth ] + 1 where [nth ] is the largest integer below nth . Resolving Eq. (141) for a0 and using Eq. (147) gives D1 tan(nth  + − =4) + tan(− =4) D0 tan(nth  + − =4) − tan(− =4)    1 D1 − 1 : = sin + D0 2 tan(− =2) tan(nth )

a0 =

(149)

The exact zero-energy solutions 0 , 1 of the Schr4odinger equation may be known for a tail-region of the potential. The amplitudes D0; 1 and phases 0; 1 of the WKB form of the exact solutions can then be derived from these wave functions, if the tail-region, where the Schr4odinger equation is accurately solved by the known forms of 0 and 1 , overlaps with the WKB region, where these wave functions can be matched to the WKB form (136). The amplitudes and phases are thus tail parameters, which depend only on the potential in the tail region and not on its shape inside the WKB region, or at even smaller r values. The factor   D1 0 − 1 (150) b= sin D0 2 in front of the square bracket in Eq. (149) has the physical dimension of a length and is a characteristic property of the potential tail beyond the WKB region. The threshold quantum number nth , on the other hand, is related to the total number of bound states supported by the well and depends

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on the whole potential via the action integral S(0). It is likely that there is an inner WKB region in the well when nth is a large (but >nite) number; this is not a necessary condition however, as shown by the example of a shallow step potential, see below. For a uniform distribution of values of nth , the second term in the square bracket in Eq. (149) will be distributed evenly between positive and negative values, so the >rst term de>nes a mean scattering length aU0 , b b ; a0 = aU0 + : (151) aU0 = tan[(0 − 1 )=2] tan(nth ) 4.1.1. Example: Sharp-step potential The radial sharp-step potential is de>ned as,  2 K0 ; ˝2 (K0 )2 ˝2 Vst (r) = − ,(L − r) = − 2M 2M 0;

for 0 ¡ r 6 L ;

(152)

for r ¿ L :

The quantal region where the WKB approximation is poor is restricted to the single point r = L where the potential is discontinuous. The WKB approximation is exact for 0 ¡ r ¡ L regardless of whether the potential be deep or shallow, and the tail region of the potential can be any interval of r values that includes the discontinuity at r = L. The zero-energy solutions of the Schr4odinger equation with the asymptotic (here: r ¿ L) behaviour (135) are given, in the WKB region 0 ¡ r ¡ L, by st 0 (r)

= cos[K0 (L − r)] ;   L st1 st cos K0 (L − r) − 1 (r) = cos(st1 =2) 2



st1 with tan 2

 =−

1 : K0 L

(153)

The zero-energy reAection phase 0 at the outer turning point (here at r = L) is zero. The tail parameter b and the mean scattering length are thus given by 1 bst = ; aUst0 = L : (154) K0 The reAection phase at the inner turning point r = 0 is  corresponding to reAection at a hard wall, so the threshold quantum number de>ned by Eq. (148) is given by nstth  = K0 L − =2. Indeed, the number of bound states in the sharp-step potential (152) is the largest integer bounded by nstth + 1. With Eq. (151) and using tan(nstth ) = −1=tan(K0 L) we obtain the well known result [98] for the scattering length of the sharp-step potential, tan(K0 L) ast0 = L − : (155) K0 4.1.2. Example: Attractive homogeneous potentials More important and realistic examples are the homogeneous potential tails, C& ˝2 (& )&−2 = − ; (156) r& 2M r& which are characterized by a power & and a strength parameter & with the physical dimension of a length. The zero-energy wave functions which solve the Schr4odinger equation with the potential V&att (r) = −

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(156) and have the asymptotic behaviour (135) are essentially Bessel functions of order ±1=(& − 2) [55,73,76], 
2(1 + ') r (&) (&) ' (r) = J (z); (r) = 2(1 − ')' & rJ−' (z) ; ' 0 1 '' &  1=(2') & 1 ; z = 2' : (157) '= &−2 r For a suSciently rapid fall-o? of V&att (r), namely & ¿ 2, the WKB approximation becomes increasingly accurate for r → 0, see Section 2.4. The WKB region corresponds to small values of r=& and hence to large arguments of the Bessel functions in Eq. (157), so we can use their asymptotic expansion [59] to write 0 and 1 in the WKB form (136). This yields the phases and amplitudes,   0(&) = + '; 1(&) = − ' ; 2 2   ˝ 2(1 + ') ˝& (&) (&) 2(1 − ')'' : ; D1 = (158) D0 = ' '& ' ' The length parameter (150) is now given by b(&) = & '2'

2(1 − ') '1+2' sin(') = & ; 2(1 + ') 2(1 + ')2

and the mean scattering length is 2(1 − ') cos(') : aU0(&) = & '2' 2(1 + ')

(159)

(160)

It is interesting to observe, that the formula (160) for the mean scattering length of the attractive homogeneous potential (156) is very similar to the formula (74) for the true scattering length of the repulsive homogeneous potential (53) discussed in Section 3.3. It simply contains an additional factor cos('). The true scattering length for a potential with an attractive tail such as (156) depends on the whole potential via the threshold quantum number nth according to Eq. (151). If we take the threshold value in (0) of the inner reAection phase in Eq. (148) to be =2, then nth  = S(0)=˝ − (1 + ')=2 and the true scattering length is    S(0) ' − : (161) a0(&) = aU0(&) 1 − tan(')tan ˝ 2 The formulas (160) and (161) for homogeneous potential tails were >rst derived by Gribakin and Flambaum [95]. Together with Harabati these authors also derived an expression for the e?ective range re? in the next term of the expansion (130) for the phase shift [96]. For the discussion in this section we assumed that the potential falls o? faster than 1=r 3 asymptotically. Indeed, for a potential behaving asymptotically as (133), the low-energy behaviour of the phase shift is given by Eq. (134), so a >nite scattering length cannot be de>ned. Note, however, that the expression (159) for the tail parameter b(&) remains >nite for ' → 1 corresponding to & → 3, and also for higher integer values of ' corresponding to powers & = 2 + 1=' between 2 and 3.

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

401

Table 3 Ratios of the tail parameter (159) and the mean scattering length (160) to the strength parameter & for attractive homogeneous potential tails (156) &

3

4

5

6

7

8

&→∞

b(&) =& aU(&) 0 =&

 —

1 0

0.6313422 0.3645056

0.4779888 0.4779888

0.3915136 0.5388722

0.3347971 0.5798855

=& 1

The mean scattering length (160) diverges with 2(1−') for positive integers ', but b(&) remains well de>ned and >nite. The ratio b(&) =& expressing the tail parameter b(&) of a homogeneous potential in units of the strength parameter & is tabulated in Table 3 for integer powers & from 3 to 8. For & = 4; : : : ; 8 Table 3 also shows the mean scattering length aU0(&) in units of & ; for homogeneous potentials (156), aU0(&) and b(&) are related by aU0(&) = b(&) =tan('). 4.2. Near-threshold quantization and level densities The generalized quantization rule as introduced in Section 3.4 reads  S(E) def 1 rout (E) in out = p(r) dr = n + + : ˝ ˝ rin (E) 2 2

(162)

This assumes that there is a WKB region between the inner classical turning point rin and the outer one rout , where WKB wave functions are accurate solutions of the Schr4odinger equation. The reAection phases in and out account for the phase loss of the WKB wave at the respective turning points. They are equal to =2 when the conditions for the conventional connection formulae are well ful>lled near the turning points, but they can deviate from =2 away from the semiclassical limit. For a potential V (r) with a deep well and an attractive tail, the reAection phase in at the inner turning point may—or may not—be close to =2; in any case it can be expected to be a smooth analytic function of the energy E, and there is nothing special about the threshold E = 0, cf. Eq. (144). The situation is di?erent near the outer turning point rout which, for a smoothly vanishing potential tail, moves to in>nity at threshold. When the potential is attractive at large distances and vanishes more slowly than 1=r 2 , then the action integral S(E) grows beyond all bounds as E → 0; the potential well supports an in>nite number of bound states and conventional WKB quantization, with out = =2 at the outer turning point, becomes increasingly accurate towards threshold. For a potential behaving asymptotically as V&att of Eq. (156) with 0 ¡ & ¡ 2, and energies E = −˝2 %2 =(2M) close enough to threshold, the action integral can be written as  rout (E)  S(E) (& )&−2 = C+ − %2 dr & r ˝ r0 √ F(&)  2(1=& − 12 ) %→0
; (163) ; F(&) = ∼ C + (%& )(2=&)−1 2& 2(1=& + 1)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

which leads to the near-threshold quantization rule, F(&) n→∞ n ∼ C

+ : (%& )(2=&)−1

(164)

The point r0 in Eq. (163) is to be chosen large enough for the potential to be accurately described by the leading asymptotic term proportional to 1=r & . The constants C, C
and C

in Eqs. (163) and (164) depend on the potential at shorter distances r ¡ r0 , but the energy dependent terms depend only on the potential tail beyond r0 , i.e. only on the power & and the strength parameter & determining the leading asymptotic behaviour of the potential tail. For a Coulombic potential tail, & = 1, F(1) = =2 we recover the Rydberg formula, En = −

R ˝2 %(n)2 =− ; 2M (n − C

)2

R=

˝2 ; 2M(21 )2

(165)

with Bohr radius 21 , Rydberg constant R and quantum defect C

− 1. The level density is de>ned as the (expected) number of energy levels per unit energy. If the quantum number n is known as a function of energy, then the level density is simply the energy derivative of the quantum number, dn=dE. Simple derivation of Eq. (164) with respect to E = −˝2 %2 =(2M) gives the near-threshold behaviour of the level density,   1 − 1  1 + 1 & 2 ˝2 dn E →0 F(&) 1 1 1 & 2 = : (166) − dE  & 2 2M(& )2 |E| For Coulombic tails, & = 1, this reduces to the well known form, √ dn E →0 1 R = : dE 2 |E|3=2

(167)

When the potential vanishes faster than 1=r 2 at large distances, then the action integral S(E) remains bounded at threshold. The number of bound states is >nite, and conventional WKB quantization deteriorates towards threshold, see Section 3.4. Based on the exact zero-energy solutions (135) introduced in Section 4.1, it is however possible to derive a modi>ed quantization rule which becomes exact in the limit E = −˝2 %2 =(2M) → 0. To order below O(E) = O(%2 ), the wave function b (r)

=

0 (r)

−%

1 (r)

r →∞

∼ 1 − %r

(168)

solves the Schr4odinger equation and, to order below O(%2 ), it has the correct asymptotic behaviour r →∞ required for a bound state at energy E, namely b ˙ exp(−%r). If there is a WKB region of moderate r values, where we can write the WKB expressions (136) for 0 (r) and 1 (r), then in this region the bound state wave function (168) has the form    ∞ 1 1 +

−; ; (169) cos p0 (r ) dr − b (r) ˙
˝ r 4 p0 (r) in analogy to Eq. (140). In Eq. (169), ; is the angle de>ned by      − D1 1 + %D1 =D0 − 2 = tan 1 + 2% tan ; = tan + O(% ) ; 1 − %D1 =D0 4 4 D0

(170)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

so ; =

+ % sin( 2− )D1 =D0 + O(%2 ), and    ∞ 1 0 1

2 − %b + O(% ) ; p0 (r ) dr − cos b (r) ˙
˝ r 2 p0 (r)

403

− 4

(171)

where b is the length parameter already introduced in Section 4.1, Eq. (150). Comparing this with the WKB wave function for the bound state at energy E,    rout (E) 1 1 out (E)

; (172) cos p(r ) dr − WKB (r) ˙
˝ r 2 p(r) yields an explicit expression for the reAection phase out at the outer turning point, namely out (E) = 0 + 2

S(E) − S(0) + 2%b + O(%2 ) : ˝

(173)

In deriving Eq. (173) we have exploited the fact, that the di?erence between the action integrals (162) at >nite energy E and (146) at energy zero is given, to order less than E, entirely by the tail parts of the integrals beyond the point r in the WKB region. Contributions from smaller distances than r to the action integral can be expected to depend smoothly and analytically on E near threshold, so their e?ect on the di?erence S(E) − S(0) will only be of order E ˙ %2 . Inserting the expression (173) for the outer reAection phase and (144) for the inner reAection phase into the quantization rule (162) yields b % + O(%2 ) ; (174)  where nth is the threshold quantum number already introduced in Section 4.1, Eq. (148). Note that the leading energy dependence of the outer reAection phase near threshold exactly cancels with the energy dependence of the action integral, so the near-threshold
quantization rule (174) has a universal form with a leading energy dependent term proportional to |E|. The parameter b determining the magnitude of the leading energy dependent term in the near-threshold quantization rule is just the tail parameter already de>ned in Eq. (150). The near-threshold quantization rule (174) applies for potential tails falling o? faster than 1=r 2 asymptotically. For potentials falling o? faster than 1=r 2 but not faster than 1=r 3 , the exact zero energy solutions (135) have next-to-leading asymptotic terms whose fall-o? is not a whole power of r faster than the leading term 1 or r. The wave function 0 remains well de>ned and its deviation from unity vanishes asymptotically. The wave function 1 may have asymptotic contributions corresponding to a nonnegative power of r (less than 1); a possible admixture of 0 would be of equal or lower order, so it seems that the de>nition of 1 has some ambiguity regarding possible admixtures of 0 . This does not a?ect the derivation of the expression (171), however, because 1 enters with a small coeScient % in the wave function (168) and only its leading term is relevant. The length parameter (150) is actually invariant with respect to possible ambiguities in the choice of 1 . From the universal form (174) of the near-threshold quantization rule, we immediately derive the leading behaviour of the near-threshold level density,  dn E →0 1 2Mb2 + O(E 0 ) : = (175) dE 2 ˝2 |E| %→0

n ∼ nth −

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

Eq. (175) is also quite universal, in that it holds for all potential tails falling o? faster than 1=r 2
. The leading contribution to the near-threshold level density is quite generally proportional to 1= |E|, and its magnitude is determined by the tail parameter b. Even though the number of bound states in a potential well with a short-ranged tail is >nite and there usually is a >nite interval
below threshold with no energy level at all, the level density at threshold becomes in>nite as 1=
|E| towards E → 0. The probability density for >nding a bound state near E = 0 diverges as 1= |E|, but the expected number of states in a small energy interval below threshold, which is obtained by integrating this
probability density, has a leading term proportional to |E|. It is worth mentioning that the considerations above, and in particular the universal formulas (174) for the near-threshold quantization rule and (175) for the near-threshold level density, apply for all potential wells with tails vanishing faster than 1=r 2 , irrespective of whether the leading asymptotic part of the tail is attractive or repulsive. The only condition is, that on the near side of the tail there be a WKB region in the well where the WKB approximation is good for near-threshold energies. The threshold properties are derived via the zero-energy solutions (135) which can be written as WKB wave functions (136) in this region and involve four independent parameters, two amplitudes D0; 1 and two phases 0; 1 . Due to the freedom to choose the overall normalization of the wave functions (143), (171), there remain three independent tail parameters, which determine the near-threshold properties of the potential and depend only on the potential tail beyond the semiclassical region. Three physically relevant tail parameters derived from D0; 1 and 0; 1 are: (i) the characteristic parameter b given by Eq. (150), which enters the universal near-threshold quantization rule (174) and determines the leading, singular contribution to the near-threshold level density (175); (ii) the mean scattering length aU0 given by Eq. (151); (iii) the zero-energy reAection phase 0 , which enters into the de>nition of the threshold quantum number nth , see Eq. (148), and is an important ingredient of the universal near-threshold quantization rule (174). The characteristic parameter b and the mean scattering length aU0 are related (151) via the di?erence − = 0 − 1 of the phases, aU0 = b=tan(− =2), so any two of the parameters b, aU0 and − contain essentially the same information when tan(− =2) is >nite. 4.2.1. Example: The Lennard-Jones potential To demonstrate the validity of the near-threshold quantization rule (174) we take a closer look at the threshold of the Lennard-Jones potential (93) which was studied in Ref. [73] and discussed in Section 3.4, 

r 6  rmin 12 min ; (176) −2 VLJ (r) = j r r it has its minimum value, −j, at r =rmin . The attractive tail of the potential is of the form (156) with & = 6, and the strength parameter 6 is given by 6 = rmin [4M(rmin )2 j=˝2 ]1=4 . The energy eigenvalues En , measured in units of j, depend only on the reduced strength parameter BLJ = 2M(rmin )2 j=˝2 , and for BLJ = 104 the potential well supports 24 bound states, n = 0; 1; : : : ; 23. In this speci>c case, the energy of the highest bound state, n=23, is near −2:7×10−6 , and its distance to the threshold is less than one seventieth of its separation to the second most weakly bound state at E22 =−0:198 : : :×10−3 , see Table 2 in Section 3.4. It is, however, still not so close to the anticlassical limit, E = 0, as can be seen by noting that the reduced classical turning point a = %rout = (%6 )2=3 , which vanishes in the anticlassical limit, is still as large as 1.56 at the energy E23 .

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

405

-5

log10|E23|

-6

-7

exact

-8

conv. WKB near-thr. qu. rule

-9

-10 0.0

0.5

a

1.0

1.5

Fig. 9. Binding energy |E23 | of the highest bound state in the Lennard-Jones potential (176) as function of the reduced classical turning point a = %rout = (%6 )2=3 , as the reduced strength parameter BLJ is varied in the range between 9800 (corresponding to a ≈ 0) and 104 (a = 1:56). The exact energies are shown as >lled circles, the >lled triangles show the prediction of conventional WKB quantization (84) and the open squares show the results obtained with the near-threshold quantization rule (174).

The immediate vicinity of the anticlassical limit can be studied by gradually reducing the depth of the potential well in order to push the highest bound state closer to threshold. When applying the near-threshold quantization rule (174), we assume the reAection phase at the inner classical turning point, which enters in the de>nition (148) of nth , to be =2, which is not exactly true. For the characteristic parameter b we ignore deviations of the potential tail from the homogeneous −1=r 6 form, i.e., we assume b = b(6) = 0:4779888 × 6 (see Table 3 in Section 4.1). The results are shown in Fig. 9 where the binding energy |E23 | of the highest bound state (in units of j) is plotted as a function of the reduced outer classical turning point a at the energy of this state. The largest a value in the >gure, a = 1:56, corresponds to a reduced strength parameter BLJ = 104 and a binding energy (in units of j) of 2:6969 × 10−6 . By gradually reducing BLJ , the exact binding energy of the n = 23 state, shown as >lled circles in Fig. 9, gets smaller and smaller, and it vanishes for BLJ ≈ 9800. At the same time, the reduced outer classical turning point a at the energy E23 decreases from its initial value 1:56 for BLJ = 10 000 to zero for BLJ ≈ 9800. The binding energy obtained via conventional WKB quantization (>lled triangles), which is based on reAection phases =2 at both inner and outer classical turning points, is almost 40% too large for a ≈ 1:5; it decreases much more slowly and reaches a >nite value near 2 × 10−7 when the exact binding energy vanishes. In contrast, the prediction of the universal near-threshold quantization rule (open squares) is not so accurate when a is larger than unity, but it improves rapidly as a decreases. For the three smallest values of a in Fig. 9 which lie between 0.05 and 0.2, the absolute error of the prediction of the near-threshold quantization rule (174) is less than 10−11 times the potential depth. This is smaller than the level spacing to the second highest state, n = 22, by a factor 10−7 and represents an improvement of

406

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

a factor of 10−4 over the performance of conventional WKB quantization. Note that applying the universal near-threshold quantization rule (174) is no more involved than applying conventional WKB quantization (84); it is in fact less so, because the action integral only has to be calculated at threshold, E = 0. The direct numerical integration of the one-dimensional Schr4odinger equation is, of course, always possible, but close to threshold it can be a nontrivial and subtle exercise, and it is de>nitely more time consuming than the direct application of a quantization rule. The universal near-threshold quantization rule (174) can thus be of considerable practical use, e.g., when a problem involves many repetitions of an eigenvalue calculation near threshold. 4.3. Nonhomogeneous potential tails The near-threshold properties of a potential tail falling o? faster than 1=r 2 are determined by three independent tail parameters, the length parameter b, the mean scattering length aU0 (which is, however, not de>ned for a potential falling o? as 1=r 3 ) and the zero-energy reAection phase 0 . This statement implies, that there is a region of moderate r values, where the WKB approximation is accurate for near-threshold energies. The tail parameters are then properties only of the potential tail beyond the WKB region; they do not depend on the potential in the WKB region or at even smaller r values. The fact that the leading asymptotic (r → ∞) behaviour of a potential is proportional to 1=r & does not necessarily mean, that the tail parameters are as given by Eqs. (158) [see also Eq. (89)], (159) and (160) for homogeneous tails (156). For these results to be valid, the homogeneous form (156) of the potential must be accurate not only in the limit of large r values, but all the way down to the WKB region. If the potential tail beyond the WKB region deviates signi>cantly from the homogeneous form (156), then the tail parameters di?er from the tail parameters of the homogeneous tails. The extent to which such nonhomogeneous contributions quantitatively a?ect the tail parameters was >rst studied by Eltschka et al. [82,97]. The tail parameters can be derived from the zero-energy solutions of the Schr4odinger equation in the tail region of the potential, which are then expressed in WKB form in the WKB region as described in Sections 4.1 and 4.2. For a given (nonhomogeneous) potential tail, the tail parameters can always be derived, at least numerically, from the known zero-energy wave functions. If the zero-energy solutions of the Schr4odinger equation are known analytically for the tail region of the potential, then the tail parameters can be derived analytically. Several examples are given in this section. Consider a potential tail consisting of two homogeneous terms,   (& )&−2 (&1 )&1 −2 C & C &1 ˝2 V&; &1 (r) = − & − &1 = − ; &1 ¿ & ¿ 2 : + (177) r r 2M r& r &1 In contrast to homogeneous potentials, potential tails consisting of two homogeneous terms contain an intrinsic, energy independent, length scale which can, e.g., be chosen as the position L, where the two contributions have equal magnitude,  L=

C &1 C&



1 &1 − &



(&1 )&1 −2 = (& )&−2



1 &1 − &

;

(178)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

407

they also have an intrinsic scale of depth, which can be chosen as the magnitude of one of the terms at r = L and expressed in terms of a depth parameter K0 , ˝2 (K0 )2 C& C&1 (& )(&−2)=2 (&1 )(&1 −2)=2 = & = &1 ; K0 = = : 2M L L L&=2 L&1 =2 The dimensionless parameter  (&−2)(&1 −2) 2(&1 −&) & < = K0 L = & 1

(179)

(180)

is a useful measure of the relative importance of the two contributions proportional to 1=r & and to 1=r &1 , respectively [82]. When the powers &, &1 in the potential (177) ful>ll the condition &1 − 2 = 2(& − 2) ;

(181)

zero-energy solutions of the Schr4odinger equation are available analytically [99], and analytical expressions for the tail parameters b, 0 and 1 are given in Refs. [82,97]. Zero-energy solutions are also available for the special case, & = 4;

&1 = 5 ;

(182)

which does not ful>ll the condition (181), and analytical expressions for the tail parameters as functions of < are given in Refs. [100,101]. A further example of a nonhomogeneous potential tail for which analytic zero-energy solutions of the Schr4odinger equation are known is,  −1 −1  3 r r4 ˝2 r 3 r4 + =− + ; (183) V1 (r) = − C3 C4 2M 3 (4 )2 which resembles a homogeneous tail for r-values either much larger or much smaller than the characteristic length L=

C4 (4 )2 = : C3 3

(184)

For rL the potential (183) resembles a −1=r 3 potential, as in the van der Waals interaction between a polarizable neutral atom and a conducting or dielectric surface; for r
L it resembles a −1=r 4 potential. The potential (183) was used by Shimizu [41] as a model for describing the e?ects of retardation in atom–surface interactions [57] and has been further studied in Refs. [56,101], see also Section 5.3. A natural de>nition of the intrinsic strength of the potential (183) is via the wave number K0 given by (3 )2 ˝2 (K0 )2 C3 C4 ; = 3 = 4 ⇒ K0 = 2M L L (4 )3

(185)

and the dimensionless parameter measuring the relative importance of the large-r and the smaller-r parts of the potential is, √ 3 2M C3 √ < = K0 L = = : (186) 4 ˝ C4

408

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

We thus have analytical results for three di?erent potential tails whose leading asymptotic (r → ∞) behaviour is proportional to −1=r & with & = 4: the two-term sum (177) with condition (181) ful>lled implying &1 = 6, the case (182) where it is not ful>lled, and Shimizu’s potential (183). A −1=r 4 potential occurs in several physically important situations, such as in the retarded atom– surface interaction mentioned above, and also as the leading contribution to the interaction between a charged projectile (electron or ion) and a neutral polarizable target (atom or molecule), when retardation e?ects are not taken into account. The strength of the leading −1=r 4 term in an ion-atom potential depends on the polarizability of the neutral atom [102]. For any two-term tail ful>lling (181), the tail parameters b, 0 and − are given as functions of the parameter (180) in Refs. [82,97] (where < is called (). For the special case (&; &1 ) = (4; 6)—for which < = (4 =6 )2 —the parameters b, aU0 = b=tan(− =2) and 0 are       2( 3 − 1 i<)  3 1 2( − i<)    4 4 − arg ; b(4; 6) = 26  41 41  sin  2( 4 − 4 i<)  4 2( 14 − 14 i<)       2( 3 − 1 i<)  2( 34 − 14 i<)   4 4  (4; 6) − arg aU0 = 26  1 1 ;  cos  2( 4 − 4 i<)  4 2( 14 − 14 i<)   

<  < 3 1 3 (4; 6) 1 − ln − 2 arg 2 − i< : (187) 0 =  + 4 2 4 4 4 For the two-term tail (182)—for which <=(4 =5 )3 —the analytical expressions for the tail parameters as functions of < are [100,101],  −1 3(5 )3 2 F with F(z) = |H 1(1) (z)|2 ; b(4; 5) = < (4 )2 3 3  F
( 23 <) (5 )3 5) aU(4; 1 + < = − ; 0 2(4 )2 F( 23 <)   J1 2 ( <) 4 5 5) (188) =  − < − 2 arctan  3 3  : (4; 0 6 3 Y1 2 ( <) 3 3

For Shimizu’s potential (183) we have < = 3 =4 according to Eq. (186), and the analytical expressions for the tail parameters are, (4 )2 F(2<)−1 with F(z) = |H1(1) (z)|2 ; 3   F
(2<) (4 )2 (V1 ) ; 1+< aU0 = − 23 F(2<)   J1 (2<) 3 (V1 ) 0 =  − 4< − 2 arctan : 2 Y1 (2<) b(V1 ) =

In Eqs. (188) and (189) H0(1) (z) = J0 (z) + iY0 (z) are the Hankel functions of order 0 [59].

(189)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

409

4,6

3

b / α1

4,5 2

V1

1

0 0

0.5

1

(a)

1.5



2

4,6 2

b / 4

4,5 V1 1

0 0 (b)

1

2



3

4

5

Fig. 10. Dependence of the length parameter b on the parameter < for two-term potential tails and Shimizu’s potential. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) [for which < = (4 =6 )2 ] and (&; &1 ) = (4; 5) [for which < = (4 =5 )3 ], respectively, and the dotted lines show the results for Shimizu’s potential (183) [for which < = 3 =4 ]. Part (a) shows b=&1 with &1 = 6, 5 and 3, respectively; part (b) shows b=4 .

The dependence of the parameter b on < is illustrated for the three potential tails in Fig. 10. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) and (&; &1 ) = (4; 5), respectively, and the dotted lines show the results for Shimizu’s potential (183). Part (a) shows b=&1 with &1 = 6; 5 and 3, respectively, and in the limit < → 0 we recover the results for the √ corresponding homogeneous potential tail as given by Eq. (159) and Table 3: b(6) =6 = 2( 34 )=[2 2 2( 54 )], b(5) =5 = 2( 23 )=[31=6 22( 43 )] and b(3) =3 = . Part (b) of Fig. 10 shows b=4 , and for < → ∞ we recover the result for a homogeneous −1=r 4 potential, namely b=4 = 1. Fig. 11 shows the dependence on < of the mean scattering lengths aU0 . Part (a) shows aU0 =&1 with &1 = 6; 5 and 3, respectively; part (b) shows aU0 =4 . For a homogeneous −1=r 4 potential tail, the mean 4 scattering length vanishes, because tan((4) − =2) is in>nite. A modi>cation of the −1=r tail involving signi>cant nonhomogeneity in the potential tail beyond the WKB region generally leads to a >nite mean scattering length, so the e?ect of such a modi>cation is appreciable. For homogeneous −1=r & tails with & ¿ 4, the mean scattering length is always positive, indicating that positive scattering lengths are more probable than negative ones [95]. Shimizu’s potential (183) is a nice example of a potential tail with a negative mean scattering length, implying that negative

410

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 3

1

(4,6) (4,5)

2

-1

a0 / 4

a0 / α1

0

(4,6)

V1 1

(4,5)

-2

0 V1 0

(a)

1

2 

3

4

0 (b)

1

2 

3

4

Fig. 11. Dependence of the mean scattering length aU0 on the parameter < for three potential tails. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) and (&; &1 ) = (4; 5), respectively, and the dotted lines show the results for Shimizu’s potential (183). Part (a) shows aU0 =&1 with &1 =6, 5 and 3, respectively; part (b) shows aU0 =4 .

scattering lengths are more probable than positive ones. As < = 3 =4 becomes smaller and smaller, the relative importance of the −1=r 3 part of the potential increases, but scattering lengths and their mean can still be de>ned, because asymptotically (r → ∞) the potential falls o? as −1=r 4 . From Eq. (189), the small-< behaviour of the mean scattering length is <→0

aU0(V1 ) ∼ 23 (ln(<) + (E ) ;

(190)

where (E = 0:5772 : : : is Euler’s constant [59]. For a >xed value of 3 the mean scattering length tends to larger and larger negative values as < = 3 =4 decreases; in the limit < → 0 the mean scattering length approaches −∞ reAecting the fact that scattering lengths cannot be de>ned for a potential asymptotically proportional to 1=r 3 . The zero-energy reAection phases 0 are displayed in Fig. 12. Again, the solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) and (4; 5), and the dotted lines show the results for Shimizu’s potential (183). For < → 0 we recover the results (5) (3) 3 5 3 for the corresponding homogeneous tail as given by Eq. (158): (6) 0 = 4 , 0 = 6 , 0 = 2 . For (4) < → ∞ we recover the result expected for a homogeneous −1=r 4 tail, 0 = . Analytical expressions for the tail parameters b, aU0 and 0 of various potentials are compiled in Table 4. 4.4. The transition from a >nite number to in>nitely many bound states, inverse-square tails The near-threshold quantization rule (164) for an attractive potential tail vanishing as −1=r & with 0 ¡ & ¡ 2 becomes meaningless as & approaches the value 2 (from below). On the other hand, the tail parameter b entering in the near-threshold quantization rule (174) for potential tails vanishing faster than 1=r 2 diverges to in>nity for a (homogeneous) tail vanishing as −1=r & , when the power & approaches 2 from above implying ' → ∞, see Eq. (159). In order to understand the transition from potential wells with tails vanishing faster than −1=r 2 , which support at most a >nite number of bound states, to those with tails vanishing more slowly than −1=r 2 , which support in>nitely many

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411

(4,6)

1.5

(4,5) 0 / π

V1 1

0

1

2 

3

4

Fig. 12. Dependence of the zero-energy reAection phase 0 on the parameter < for three potential tails. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) [for which < = (4 =6 )2 ] and (&; &1 ) = (4; 5) [for which < = (4 =5 )3 ], respectively, and the dotted lines show the results for Shimizu’s potential (183) [for which < = 3 =4 ].

bound states, it is necessary to look in some detail at potentials with tails asymptotically proportional to the inverse square of the distance, 2 ( def ˝ r →∞ : (191) V (r) ∼ V( (r) = 2M r 2 Inverse-square potentials of the form (191) with positive or negative values of the strength parameter ( occur in various physically relevant situations. Separating radial and angular motion in the twoor more-dimensional Schr4odinger equation leads to a centrifugal potential of the form (191) with ( as given by Eqs. (58) and (59) for three and two dimensions, respectively. Attractive and repulsive inverse-square potentials occur in the interaction of an electrically charged particle with a dipole, e.g. in the interaction of an electron with a polar molecule or with a hydrogen atom in a parity-mixed excited state [103–106]. If the inverse-square tail (191) is suSciently attractive, more precisely, if 1 1 def (192) ( = − g¡ − ; g¿ ; 4 4 then the potential supports an in>nite number of bound states, usually called a “dipole series”, and towards threshold, E = 0, the energy eigenvalues of the bound states depend exponentially on the quantum number n, [103,107]   2n  n→∞ En ∼ − E0 exp−  : (193) g − 14 The strength g of the attractive inverse-square tail determines the asymptotic value of the ratio of successive energy eigenvalues,   En n→∞ 2  ; (194) ∼ exp  En+1 1 g− 4

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Table 4 Characteristic length b, mean scattering length aU0 and zero-energy reAection phase 0 for several attractive potential tails, V (r) = −v(r) × ˝2 =(2M) v(r)

b

aU0

(K0 )2 ,(L − r)

1=K0

& '1+2' 2(1+')2

L & '2'

3

—

(& )&−2 r& 3 r3

(& )&−2 + r& (&1 )&1 −2 r &1

  ' 2(1−')  2(z+ )  &1 (2')2(1+')  2(z− )  ×  

2(z+ ) sin '2 − arg 2(z −)

2(1−') cos(') 2(1+')

  ' 2(1−')  2(z+ )  &1 (2')2(1+')  2(z− )  ×  

2(z+ ) cos '2 − arg 2(z −)

0

Comments

0 (' + 12 )

'=

1 &−2

¡1

(1 + ') 2 +

'=

1 &−2

¡1

<'[1 − ln( <'2 )]

&1 2

−2 arg(z+ )

<=

3 2



=&−1

z± = 3 r3

(4 )2 r4

+

3 [1 2

+ coth( < )] 2

 + <(1 − ln <2 )

—

<=

& &1

1='

1±'−i<' 2

3 4

−2 arg 2(1 − i <2 ) (4 )2 r4

+

(5 )3 r5

3(5 )3 (4 )2

F(z)−1

3

(5 ) − 2( (1 + < )2 4

F  (z) ) F(z)

5 6

−

4 3

2 arctan

3

[ r 3 +

r 4 −1 ] (4 )2

(K0 )2 1+exp[(r−L)=]

(K0 )2 exp(− r ) a

(4 )2 3

F(z)−1

 coth(K0 )



2

− (24 )3 (1 + <

F  (z) ) F(z)

L − 2((E + R(z))

2 ln(K0 )

<− 

J1=3 (z) Y1=3 (z)



3 2

 − 4<−   2 arctan YJ11(z) (z)  + 4K0  ln 2   2 0 ) +2 arg 2(iK 2(2iK0 ) 1 2

z = 23 <    (1) 2 F = H1=3 (z) <=

3 4

z = 2< F = |H1(1) (z)|2 z = 2 (iK0 ) a a



Here the semiclassical region of “small” r is r → −∞. (E is Euler’s constant and

< = ( 45 )3

2

the digamma function.

but not the explicit positions of the energy levels, which are >xed by the constant E0 in Eq. (193). This reAects the fact that there is no energy scale in a Schr4odinger equation with a kinetic energy and an inverse-square potential; if (r) is a solution at energy @, then (sr) is a solution at energy s2 @. For a pure −1=r 2 potential one can obtain a discrete bound state spectrum corresponding to the right-hand side of Eq. (193) by requiring orthogonality of the bound state wave functions at di?erent energies, but the resulting spectrum is unbounded from below, En → −∞ for n → −∞ [107]. In a realistic potential well with a suSciently attractive inverse-square tail, Eq. (192), the actual positions of the bound state energies are determined by the behaviour of the potential at small distances, where it must necessarily deviate from the pure −1=r 2 form. If the potential tail contains

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413

a further attractive term proportional to −1=r m , m ¿ 2, then as r decreases this term becomes dominant and the WKB approximation becomes increasingly accurate. Potential wells with two-term tails,   1 ˝2 (m )m−2 g r →∞ (195) V (r) ∼ Vg; m (r) = − + 2 ; m ¿ 2; g ¿ ; m 2M r r 4 support an in>nite dipole series of bound states and there may be a WKB region at moderate r values if the well is deep enough. In this case, near-threshold properties of the bound states can be derived [84,108] by matching the asymptotic (r → ∞) solutions of the Schr4odinger equation with the inverse-square term alone to zero-energy solutions of the tail (195), which are then expressed as WKB waves in the WKB region in a procedure similar to that used in Section 4.2. This results [108] in an explicit expression for the factor E0 , which asymptotically (n → ∞) determines the positions of the energy levels in the dipole series (193), 4

2˝2 (m − 2) m−2 E0 = exp M(m )2



    tan(S˜ 0 =˝) 2 C + D + + arctan : B 2 tanh(;=2)

The parameters B, ;, C and D appearing in Eq. (196) are,  1 2B B= g− ; ;= ; C = arg 2(iB); D = arg 2(i;) ; 4 m−2

(196)

(197)

and S˜ 0 is essentially the threshold value of the action integral from the inner classical turning point rin (0) to a point r in the WKB region,  m−2  m 2 S˜ 0 2 in (0)  1 r

p0 (r ) dr + − = − : ˝ ˝ rin (0) m−2 r 2 4

(198)

A condition for the applicability of the formula (196) is, that near the point r in the WKB region the potential must be dominated by the −1=r m term so that the inverse square contribution can be neglected; the sum of the integral and the term proportional to 1=r (m−2)=2 in (198) is then independent of the choice of r. More explicit solutions are available when the whole potential consists of an attractive inversesquare tail and a repulsive 1=r m core,   g ˝2 (m )m−2 (199) − 2 ; m¿2 : V (r) = 2M rm r The existence of a WKB region in the well is not necessary in this case, because analytical zero-energy solutions of the Schr4odinger equation are available for the whole potential. The zeroenergy solution which vanishes at r = 0 approximates >nite energy solutions to order less than O(E) in the region of small and moderate r values, and it can be matched to the solution which vanishes asymptotically in the presence of the attractive inverse-square tail. This yields the following

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expression for the factor E0 de>ning the energies of the near-threshold bound states of the dipole series (193) [108], 4   2˝2 (m − 2) m−2 C+D ; E0 = exp 2 M(m )2 B

(200)

where B, C and D are as already de>ned in Eq. (197). Note that E0 is only de>ned to within a factor consisting of an integer power of the right-hand side of Eq. (194); multiplying E0 by an integer  

power of exp 2= g − 14 does not a?ect the energies in the dipole series (193) except for an appropriate shift in the quantum number n labelling the bound states. One interesting feature of dipole series is that, for a given strength ( ¡ − 14 of the attractive potential tail, the limit of large quantum numbers, n → ∞, does not coincide with the semiclassical limit. As discussed in Section 2.1, the semiclassical limit for an inverse-square potential can be approached for large absolute values of the potential strength, but not by varying the energy. As a consequence, the energy eigenvalues of a dipole series obtained via conventional WKB quantization (84) do not become more accurate in the limit n → ∞, not even when the Langer modi>cation (64) is used in the WKB calculation. The WKB energies obtained with the Langer modi>cation acquire a constant relative error in the limit n → ∞, whereas the error grows exponentially with n if they are calculated without the Langer modi>cation [109]. A potential with an attractive inverse-square tail (191) no longer supports an in>nite series of bound states, when the strength parameter ( is equal to (or larger than) − 14 . This can be expected from the breakdown of formulas such as (193), (196) and (200) when −( = g = 14 . It is also physically reasonable, considering that the inverse-square potential V(=−1=4 is the s-wave (l2 = 0) centrifugal potential for a particle moving in two spatial dimensions, see Eq. (59). The fact that the radial Schr4odinger equation for a particle moving in a plane with zero angular momentum actually contains an attractive centrifugal force has lead Cirone et al. [110] to investigate possibilities of a state being bound by the associated force, which the authors call “anticentrifugal”. Kowalski et al. [111] have contributed to a clari>cation of the issue by drawing attention to the topological e?ects occurring when a singular point is extracted from the plane as an origin for the de>nition of the polar coordinates. It is diScult to imagine a physical mechanism that would bind a free particle in a Aat plane, so the discontinuation of dipole series of bound states at the value − 14 of the strength parameter ( seems more than reasonable. A potential well with a weakly attractive inverse-square tail, i.e. with a strength parameter in the range −

1 6(¡0 ; 4

(201)

can support a (>nite) number of bound states if supplemented by an additional attractive potential. If the additional potential is regular at the origin, then the action integral from the origin to the outer classical turning point diverges because of the −1=r 2 singularity of the potential at r = 0, so a naive application of the generalized quantization rule (162) does not work. This can be overcome by shifting the inner classical turning point to a small positive value and adjusting the reAection phase in accordingly [70].

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415

The near-threshold quantization rule for a weakly attractive inverse-square tail has been studied in some detail by Moritz et al. [112], and analytical results have been derived for tails of the form   1 ˝2 (m )m−2 g r →∞ (weak) V (r) ∼ Vg; m (r) = − + 2 ; m ¿ 2; g 6 : (202) m 2M r r 4 For g ¡ 14 (i.e., excluding the limiting case g = 14 ), the near-threshold quantization rule is [112], n = nth − with

(%m =2)20 + O((%m )40 ) + O(%2 ) ; 2 2' sin(0)(m − 2) 0'[2(0)2(')]

(203)

 20 1 1 −g= (+ and ' = : (204) 0= 4 4 m−2 The >nite but not necessarily integer threshold quantum number nth in Eq. (203) is given by  m−2  m 2 1 r 2 in (0)  '

− −  : (205) p0 (r ) dr + − nth  = ˝ rin (0) m−2 r 2 4 2 

As in the discussion of Eqs. (196) and (198), the point r de>ning the upper limit of the action integral must lie in a region of the potential well where the WKB approximation is suSciently accurate and the potential is dominated by the −1=r m term, so the inverse-square contribution can be neglected; the sum of the integral and the term proportional to 1=r (m−2)=2 in (205) is then independent of the choice of r. When we express % in terms of the energy E = −˝2 %2 =(2M), the near-threshold quantization rule (203) becomes n = nth − B(−E)0 ;

(206)

with B=

(M(m )2 =(2˝2 ))0 : sin(0) (m − 2)2' 0'[2(0)2(')]2

(207)

The limiting case (=− 14 corresponding to 0=0 and '=0 requires special treatment; the near-threshold quantization rule in this case is [112],   ˝2 2=(m − 2) 1 n = nth + ; B = : (208) +O ln(−E=B) 2M(m )2 [ln(−E=B)]2 Again, nth is given by the expression (205); note that ' vanishes in this case. We now have a very comprehensive overview of near-threshold quantization in potential wells with attractive tails. Potentials falling o? as −1=r & with a power 0 ¡ & ¡ 2 support an in>nite number of bound states, and the limit of in>nite quantum numbers is the semiclassical limit. The near-threshold quantization rule (164) contains a leading term proportional to 1=(−E)1=&−1=2 in the expression for the quantum number n. For & = 2, the threshold E = 0 no longer represents the semiclassical limit of the Schr4odinger equation, but the potential still supports an in>nite number of bound states, if the attractive inverse-square tail is strong enough, Eq. (192); the near-threshold quantization rule now contains

g − 14 ln(−E) in the expression for the quantum number n, see Eq. (193). The attractive

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

inverse-square tail ceases to support an in>nite series of bound states at the value g = −( = 14 of the strength parameter, which corresponds to the strength of the (attractive) s-wave centrifugal potential for a particle in a plane. In the near-threshold quantization rule, the leading term in the expression for the quantum number now is a >nite number nth related to the total number of bound states,
and 1

the next-to-leading term contains the energy as 1=ln(−E) for ( = − 14 [Eq. (208)], or as (−E) (+ 4 for ( ¿ 14 , see Eq. (206). It is interesting to note, that the properties of potential wells with short-ranged tails falling o? faster than 1=r 2 >t smoothly into the picture elaborated for inverse-square tails when we take the strength of the inverse-square term to be zero. The near-threshold quantization rule (203) acquires the form (174) when ( = 0, 0 = 12 , and the coeScient of % becomes b(m) = with b(m) given by Eq. (159) when we also insert ' = 1=(m − 2). The discussion of weakly attractive inverse-square tails, de>ned by the condition (201), can be continued without modi>cation into the range of weakly repulsive inverse-square tails, de>ned by strength parameters in the range 3 0¡(¡ : (209) 4  The parameter 0 = ( + 14 determining the leading energy dependence on the right-hand sides of Eqs. (203) and (206) then lies in the range 1 ¡0¡1 ; (210) 2 and the leading energy dependence (−E)0 expressed in these equations is still dominant compared to the contributions of order O(E), which come from the analytical dependence of all short-ranged features on the energy E and were neglected in the derivation of the leading near-threshold terms. We can thus complete the comprehensive overview of near-threshold quantization by extending it to repulsive potential tails. For weakly repulsive inverse-square tails (209), the formulas (203) and (206) remain valid. The upper boundary of this range is given by  3 1 (= ; 0= (+ =1 ; (211) 4 4 which corresponds to the p-wave centrifugal potential in two spatial dimensions, l2 = ±1, see Eq. (59). At this limit, the near-threshold quantization rule has the form, n = nth − O(E) ;

(212)

and this structure prevails for more strongly repulsive inverse-square tails, ( ¿ 34 , and for repulsive potential tails falling o? more slowly than 1=r 2 . Repulsive tails falling o? more rapidly than 1=r 2 comply with the case ( = 0, i.e. of vanishing strength of the inverse-square term in the potential, and, provided there is a suSciently attractive well at moderate r values, the quantization rule has the form (174) with a tail parameter b and a threshold quantum number nth which also depends on the shorter-ranged part of the potential. Note that the condition (211) also de>nes the boundary between systems with a singular and a regular level density at threshold. For attractive potential tails and for repulsive potential tails falling o? more rapidly than 1=r 2 or as an inverse-square potential with ( ¡ 34 , the level density dn=dE

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417

Table 5 Summary of near-threshold quantization rules for attractive and repulsive potential tails. The second column gives the leading term(s) to the quantization rule in the limit of vanishing energy, E = −˝2 %2 =(2M) → 0. The third column lists equations where explicit expressions for the constants appearing in the second column can be found; these can apply quite generally, as in the >rst row, or to special models of potential tails with the asymptotic behaviour given in the >rst column V (r) for r → ∞ ˝2 − 2M (& )&−2 =r & , ˝2 2M

(=r 2 , ( ¡ −

Quantization rule for E → 0 0¡&¡2

1 4

(¿

3 4

3 4

˙ ±1=r , & ¿ 2



|(| −

n ∼ nth + A=ln(−E=B)

˙ +1=r & , 0 ¡ & ¡ 2 &

F(&)=(%& )

√

n ∼ nth − B(−E) n ∼ nth − O(E) n ∼ nth − O(E) 1 n ∼ nth − b% 

Refs. for constants

(2=&)−1

1 n ∼ − 2 ln(−E=E0 )=

( = − 14 − 14 ¡ ( ¡

n∼

1 

(+1=4

F(&): Eq. (163) 1 4

E0 : (196), (200) nth : (205) A, B: (208) nth : (205) B: (207) nth : (205)

nth : (148) b: (150), (154), (159), (187), (188), (189), Tables 3, 4

is singular at threshold, and the leading singular term is determined by the tail of the potential. For a repulsive inverse-square tail with ( ¿ 34 , and for a repulsive tail falling o? more slowly than 1=r 2 , the level density is regular at threshold, and the leading (constant) term depends also on the short-ranged part of the potential. A summary of the near-threshold quantization rules reviewed in the last three subsections is given in Table 5. 4.5. Tunnelling through a centrifugal barrier A potential with a repulsive tail at large distances and a deep attractive part at small distances forms a barrier through which a quantum mechanical particle can tunnel. If the repulsive tail falls o? more slowly than 1=r 2 , then the threshold represents the semiclassical limit (in the tail region), and the conventional WKB formula for the tunnelling probability (114),  1 rout (E) −2I (E) WKB ; I (E) = |p(r)| dr ; (213) PT (E) = e ˝ rin (E) involving the action integral between the inner classical turning point rin (E) and the outer classical turning point rout (E) is expected to work well near threshold, provided the conditions of the semiclassical limit are also well ful>lled around rin (0). As we saw in Section 3.5, the WKB formula (213) fails near the base of a barrier with a tail falling o? faster than 1=r 2 , because this corresponds to the anticlassical limit of the Schr4odinger equation; the exact tunnelling probability vanishes at the base whereas Eq. (213) produces a >nite result. Potential barriers involving the centrifugal term in

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

the radial Schr4odinger equation fall o? as 1=r 2 for large r, and hence lie on the boundary between these two regimes. When the potential on the near side (smaller r) of the barrier supports a WKB region where WKB wave functions are good approximate solutions of the Schr4odinger equation for near-threshold energies, then the amplitudes for transmission through and reAection by the barrier can be obtained in a way quite analogous to the methods for deriving the near-threshold quantization rules as described in Section 4.4, see Refs. [84,108,112]. Analytical results have been derived for a potential tail of the form   ( ˝2 (m )m−2 r →∞ V (r) ∼ V(; m (r) = ; m¿2 : (214) − 2M r 2 rm For suSciently large r values, the −1=r m term in the potential tail (214) can be neglected, and the wave function on the far side of the barrier at energy E = ˝2 k 2 =(2M) ¿ 0 can be approximated by the analytically known solution of the Schr4odinger equation with the 1=r 2 potential alone. In the barrier region, the analytically known zero-energy solutions for the potential (214) solve the Schr4odinger equation to order less than O(E), and a unique solution is obtained by matching to the asymptotic wave function just mentioned. In the WKB region on the near side of the barrier, the unique solution constructed as above can be written as a superposition of inward and outward travelling WKB waves. Comparing the amplitudes of inward and outward travelling waves on both sides of the barrier yields the transmission amplitude to order less than O(E), and the leading contribution to the transmission probability PT is, k →0

PT ∼ P(m; ()(km )20 ;

(215)

with the coeScient P(m; () =

42 =220 ; (m − 2)2' 0'[2(0)2(')]2

(216)

 0 = ( + 14 and ' = 20=(m − 2) are already de>ned in Eq. (204), but now, in Eqs. (215) and (216), ( can also assume nonnegative values. When the inverse-square tail originates from a centrifugal potential in three dimensions, its strength parameter ( is related to the angular momentum quantum number l3 by,  1 1 (217) ( = l3 (l3 + 1); 0 = ( + = l3 + ; 4 2 and the energy dependence of the transmission probability (215) is simply an expression of Wigner’s threshold law [113], according to which probabilities Pl3 which are suppressed by a centrifugal barrier of angular momentum quantum number l3 generally behave as E →0

Pl3 ˙ E 0 = E l3 +1=2 :

(218)

Note that the formulae (215) and (216) can be continued to negative strength parameters in the range of weakly attractive (201) or vanishing inverse-square tails; in fact, they hold for any ( ¿ − 14 . For − 14 ¡ ( 6 0, the potential tail (214) no longer contains a barrier, so transmission between the asymptotic (large r) region and the inner WKB region is classically allowed at all (positive) energies. The probability for this classically allowed transmission is, however, less than unity, because incoming

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waves can be reAected in the region of high quantality, where the WKB approximation is not good, see Section 2.3. Such “quantum reAection” will be discussed in more detail in Section 5. The probability for transmission from the outer asymptotic region to the inner WKB region actually goes to zero at threshold, even for weakly attractive inverse-square tails, − 14 ¡ ( 6 0, for which there is no barrier. Wigner’s threshold law (218) can be formally extended down to negative angular momentum quantum numbers in the range − 12 ¡ l3 6 0. We now discuss the accuracy of the WKB formula (213) for tunnelling probabilities through a centrifugal barrier at near-threshold energies. For an inverse-square tail (191) with ( ¿ 0, the WKB √ integral I (E) is dominated by the tail near outer classical turning point rout = (=k at near-threshold √ energies, and it diverges as ln( (=k) for k → 0. This means that the leading contribution to the √ √ WKB tunnelling probability (213) is proportional to k 2 ( ˙ E ( , in contradiction to the exact result (218) which obeys Wigner’s threshold law. This contradiction can be resolved by invoking the Langer modi>cation (64) when applying the WKB formula, but a residual error remains, because the WKB expression does not necessarily give the right coeScient of the E 0 term. For potential tails of the form (214), Moritz [84,108] found an upper bound for the WKB integral entering the expression (213), namely     20 m0 2−4=m ln Iapprox = − 1 + O((km ) ) ; (219) m−2 (km )1−2=m so the leading term for the corresponding tunnelling probability,  2m0=(m−2) e (km )20 ; PTWKB; approx = e−2Iapprox = 20

(220)

is a lower bound for the leading term of the WKB tunnelling probability (213). It turned out [114] that the expression (220) is not only a lower bound but becomes equal to the WKB tunnelling probability (213) in the limit E → 0. In the near-threshold limit, the WKB tunnelling probability (213) thus overestimates the exact tunnelling probability, which is given by Eqs. (215) and (216), by a factor G,   0+' e−2Iapprox e 2(0)2(') 2 def G = lim = : (221) k →0 PT 200−1=2 ''−1=2 For large values of the strength ( of the 1=r 2 term in the potential, 0 and ' are also large and we can express the gamma functions in Eq. (221) via Stirling’s formula [59]. This gives   1 m (→∞ +O ; (222) G ∼ 1+ 120 02 showing that the WKB treatment gives the correct leading behaviour of the near-threshold tunnelling probabilities in the limit of large angular momenta. The dependence of the factor G on 0 ≡ l3 + 12 is illustrated in Fig. 13 for powers m of the attractive term in the potential tail (214) ranging from m = 3 to m = 7. The WKB results become worse as 0 decreases and as m increases. For the realistic example 0 =3=2; m=6, which corresponds to angular momentum quantum number l3 =1 and an inter-atomic van der Waals attraction, the WKB formula (213) overestimates the exact tunnelling probabilities by 38% near threshold. Transmission probabilities can also be calculated for attractive inverse-square tails, if there is a WKB region of moderate r values where the WKB approximation is good. The equations (215)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

2.5 m=3 m=4 m=5 m=6 m=7

2.0 G

1.5

1.0 0

2

4

µ

6

8

10

Fig. 13. Behaviour of G [Eq. (221)] as function of 0 for m = 3; : : : ; 7. For a potential barrier (214) consisting of an attractive −1=r m potential and a centrifugal term corresponding to angular momentum quantum number l3 = 0 − 12 ; G gives the factor by which the conventional calculation of transmission probabilities via the WKB formula (213), including the Langer modi>cation (64), overestimates the exact result (215), (216) for near-threshold energies. From [108].

and (216) for the exact transmission probabilities through the potential tail (214) are also valid for weakly attractive or vanishing inverse-square terms, − 14 ¡ ( 6 0. In this range of values of (, the Langer modi>cation (64) actually produces an asymptotically repulsive potential with a barrier to tunnel through, so the conventional WKB formula (213) can be applied for near-threshold energies. Note, however, that the factor (221) by which the conventional WKB result overestimates the exact result is quite large in the range − 14 ¡ ( 6 0 corresponding to 0 ¡ 0 ¡ 12 and 0 ¡ ' ¡ 1=(m − 2), and diverges to +∞ for ( → − 14 corresponding to 0; ' → 0. For vanishing strength of the inverse-square term, ( = 0; 0 = 12 ; ' = 1=(m − 2), the near-threshold probability (215), (216) for transmission through the potential tail (214) reduces to, 4'1+2' PT = km P(m; ( = 0) = km = 4kb(m) ; (223) 2(1 + ')2 where b(m) is the length parameter (159), which determines the near-threshold quantization rule (174) and the level density just below threshold for a homogeneous −1=r m potential tail. Eq. (223) thus formulates a connection between the near-threshold properties of bound states at negative energies and those of continuum states at positive energies. This connection applies not only for the special case of vanishing strength of the inverse-square term, but to all strengths within the range of weak inverse-square tails [112],  1 0¡0 = ( + ¡1 : (224) 4 If we write the near-threshold quantization rule (203) as %→0

n ∼ nth  − C¡ (%)20

(225)

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421

for small negative energies E = −˝2 %2 =(2M), and the expression for near-threshold transmission probabilities (215), (216) at small positive energies E = +˝2 k 2 =(2M) as k →0

PT ∼ C¿ (k)20

(226)

then the constants C¡ and C¿ are related by C¿ = 4 sin(0)C¡ :

(227)

The constant  in Eqs. (225), (226) can be any (common) length; it is included so that the coef>cients C¡ and C¿ are dimensionless. For vanishing strength of the inverse-square term, i.e. for potentials falling o? faster than 1=r 2 , we have 20 = 1, and the product C¡  is simply the length parameter b of the potential tail as de>ned in Section 4.1, Eq. (150), see also the bottom block of Table 5. In this case, Eqs. (225) and (226) reduce to %→0

n ∼ nth  − %b;

k →0

PT ∼ 4kb ;

(228)

with the same length parameter b appearing in both equations. The relation (227) is independent of the power of the shorter-ranged attractive contribution to the potential tail (214). It seems reasonable to assume that it is a universal relation connecting the near-threshold states at positive and negative energies for potentials with weak inverse-square tails, and that this is also true for the special case (228) of potentials falling o? faster than 1=r 2 . The behaviour for this latter case is con>rmed in Section 5.1, see Eq. (241). 5. Quantum re ection Just as quantum mechanics can allow a particle to tunnel through a classically forbidden region, it can also lead to the reAection of a particle in a classically allowed region where there is no classical turning point. The term “quantum reAection” refers to such classically forbidden reAection. Quantum reAection can only occur in a region of appreciable quantality, i.e. where the condition (36) is violated. In regions where Eq. (36) is well ful>lled, motion is essentially (semi-) classical, and, in the absence of a classical turning point, the particle does not reverse its direction. Quantum reAection can occur above a potential step or barrier, or in the attractive long-range tails of potentials describing the interaction of atoms and molecules with each other or with surfaces. For potentials falling o? faster than 1=r 2 , the probability for quantum reAection approaches unity at threshold, so it is always an important e?ect at suSciently low energies. Quantum reAection had been observed to reduce the sticking probabilities in near-threshold atom-surface collisions more than twenty years ago [115–120], and the recent intense activity involving ultracold atoms and molecules has drawn particular attention to this phenomenon [121–125]. As described in Section 3.5, transition and reAection amplitudes, T and R, can be de>ned using WKB wave functions for the incoming, transmitted and reAected waves in the semiclassical regions, or plane waves if the potential tends to a constant asymptotically. The transmission and reAection probabilities (103), PT = |T |2 ;

PR = |R|2 ;

(229)

are independent of the choice of WKB waves or plane waves, but the phases of the transmission and reAection amplitudes depend on this choice [see Eqs. (104), (105)] and also on the choice of

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reference points, at which the reference waves have vanishing phase. Throughout this section, the reAection amplitude is always de>ned with respect to incoming plane waves incident from the right as in Eq. (102). Whenever the phase of R is important, we retain the subscript “r” to remind us of this choice. The point of reference is taken to be at r = 0 unless explicitly stated otherwise. For brevity, we shall refer to the absolute value of the reAection amplitude, i.e. to the square root of the reAection probability, as the re?ectivity. 5.1. Analytical results A fundamentally important example [46] is the Woods-Saxon step potential, V (r) = −

˝2 (K0 )2 =(2M) ; 1 + exp(r=)

(230)

which is treated in detail in the textbook of Landau and Lifshitz [43]. The reAection amplitude, de>ned with reference to plane waves (102) is,  
2(2ik) 2(−ik − iq) 2 q + k Rr = (231) ; q = (K0 )2 + k 2 : 2(−2ik) 2(ik − iq) q−k From the properties of the gamma functions of imaginary argument [59] the absolute value of the reAection amplitude (231) is sinh[(q − k)] |R| = ; (232) sinh[(q + k)] as already used in Section 2.1, Eq. (14). For small values of the di?useness,  → 0, the Woods-Saxon potential (230) approaches the sharp step potential already discussed in Sections 2.4 and 4.1, and the reAection amplitude becomes [cf. Eq. (39)] q−k Rr = − : (233) q+k The large-r tail of the Woods-Saxon potential (230) is an exponential function, ˝2 (K0 )2 exp(−r=) : (234) 2M The exponential potential (234) is an interesting example in itself, because the semiclassical approximation becomes increasingly accurate for r → −∞, although the r-dependence of the potential gets stronger and stronger. The Schr4odinger equation for the potential (234) is solved analytically [59] by any Bessel function of order ' = 2ik and argument z = 2K0  exp[ − r=(2)]. The solution which merges into a leftward travelling WKB wave for r → −∞ is the Hankel function (r) ˙ H'(1) (z), and matching this solution to a superposition of incoming and reAected plane waves yields the reAection amplitude, 2(2ik) (K0 )−4ik exp(−2k) : Rr = (235) 2(−2ik) V (r) = −

For any given value of k, the reAection amplitude (231) of the Woods-Saxon step actually becomes equal to the result (235) in the limit that the relative diBuseness [63] K0  is large, K0  → ∞.

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

423

In the near-threshold region, the leading behaviour of the reAection amplitude (231) is k →0

Rr ∼ − [1 − 2k coth(K0 ) − 4ik((E + R{ where (E = 0:57721 : : : is Euler’s constant and contributions to the reAectivity are, |R| = 1 − 2kb + O((k)2 );

2

2 (−iK0 )})]

;

(236)

= 2
=2 is the digamma function [59]. The leading

b =  coth(K0 ) :

(237)

This means, that the reAectivity is unity at threshold, and its initial decrease from unity is linear in k, i.e. in the asymptotic velocity of the particle on the side where this velocity goes to zero. This threshold behaviour of the quantum reAectivity is very general and holds for all potential tails falling o? faster than 1=r 2 [55,119] as can be shown using the methods applied in Sections 4.1 and 4.2. Consider an attractive potential going to zero faster than 1=r 2 for large r with a semiclassical WKB region at moderate or small r values. Using the zero-energy solutions (135) of the Schr4odinger equation, which go unity resp. r for large r and behave as (136) for small r, we can construct a wave function (r) =

D0

1 (r)exp(i0 =2)

− D1 0 (r)exp(i1 =2) ; D0 D1 sin[(0 − 1 )=2]

(238)

which is proportional to a leftward travelling WKB wave of the form (100) in the semiclassical region r → 0. Matching the asymptotic (r → ∞) form of the wave function (238) to the superposition 1 kr →0 √ [exp(−ikr) + Rr exp(ikr)] ˙ 1 + Rr − ikr(1 − Rr ) ˝k

(239)

gives the leading near-threshold contribution to the reAection amplitude, k →0

Rr ∼ −

1 − ik exp[ − i(0 − 1 )=2]D1 =D0 : 1 + ik exp[ − i(0 − 1 )=2]D1 =D0

For the reAectivity |R|, Eq. (240) implies k →0

2

|R| ∼ 1 − 2kb = exp(−2kb) + O(k );

  D1 0 − 1 : b= sin D0 2

(240)

(241)

Eq. (241) implies that the probability for quantum reAection behaves as 1 − 4kb near threshold, and k →0 this is consistent with the result PT ∼ 4kb given for the transmission probability in Eq. (228) at the end of Section 4. The characteristic length parameter of the quantal region of the potential tail, namely b as de>ned in Eq. (150), determines not only the near-threshold quantization rule (174) and the near-threshold level density (175), but also the reAection and transmission properties of the potential tail near threshold. The near-threshold behaviour of the phase of the reAection amplitude also follows from (240) and the result is, k →0

arg(Rr ) ∼  − 2k aU0 ;

aU0 =

b : tan [(0 − 1 )=2]

(242)

The parameter aU0 determining the near-threshold behaviour of the phase of the reAection amplitude is just the mean scattering length de>ned in Eq. (151). Scattering lengths and mean scattering lengths

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

are only de>ned for potentials falling o? faster than −1=r 3 . For a potential proportional to −1=r 3 , V (r) = −

˝ 2 3 C3 = − ; r3 2M r 3

(243)

the wave function which is proportional to a leftward travelling
WKB wave of the form (100) in the semiclassical region r → 0 is (r) ˙ H1(1) (z)=z with z = 2 3 =r, and matching to the asymptotic waves (239) gives k →0

arg(Rr ) ∼  − 2k3 ln(k3 ) :

(244)

Note that the formula (241) for the near-threshold reAectivity holds for all potentials falling o? faster than −1=r 2 , even for those such as (243), where the phase of the reAection amplitude becomes divergent at threshold. The energy dependence of the phase of the reAection amplitude can be related to the time gain or delay of a wave packet during reAection [126]. If the momentum distribution of the incoming wave packet is sharply peaked around a mean momentum ˝k0 , then the shape of the reAected wave packet is essentially the same and the time shift can be calculated in the same way as in partial-wave scattering [127,128] where the reAection amplitude R is replaced by the partial-wave S-matrix. The derivative of arg[R(k)] with respect to k, taken at k0 , describes an apparent shift Wr in the point of reAection, Wr = −

1 d [arg(Rr )]k=k0 : 2 dk

(245)

The time evolution of the reAected wave packet corresponds to reAection of a free wave at the point r = Wr rather than at r = 0. For a free particle moving with the constant velocity v0 = ˝k0 =M this implies a time gain Wt =

M d d 2Wr [arg(Rr )]k=k0 = −˝ [arg(R)]E=˝2 k02 =(2M) : =− v0 ˝k0 d k dE

(246)

For a positive (negative) value of Wr the reAected wave packet thus experiences a time gain (delay) relative to a free particle (with the same asymptotic velocity v0 ) travelling to r = 0 and back. Note however, that the classical particle moving under the accelerating inAuence of the attractive potential is faster than the free particle; the quantum reAected wave packet may experience a time gain with respect to a free particle but nevertheless be delayed relative to the classical particle moving in the same potential (see Section 5.2). Eq. (242) implies that the near-threshold behaviour of the space shift (245) and of the time shift (246) is k →0

Wr 0∼ aU0 ;

k →0

Wt 0∼

2M aU0 : ˝k0

(247)

The near-threshold behaviour of the time shift due to reAection for a wave packet with a narrow momentum distribution is determined by the mean scattering length aU0 . Near threshold, the quantum reAected wave packet evolves as for a free particle reAected at r = aU0 .

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

425

0 α=3 α=4

-2

α=5 α=6 log Rα

-4

-6

-8

5

10

15

20

25

kα Fig. 14. Natural logarithm of the reAectivity of the homogeneous potential (248) as function of k& for various values of &. From [56].

5.2. Homogeneous potentials The threshold behaviour of the reAection amplitude as summarized by Eqs. (241) and (242) is determined by the two parameters b and aU0 which were obtained analytically for a variety of attractive potential tails in Section 4. They are given for homogeneous potentials, V&att (r) = −

C& ˝2 (& )&−2 = − r& 2Mr &

(248)

in Eqs. (159) and (160) and are tabulated in Table 3. For homogeneous potentials (248) the properties of the Schr4odinger equation do not depend on the energy E = ˝2 k 2 =(2M) and potential strength parameter & independently, but only on the product k& . For energies above the near-threshold region, analytical solutions of the Schr4odinger equation are not available (except for & = 4), and the reAection amplitudes have to be obtained numerically. Figs. 14 and 15 show the real and imaginary parts of ln Rr , namely ln |R| and arg Rr , as functions of k& for various values &. For attractive potential tails such as (248) or the Casimir-Polder-type potentials studied in the next section, the quantality function (36) tends to have its maximum absolute value near the position rE where the absolute value of the potential is equal to the total energy [56], |V (rE )| = E =

˝2 k 2 : 2M

(249)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 4 3

φ(k)

2 1 0 -1 -2 0

1

2

3

4

5

kβα

Fig. 15. Phase  = arg Rr of the quantum reAection amplitude for the homogeneous potential (248). From top to bottom the curves show the results for & = 3, 4, 5, 6 and 7. From [126].

In the corresponding repulsive potential −V (r), the point rE is the classical turning point. For the homogeneous potential (248) we have rE = & (k& )−2=& :

(250)

In the limit of large energies, we may use a semiclassical expression for the reAection amplitudes which was derived by Pokrovskii et al. [129,130]. We use the reciprocity relation (101) to adapt the formula of Refs. [129,130] to the reAection amplitude Rr de>ned via the boundary conditions (102),   rt 2i k →∞ p(r) dr  : (251) Rr (k)∗ ∼ i exp ˝ Here rt is the complex turning point with the smallest (positive) imaginary part. For a homogeneous potential (248) it can be written as rt = (−1)1=& rE = ei=& rE ;

(252)

where rE is de>ned by Eq. (250) and lies close to the maximum of |Q(r)|. Real values of the momentum p(r) only contribute to the phase of the right-hand side of Eq. (251), so the reAectivity |R| is una?ected by a shift of the lower integration point anywhere along the real axis. Integrating along the path r=rE = cos(=&) + i; sin(=&) with ; = 0 → 1 gives the result [56] k →∞

|R| ∼ exp(−B& krE ) = exp[ − B& (k& )1−2=& ] ;   

 −&

   1 R + i; sin B& = 2 sin 1 + cos d; : & & & 0

(253)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

427

Table 6 The coeScients B& , which are given by Eq. (253) and appear before (k& )1−2=& in the exponents describing the high-energy behaviour of the reAectivities of attractive homogeneous potential tails &

3

4

5

6

7

8

&→∞

B&

2.24050

1.69443

1.35149

1.12025

0.95450

0.83146

2=&

0 α=3 α=4

-2

α=5 α=6 log Rα

-4

-6

-8

2

4

6

8

(kβα) 1− 2/α

Fig. 16. Natural logarithm of the reAectivity of the homogeneous potential (248) as function of (k& )1−2=& for various values of &. The straight lines correspond to the behaviour (253) with the values of B& as listed in Table 6. From [56].

In terms of the energy E, the particle mass M and the strength parameter C& of the potential (248), the energy-dependent factor in the exponent is √ 1 1 1 pas rE ; (254) (k& )1−2=& = E 2 − & (C& )1=& 2M = ˝ ˝ where pas = ˝k is the asymptotic (r → ∞) classical momentum. The high-energy behaviour (253) of the reAectivity as function of ˝ is an exponential decrease typically expected for an analytical potential which is continuously di?erentiable to all orders, see the discussion in Section 2.4. Numerical values of the coeScients B& are listed in Table 6. Fig. 16 shows a plot of the logarithm of |R| as function of (k& )1−2=& for various values &, and the obvious convergence of the curves to the straight lines is strong evidence in favour of the high-energy behaviour (253). The phase of the right-hand side of Eq. (251) depends more sensitively on the choice of lower integration limit, which is not speci>ed in Refs. [129,130]. The k-dependence of the integral in the exponent is determined by the complex classical turning point (252), rt = rE [cos(=&) + i sin(=&)].

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

4 2

φ(k)

0 -2 -4 -6 0

1

2 (kβα)

3

4

5

1- 2/α

Fig. 17. Phase  = arg Rr of the reAection amplitude for the homogeneous potential (248) as function of (k& )1−2=& for various values of &. From top to bottom the curves show the results for & = 3, 4, 5, 6 and 7.

If we assume that the real part of the integral becomes proportional to ˝k × R(rt ) = ˝krE cos(=&) for large k, then the high-energy behaviour of the phase of the reAection amplitude is k →∞

arg Rr ∼ c − c0 krE = c − c0 (k& )1−2=&

(255)

with real constants c; c0 . This conjecture is supported by numerical calculations as demonstrated in Fig. 17. Eq. (255) implies that the space shift (245) is given for large energies by   2 k0 →∞ c0 1− rE : (256) Wr ∼ 2 & The space shifts (245) obtained from the numerical solutions of the Schr4odinger equation are plotted in Fig. 18 as functions of k0 & for & = 3, 4, 5, 6 and 7. Except for & = 3 and values of k0 3 less than about 0.15, the space shifts are always positive: according to Eq. (246) this corresponds to time gains relative to the free particle reAected at r =0. For & =3 and energies close to threshold there are signi>cant time delays. Note, however, that the classical particle accelerated under the inAuence of the attractive potential is faster than the free particle [with the same asymptotic velocity v0 =˝k0 =M], and its time gain is   ∞ 2M 1 1 dr = − B(&)rE ; (257) (Wt)cl = 2M ˝k0 p(r) ˝k0 0 where B(&) depends only on &,     1 1 1 1 + 2 1− : B(&) = √ 2 2 & &  Numerical values of B(&) are given in Table 7.

(258)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

429

0.6 0.4

∆r/βα

0.2 0 -0.2 -0.4 -0.6 -0.8

0

1

2

3

4

5

k0βα Fig. 18. Space shift (245) for quantum reAection by the homogeneous potential (248) as function of k0 & for various values of &. From bottom to top the curves show the results for & = 3, 4, 5, 6 and 7. From [126]. Table 7 Numerical values of B(&) as de>ned in Eq. (258) &

3

4

5

6

7

8

&→∞

B(&)

0.862370

0.847213

0.852623

0.862370

0.872491

0.881900

1

The time gain (257) corresponds to the space shift v0 (Wt)cl = B(&)rE ; (259) (Wr)cl = 2 the classical particle which is accelerated in the potential and reAected at r = 0 eventually returns at the same time as a free particle reAected at (Wr)cl . The classical space shifts (259) are generally larger than the space shifts of the quantum reAected wave, as illustrated in Fig. 19 for the example & = 4. At high energies both the classical space shifts (259) and the quantum space shift (256) show the same dependence on k0 & , i.e., proportionality to rE , but the coeScient B(&) in the classical case is larger than the coeScient in the quantum case. At small energies, the classical space shift diverges as rE [Eq. (250)], whereas the quantum space shift remains bounded by a positive distance of the order of the potential strength parameter & , see Figs. 18, 19. Although the quantum reAected wave may experience a time gain relative to the free particle reAected at r = 0, it is always delayed relative to the classical particle which is accelerated in the attractive potential [126]. 5.3. Quantum re?ection of atoms by surfaces The probability for quantum reAection of atoms by a surface is directly accessible to measurement, because the (elastically) reAected atoms return with their initial kinetic energy, whereas those atoms

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

1.25 1

∆r/βα

0.75 0.5 0.25 0 0

1

2

k0βα

3

4

5

Fig. 19. Space shift (245) for quantum reAection by the homogeneous potential (248) with & = 4. The solid line shows the space shift of the quantum reAected wave while the dot-dashed line shows the classical space shift (259). Similar results are obtained for other powers & ¿ 3. From [126].

which are transmitted through the quantal region of the potential tail and hence approach the surface to within a few atomic units are usually inelastically scattered or adsorbed (sticking). Quantum reAection has been observed in the scattering of thermal hydrogen atoms from liquid helium surfaces [118,120], and in the scattering of laser-cooled metastable neon atoms from smooth [41] or structured [131] silicon surfaces. Beyond the region of very small distances of a few atomic units, the interaction between a neutral atom (or molecule) and a conducting or dielectric surface is well described by a local van der Waals potential, corrected for relativistic retardation e?ects as described in the famous paper by Casimir and Polder in 1948 [57]. A compact expression has been given in Refs. [132,133]; in atomic units, the atom-surface potential is   ∞ (&fs )3 ∞ V@ (r) = − &d (i!)!3 exp(−2!rp&fs )h(p; @) dp d! ; (260) 2 0 1 where h(p; @) =

s − @p s−p + (1 − 2p2 ) ; s+p s + @p

with s =




@ − 1 + p2 ;

(261)

&fs ≡ 1=c=0:007297353 : : : is the >ne-structure constant and @ is the dielectric constant of the surface; &d is the frequency-dependent dipole polarizability of the projectile atom in its eigenstate labelled n0 with energy En0 [49],   | n0 | Zj=1 zj | n |2 2(En − En0 ) : (262) &d (i!) = (En − En0 )2 + !2 n

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

431

For a perfectly conducting surface, a simpler formula is obtained by taking @ → ∞ in Eq. (261) and integrating over p in Eq. (260),  ∞ 1 V∞ (r) = − &d (i!)[1 + 2&fs !r + 2(&fs !r)2 ]exp(−2&fs !r) d! 4r 3 0   ∞  x 1 [1 + 2x + 2x2 ]exp(−2x) d x : =− &d i (263) 4&fs r 4 0 &fs r For small r values, we can put r = 0 in the upper line of Eq. (263) and obtain the van der Waals potential between the atom and a conducting surface,  ∞ C3 (∞) 1 vdW V∞ (r) = − ; C (∞) = &d (i!) d! : (264) 3 r3 4 0 For >nite values of the dielectric constant @, the derivation of the small-r behaviour of the potential is a bit more subtle, but the result is quite simple [132,134], V@vdW (r) = −

C3 (@) ; r3

C3 (@) =

@−1 C3 (∞) : @+1

(265)

For large r values, we can assume the argument of &d in the lower line of Eq. (263) to be zero and perform the integral over x. This gives the highly retarded limit of the Casimir-Polder potential between the atom and a conducting surface, ret V∞ (r) = −

C4 (∞) ; r4

C4 (∞) =

3 &d (0) : 8 &fs

(266)

For >nite values of the dielectric constant @, we have [133,134] V@ret (r) = −

C4 (@) ; r4

C4 (@) =

@−1 (@)C4 (∞) ; @+1

(267)

∞ where (@) = 12 ((@ + 1)=(@ − 1)) 0 h(p + 1; @)(p + 1)−4 dp is a well de>ned smooth function which for @ = 1 to unity for @ → ∞. Explicit expressions for increases monotonically from the value 23 30 (@) and a table of values are given in Ref. [133]. The atom-surface potential behaves as −C3 =r 3 for “small” distances [Eqs. (264), (265)] and as −C4 =r 4 for large distances [Eqs. (266), (267)]. The ratio L=

C4 (4 )2 = C3 3

(268)

de>nes a length scale separating the regime of “small” r values, rL, from the regime of large r values, r
L. In Eq. (268) we have introduced the parameters 3 and 4 which express the potential strength in the respective limit in terms of a length, as for the homogeneous potentials (248). The expressions (260) and (263) have been evaluated for the interaction of a hydrogen atom with a conducting surface by Marinescu et al. [135] and for the interaction of metastable helium 21 S and 23 S atoms with a conducting surface (@ = ∞) and with BK-7 glass (@ = 2:295; (@) = 0:761425) and fused silica (@ = 2:123; (@) = 0:760757) surfaces by Yan and Babb [134]. A list of the potential parameters determining the “short”-range and the long-range parts of the respective potentials is given in Table 8.

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

Table 8 Parameters determining the “short”-range behaviour (264), (265) and the long-range behaviour (266), (267) of the atom-surface potentials calculated by Marinescu et al. [135] for hydrogen and by Yan and Babb [134] for metastable helium. The length L is the distance (268) separating the regime of “small” distances from the regime large distances; < is the parameter (273) determining the relative importance of the “small”-distance regime and the large-distance regime for quantum reAection. All quantities are in atomic units Atom

H

He(21 S)

He(23 S)

@

∞

∞

2.295

2.123

∞

2.295

2.123

C3 C4 3 4 L <

0.25 73.61 919 520 294 1.77

2.6712 13091 38980 13820 4901 2.82

1.0498 3918 15320 7561 3732 2.03

0.9605 3582 14017 7230 3729 1.94

1.9009 5163 27740 8680 2716 3.20

0.7471 1545 10902 4748 2068 2.30

0.6836 1413 9975 4540 2067 2.20

The lengths 3 and 4 are natural length scales corresponding to typical distances, where quantum e?ects associated with the “short”- or long-range part of the potential are important. These distances are of the order of hundreds or thousands or even tens of thousands of atomic units. The words “small” or “short” refer to length scales small compared to these very large distances, a few tens of atomic units, say. In this regime, the Casimir-Polder potential (260), (263) is a good description of the atom–surface interaction and (semi-) classical approximations are well justi>ed, but it still lies beyond the regime of really small distances of a few atomic units, where more intricate details of the atom–surface interaction involving the microscopic structure of the atom and of the surface become important. The energies at which quantum reAection becomes important are given by wave numbers of the order of 1=3 and 1=4 , i.e. typically below 10−4 atomic units for metastable helium atoms. This corresponds to velocities of the order of centimetres per second, which are very small indeed, but not beyond the range of modern experiments involving ultra-cold atoms [41,42,131,136,137]. The “high”-energy behaviour of the reAection amplitude discussed in Section 5.2 in connection with Eqs. (251), (253), (255), (256) and Figs. 16, 17 refers to high energies relative to this near-threshold regime; these can still be well within the range of ultra-cold atoms. For the potential (263) between the atom and a conducting surface, we can also make some general statements about the next-to-leading terms at large and small separations. For large separations we can exploit the fact that the dipole polarizability (262) is an even function of the imaginary part of its argument, so V∞ (r) as given in the second line of Eq. (263) is an even function of 1=r and the next term in the large-distance expression (266) must fall o? at least as 1=r 6 ,   C4 1 r →∞ V∞ (r) ∼ − 4 + O 6 : (269) r r For small distances r we can calculate a correction to the expression (264) via a Taylor expansion of the integral in the >rst line of Eq. (263), and prudent use of the Thomas-Reiche-Kuhn sum rule

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

433

[49] yields [138], Z&fs r ; (270) 4 where Z is the total number of the electrons in the atom. The second term on the right-hand side of Eq. (270) is the leading retardation correction to the van der Waals potential at “small” distances, but “small” means small compared to the lengths L listed in Table 8, and this can still be quite large in atomic units. It was >rst derived by Barton for one-electron atoms in 1974 [139]. An intriguing feature of the correction (270) to the van der Waals potential between an atom and a conducting surface is, that it is universal: it depends only on the number Z of electrons in the atom and not on its eigenstate n0 . The shape of the atom-surface potential in between the “short”-range behaviour (264), (265) and the long-range behaviour (266), (267) can be expressed in terms of a shape function vshape (r=L),

r

r C4 C3 = − 4 vshape ; (271) V (r) = − 3 vshape L L L L r →0

r 3 V∞ (r) ∼ − C3 +

x→0

x→∞

whose “short”- and long-range behaviour is given by vshape (x) ∼ 1=x3 and vshape (x) ∼ 1=x4 . Shimizu [41] analyzed his experimental data with the simple shape function v1 (x) = 1=(x3 + x4 ). This gives the potential V1 (r) [see Eq. (183)], whose near-threshold properties were already discussed in Section 4.3. The potential between a hydrogen atom and a conducting surface as calculated by Marinescu et al. [135] is well represented by a rational approximation, vH (x) =

1 + Jx ; x3 + 8x4 + Jx5

8 = 1; J = 0:31608 :

(272)

The coeScients 8 and J are actually determined by the expressions (269) and (270) respectively, so the formula (272) contains no adjusted parameters; it reproduces the tabulated values [135] of the exact hydrogen-surface potential to within 0.6% in the whole range of r values. 1 In numerical calculations it is generally advisable to work with smooth potentials which are continuous in all derivatives. Otherwise, e.g. when using a spline interpolation of tabulated potential values, discontinuities in higher derivatives of the potential can lead to remarkably irregular spurious contributions to the quantum reAection amplitude, see e.g. the discussion in Section 2.4. A detailed study of quantum reAection probabilities for potentials behaving as (264), (265) for “small” distances and as (266), (267) for large distances has been given in Ref. [56]. Which part of the potential dominantly determines the reAection probability depends on a crucial parameter √ 2MC3 3 <= √ = : (273) 4 ˝ C4 For small values of <, the quantum reAection probability resembles that of a homogeneous −1=r 3 potential (248) [with the appropriate value of 3 ] at all energies. For large values of <, the quantum reAection probability resembles that of a homogeneous −1=r 4 potential [with the appropriate value of 4 ] up to quite high energies. At very high energies the behaviour will eventually become that of 1

In Ref. [56] a numerical >t ignoring the boundary conditions imposed by Eqs. (269) and (270) led to 8=0:95; J=0:22, but did not give better results.

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0

log10PR

-1

-2

-3

-4

0.0000

0.0005

0.0010

0.0015

0.0020

k Fig. 20. Probability PR for quantum reAection of metastable 23 S helium atoms from a fused silica surface (@ = 2:123). The solid, dotted and dashed lines show the results obtained with the atom-surface potential of Yan and Babb [134], its highly retarded limit (267) and its nonretarded form (265) respectively. From [137].

a homogeneous −1=r 3 potential; this transition can be expected near the energy where the reAection probabilities predicted by Eq. (253) are the same for the homogeneous −1=r 3 potential and for the homogeneous −1=r 4 potential (with the appropriate values of 3 and 4 ), i.e. for k = (B3 =B4 )6 <=L ≈ 5:345<=L. For this value of k Eq. (253) predicts [56] a reAection probability near exp(−7:8<) for the homogeneous potentials, which is quite small when < ¿ 1. As a realistic example we refer to the quantum reAection of metastable 23 S helium atoms by fused silica surfaces, which are used in hollow optical >bres serving as atom-optical devices [136,137,140]. The atom-surface potential for this case was calculated by Yan and Babb [134], and the resulting probabilities for quantum reAection are shown as function of k as a solid line in Fig. 20. Here k = 0:001 atomic units corresponds to a perpendicular velocity of the incoming helium atoms near 30 cm=s; these and lower perpendicular velocities can be achieved experimentally when projectile atoms with somewhat larger velocities are incident at small, almost grazing, angles [41,137]. The dotted line in Fig. 20 shows the quantum reAection probability for a homogeneous −1=r 4 potential, corresponding to the highly retarded limit (267) of the atom-surface potential, with 4 = 4540 a.u. according to Table 8. The dashed line shows the results for the nonretarded van der Waals potential (265) with the appropriate strength parameter 3 = 9975 a.u. The reAectivity of the full potential corresponds quite closely to that given by the highly retarded limit of the potential up to momenta k ≈ 0:1 a.u., where the reAection probability has fallen to about 0.1%; in contrast, the unretarded van der Waals potential would lead to quantum reAection probabilities an order of magnitude smaller. This shows that the dominant contributions to the quantum reAection are generated in the highly retarded tail of the potential at distances much larger than L ≈ 2000 a.u. In order to understand the reAection of an atom beam from a surface, it is helpful to recall the space and time shifts discussed in Section 5.2. As long as the atom-surface potential is translationally

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

435

1.1

1

b 0.9 β4

v1 v2 vH

0.8

0.7 2

4

6

8

10

ρ Fig. 21. Characteristic parameter b (in units of 4 ) as function of the crucial parameter (273) for an atom-surface potential with di?erent shape functions determining the shape of the potential between the “short”- and long-range limits according to Eq. (271). The dotted line shows the results obtained with Shimizu’s [41] potential, v1 (x) = 1=(x3 + x4 ), the dot-dashed curve shows the results obtained with the smoother interpolating function, v2 (x) = [2=(x3 )] arctan[=(2x)], and the dashed line shows the results obtained with the shape function vH , Eq. (272), corresponding to the potential calculated by Marinescu et al. [135] for a hydrogen atom and a conducting surface. From [56].

invariant parallel to the surface, the parallel momentum component is conserved. For a free particle reAected at the position Wr de>ned in Eq. (245), the magnitude of the perpendicular component of momentum is conserved, but its direction is reversed at reAection. The atom beam thus appears to follow straight-line trajectories which are specularly reAected at a plane lying Wr in front of the surface. For the case above, corresponding closely to the reAection by a homogeneous −1=r 4 potential, the space shift describing the position of the apparent plane of reAection relative to the reAecting surface can be deduced from Fig. 19. The largest space shifts occur for wave numbers between 0:5=4 and 1=4 and amount to about 0:34 , i.e. the apparent plane of reAection is shifted by up to 1400 a.u. in front of the actual surface. For heavier alkali atoms reAected by a conducting surface, quantum reAection is also dominantly due to the −1=r 4 part of the atom-surface potential, and the values of 4 can be ten times larger [56] leading to maximal space shifts of the order of 104 a.u. An example for the inAuence of the shape of the potential can be obtained by studying the e?ect of di?erent shape functions (271) on the characteristic parameter b, which determines the near-threshold reAectivity according to Eq. (241). Fig. 21 shows the dimensionless ratio b=4 as function of the crucial parameter (273). The dotted line shows the results obtained with Shimizu’s potential [41], v1 (x) = 1=(x3 + x4 ), the dot-dashed curve shows the results obtained with the smoother interpolating function, v2 (x) = [2=(x3 )] arctan[=(2x)], and the dashed line shows the results obtained with the shape function vH , Eq. (272), corresponding to the potential calculated by Marinescu et al. for

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

1.5

log (-logR)

1 0.5 0

v1, =1 v1, =10 vH, =1 vH, =10

–0.5 –1

-11

-10.5

-10

-9.5 -9 log (k) (a.u.)

-8.5

-8

-7.5

Fig. 22. Quantum reAectivities |R| as observed by Shimizu [41] for the scattering of metastable neon atoms by a silicon surface (>lled dots). The >gure shows ln(−ln |R|) as function of ln k (natural logarithms) with k measured in atomic units. The straight line in the top-right part of the >gure shows the “high”-energy behaviour expected for a homogeneous −1=r 4 potential according to Eq. (253) with 4 = 11400 a:u. The straight line in the bottom-left part of the >gure shows the near-threshold behaviour (241) for b = 4 = 11400 a:u. The curves were obtained by numerically solving the Schr4odinger equation with potential shapes given by the shape function v1 de>ning Shimizu’s potential (183) and with the shape function vH for the potential of Marinescu et al. [135] for the interaction of a hydrogen atom with a conducting surface, Eq. (272); the value of 3 de>ning the “short”-range van der Waals part of the potential was either 11400 a:u: (< = 1) or 114000 a:u: (< = 10). From [56].

a hydrogen atom and a conducting surface. The values of b lie within 5% of the large-< limit 4 when < ¿ 2 and approach the small-< limit 3 = 4 =< for < → 0. The most pronounced shape dependence is observed around < = 1. Recent measurements of quantum reAection were carried out by Shimizu for metastable neon atoms reAected, e.g., by a silicon surface [41]. The transition from the linear dependence of ln |R| on k near threshold to the proportionality of −ln |R| to k 1−2=& at “high” energies is nicely exposed by plotting ln(−ln |R|) as a function of ln k, see Fig. 22. At the “high”-energy end of the >gure, the data clearly approximate a straight line with gradient near 12 corresponding to & = 4. Fitting a straight line of gradient 12 through the last six to ten data points yields ln(−ln |R|) = 5:2 + 12 ln k (straight line in top-right corner of Fig. 22), and comparing this with ln(−ln |R|) = ln B4 + 12 ln k + 12 ln 4 according to Eq. (253) yields 4 ≈ 11400 a.u. The corresponding near-threshold behaviour (241), ln(−ln |R|) = ln(24 ) + ln k is shown as a straight line in the bottom-left corner of Fig. 22 and >ts the data well within their rather large scatter. Also shown in Fig. 22 are the results obtained by numerically solving the Schr4odinger equation with a potential given by two of the three shapes already introduced in connection with Fig. 21, and with the above value of 4 and two di?erent choices of the crucial parameter (273), namely <=1 implying 3 =4 and <=10 implying 3 =104 = 114000 a.u. The large-< curves clearly >t the data well con>rming that the quantum reAectivity is essentially that of the highly retarded −1=r 4 potential (267) in this case too. In fact, large values of

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

1E-4

2

En e

0

rg y

1 Lo w

ln(-ln|R|2)

Reflection Coefficient, |R|2

1E-3

E igh

H

1

0.01

gy

ner

3

0.1

437

-1 -7

-6

-5

-4 -3 ln(kia)

-2

-1

0

1E-5 1E-6 1E-7 1E-8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 kia

Fig. 23. Quantum reAection probabilities |R|2 observed by Druzhinina and DeKieviet in the scattering of ground state 3 He atoms by a rough quartz surface. The solid line shows the reAection probabilities calculated with Shimizu’s potential (183) with 3 = 650 a:u: and 4 = 350 a:u. The straight line in the bottom left-hand corner of the inset shows the low-energy behaviour (241), and the straight line with gradient 1=3 in the top right-hand corner shows the “high”-energy behaviour (253) expected for the nonretarded −1=r 3 part of the potential. From [42], courtesy of M. DeKieviet.

the crucial parameter (273) are ubiquitous in realistic systems, so quantum reAection data provide a conspicuous and model-independent signature of retardation e?ects in atom-surface potentials [56]. In a more recent experiment, Druzhinina and DeKieviet [42] measured the probability for quantum reAection of (ground state) 3 He atoms by a rough quartz surface. The helium atoms which are transmitted all the way to the surface are reAected di?usely because of the surface roughness and contribute only negligibly to the yield of specularly reAected atoms; the specularly reAected atoms thus represent the quantum reAection yield. The results of Ref. [42] are reproduced in Fig. 23. In the label of the abscissa, ki is the incident wave number perpendicular to the surface and a=5 a.u. is the location of the minimum of a realistic atom-surface potential [141]. The authors analyzed their data using Shimizu’s potential (183); the strength parameter 4 was >xed via the known polarizability of the helium atoms [cf. Eq. (266)] and the dielectric constant of the quartz surface [cf. Eq. (267)] to be 4 = 350 a.u., and the strength parameter 3 was determined by >tting the calculated probabilities to the experimental data. This gave 3 = 650 a.u. corresponding to L = 190 a.u. and a crucial parameter < = 1:9. The straight line in the bottom left-hand corner of the inset in Fig. 23 is close to the near-threshold reAectivity (241) with b ≈ 4 = 350 a.u. as expected for the −1=r 4 part of the potential, and the straight line with slope 1=3 in the top right-hand corner shows the “high”-energy behaviour (253) expected for the −1=r 3 part of the potential with 3 = 650 a.u.

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0 -1

log10PR

-2 -3 -4 -5 -6 -7 -8 0.0

0.1

0.2 ka

0.3

Fig. 24. Quantum reAection probabilities for Shimizu’s atom surface potential (183) with 3 = 650 a:u: and 4 = 350 a:u: (solid line) in comparison with the predictions of the associated homogeneous potentials (248) proportional to −1=r 3 (dashed line) and to −1=r 4 (dotted line).

An important aim of the work in Ref. [42] was to measure quantum reAection probabilities so far above the threshold, that they are signi>cantly inAuenced by the nonretarded van der Waals part of the atom-surface potential. The high-energy data can actually be seen to approach the straight line in the upper right-hand corner of the inset in Fig. 23, but the quantum reAection probabilities are still substantially larger than the predictions for a pure −1=r 3 potential. This is illustrated in Fig. 24 comparing the quantum reAection probabilities predicted for the realistic (Shimizu’s) potential (solid line), which >t the data, with those obtained for the −1=r 3 potential alone (dashed line) and for the −1=r 4 potential alone (dotted line). The transition point k = (B3 =B4 )6 <=L ≈ 5:345<=L, where the predictions for the homogeneous potentials cross, lies at ka ≈ 2:7, which is near the highest energy for which data are included in Fig. 23. At these energies, the k-dependence of the probabilities obtained for the realistic potential is similar to that obtained for the −1=r 3 potential alone, but the absolute values are still an order of magnitude larger. The high-energy data in Fig. 23 are certainly inAuenced by the nonretarded van der Waals part of the potential, but they are also still far from the asymptotic limit where the reAection probabilities are expected to be quantitatively determined only by this nonretarded van der Waals potential. The theory of quantum reAection described in this section is based on the assumption that the quantal region is localized in the tail of the potential and that there is a semiclassical WKB region at smaller r values. These conditions are well ful>lled for suSciently attractive potential tails more singular than 1=r 2 for “small” r values, as typically occur in the interaction of atoms and molecules with each other and with surfaces. In this case the properties of quantum reAection depend only on the quantal region of the potential tail, and the quantum reAection probability approaches unity at threshold as described by Eq. (241). When the potential well is very shallow, and/or when the tail falls o? more slowly than 1=r 2 towards “small” r-values, as is the case with Coulombic tails,

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

439

the quantal region of coordinate space may extend to very small r values, where more complex ingredients of the atom–surface interaction lead to a loss of atoms from the incoming (elastic) channel via inelastic reactions or adsorption (sticking). In this case, the quantum reAection amplitude can be expected to depend sensitively on the nature of the interaction at very small distances, and, due to loss of Aux into inelastic channels, the quantum reAection probability may remain smaller than unity even in the limit of vanishing energy. 5.4. Coupled channels Possible excitations of the projectile atoms or molecules or of the reAecting surface during the process of quantum reAection can be described by generalizing the one-dimensional Schr4odinger equation (1) via a coupled channel ansatz for a vector of channel wave functions [142],   1 (r)  .   K(r) =  (274)  ..  : N (r)

The Schr4odinger equation now has the form  V11 (r)   2 2  ˝ d . − + V K = EK; V ≡   .. 2M dr 2 VN 1 (r)

···

V1N (r)

..

.. .

.

···

   ; 

(275)

VNN (r)

with the diagonal channel potentials Vii (r) and the nondiagonal coupling potentials Vij (r); j = i. We assume that for r → ∞ the diagonal potentials approach constant values Ei and that the coupling potentials as well as the di?erences Ei − Vii (r) vanish more rapidly than 1=r 2 . The coupled equations (275) then become decoupled free-particle equations for large r, and the appropriate generalization of Eq. (102) to describe an incoming wave in the channel labelled 1 and reAected waves in all open channels, E − Ei ¿ 0, is, 1 (r)

r →∞

i (r)

r →∞

∼ √

1 [exp(−ik1 r) + R11 exp(ik1 r)] ; ˝k1

1 ∼ √ R1i exp(iki r); ˝ki

i = 1 ;

(276)


where ki is the asymptotic (r → ∞) wave number in the ith channel, ˝ki = 2M(E − Ei ). One approach towards solving the coupled-channel equations (275) is to introduce a local unitary transformation L(r) = U(r)K(r) such that the appropriately transformed potential is diagonal,   W11 (r) · · · 0   . .. ..  : (277) W = UVU−1 ≡  . .   .. 0

···

WNN (r)

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The Schr4odinger equation (275) then becomes an equation for the “adiabatic” channel wave functions 1 (r); : : : ; N (r) making up the vector L,   ˝2 d 2 ˝2 U[(U−1 )

L + 2(U−1 )
L
] = EL : − + W L − (278) 2M dr 2 2M For a meaningful discussion of the concept of quantum reAection for coupled channels, it is helpful that Eq. (278) decouple and become amenable to semiclassical approximations for small r values, r → 0 (or r → −∞). The Eq. (278) decouple if the square bracket containing the >rst and second derivatives of U vanishes. This is, e.g. the case if the coupling potentials Vij ; j = i vanish for small r, or if the diagonal potentials become dominant with respect to the coupling potentials, r →0

e.g. if Vii ˙ − 1=r & whereas the coupling potentials remain bounded. In these cases, U becomes the unit matrix at small distances. If both the diagonal potentials and the coupling terms behave as an inverse power of r for small r, then the asymptotic (r → 0) decoupling of the channels depends on the relation of the powers involved. If, e.g. in a two-channel example, V11 (r) and V22 (r)  are proportional to −1=r & for r → 0, and V12 (r) and V21 (r) proportional to −1=r & , then we obtain decoupled channels for small r, if and only if |&
− &| = 1 [143]. If the diagonal potentials and the coupling potentials have the same spatial dependence for small r, dfij =0 ; (279) Vij (r) = f(r) × fij ; dr then there is a decoupling of channels, but U = 1 so the decoupled channels will be superpositions of the diabatic channels i which are uncoupled for r → ∞. Decoupling of channels occurs in the limit r → 0 for many-body systems described in hyperspherical coordinates. In these coordinates, the hyperradius r stands for the root-mean-square average of the radial coordinates of all particles involved, and the remaining coordinates, the hyperangles, encompass not only all angular degrees of freedom, but also “mock angles” de>ned by the ratios of the individual radial coordinates [144]. Note however, that in applications to many-electron atoms, the diagonal potentials are proportional to 1=r at small values of the hyperradius r, and there need not be a semiclassical WKB region in the regime where the channels decouple (see the discussion in the last paragraph of Section 5.3). For the waves transmitted to small r values, r → 0 or r → −∞, the appropriate generalization of the one-channel boundary conditions (100) is,    r 1 i

j (r) ∼ T1j
(280) qj (r ) dr ; exp − ˝ rl qj (r)
where ˝qj (r) = 2M[E − Wjj (r)] is the local classical momentum in the adiabatic channel j. The form (280) implies, that the adiabatic channel j is open for transmission, i.e., that E − Wjj (r) ¿ 0 for small r. Some general statements can be made about the threshold behaviour of the quantum reAection amplitudes when potentials and coupling terms approach their asymptotic limits faster than 1=r 2 . If the incoming channel is energetically lowest and nondegenerate (E1 ¡ Ei for all i = 1) then the near-threshold behaviour of the elastic reAectivity |R11 | follows the pattern (241), k →0

|R11 | 1∼ 1 − 2bk1 = exp(−2bk1 ) + O((k1 )2 ) ;

(281)

with a characteristic length parameter b, depending only on the tails of the potentials and coupling terms Vij in the quantal region of coordinate space. SuSciently close to threshold only the elastic

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441

reAection channel and a certain number of transmission channels are open, and particle number conservation requires the sum of the transmission probabilities to grow proportional to 1 − |R11 |2 ∼ 4bk1 + O((k1 )2 ); this actually holds for each transmission probability individually, k1 →0 k1 →0
(PT )ij ˙ k1 ; T1j ˙ k1 : (282) This is a straightforward generalization of the second equation (228) to the coupled-channel situation. Analytical solutions for two coupled step potentials with a step-like coupling term are given in Ref. [142], and smoother Woods-Saxon steps as well as inverse-power potentials are discussed in Ref. [143]. Exponential potentials and coupling terms have been studied in considerable detail for isolated special cases by Osherov and Nakamura [145,146], while more general cases were treated in Refs. [147,148] using semiclassical methods. When two diagonal diabatic potentials, V11 (r) and V22 (r) cross at a point r0 , then coupling of the channels 1 and 2 leads to an avoided crossing of the corresponding adiabatic potentials W11 and W22 . In an adiabatic process, the incoming and reAected or transmitted waves remain on the potential energy curve Wii associated with the respective adiabatic channel, but the avoided crossing can be overcome by a nonadiabatic transition. The probabilities for such nonadiabatic transitions have been a topic of great interest for more than seventy years [149–152]. With the assumptions that the diabatic potential curves are linear at the crossing and that the coupling potential is constant one obtains the semiclassical Landau-Zener formula,   V12 (r0 )2 ; (283) (P1→1 )LZ = 1 − exp −2 ˝v0 WF where v0 is the velocity at the crossing point, Mv02 =2 = E − V11 (r0 ) and WF is the di?erence of the slopes of the two crossing curves, WF = |V11
(r0 ) − V22
(r0 )|. Eq. (283) actually describes the probability that the incoming wave in the channel labelled 1 (wave function 1 (r) for large r) remains on the adiabatic potential curve and is transmitted to the transmission channel 1
, which is the channel labelled 2 in the diabatic basis. Many improvements of the simple Landau-Zener formula (283) have been proposed over the years [152], but we shall focus on one aspect, namely the quenching of the curve crossing probability due to quantum reAection [153]. Consider a system of two Woods-Saxon step potentials, Ui + (i − 1)E0 ; i = 1; 2 ; Vii (r) = − (284) 1 + exp(ai r) with a Gaussian coupling potential, V12 (r) = V21 (r) = U12 exp[ − (a12 )2 (r − r0 )2 ] ;

(285)

as illustrated in Fig. 25. The performance of the simple Landau-Zener formula is illustrated in Fig. 26 for an example of very small coupling. The transition probability P1→1 = |T12 |2 , obtained by numerically solving the two-channel Schr4odinger equation (275) with the boundary conditions (276), (280) is plotted as function of k1 ≡ k. The solid line shows the exact result, which goes to zero at threshold according to Eq. (282). This vanishing transmission probability is not accounted for in the conventional Landau-Zener formula (283)—illustrated as dashed line in Fig. 26—nor in the numerous improvements introduced over the years [152]. It is due to the fact, that the incoming wave only reaches

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 5

Channel 2 Coupling Potential

Potential

0

Channel 1

Channel 2’

-5 -10 -15 -20

Channel 1’ -20

-10

0 r

10

20

Fig. 25. Woods-Saxon potentials (284) with the coupling potential (285) (dashed line). The parameters are U1 = 5:5, U2 = 26, U12 = 0:5, E0 = 3 and a1 = a2 = a12 = 1. The dotted line shows the quantality function (36) for the elastic channel just above threshold (k = 0:2). From [153].

QM LZ

0.00012

P1→1′

0.0001 8e-05 6e-05 4e-05 2e-05 0 0

0.2

0.4

0.6

0.8

1

k Fig. 26. Transition probability P1→1 = |T12 |2 for the coupled Woods-Saxon potentials (284), (285). The parameters are U1 = 5:5, U2 = 26, U12 = 0:01, E0 = 3, a1 = a2 = 1, a12 = 0:05. The solid line shows the exact result and the dashed line is the prediction of the simple Landau-Zener formula (283). From [153].

the curve-crossing region with a small probability, because quantum reAection in the quantal region of the incoming-channel potential becomes dominant towards threshold. This region is indicated by the dotted line in Fig. 25, which shows the quantality function (36) for the potential V11 at a near-threshold momentum, k = 0:2. For k larger than about 0.4, the e?ect of quantum reAection is negligible, and the conventional Landau-Zener formula reproduces the exact result to within a rather constant error of a few per cent. A straightforward improvement of the Landau-Zener formula (283) is to account for the e?ect of quantum reAection by multiplying the probability (283) by the probability for transmission through

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

443

0.00012 0.0001

P1→1′

8e-05 6e-05

QM LZM

4e-05 2e-05 0

0

0.2

0.4

0.6

0.8

1

k Fig. 27. Transition probability P1→1 = |T12 |2 for the coupled Woods-Saxon potentials (284), (285). The parameters are U1 = 5:5, U2 = 26, U12 = 0:01, E0 = 3, a1 = a2 = 1, a12 = 0:05. The solid line shows the exact result and the dashed line is the prediction of the modi>ed Landau-Zener formula (286). From [153].

the quantal region of the potential tail, (P1→1 )LZM = (P1→1 )LZ (1 − |R|2 ) :

(286)

In the very-weak-coupling example in Fig. 26, the quantum reAectivity in the elastic channel is insensitive to the coupling, so we can take |R| to be given by the reAectivity (232) of an uncoupled Woods-Saxon potential. As shown in Fig. 27, the modi>ed Landau-Zener formula (286) does indeed account correctly for the e?ects of quantum reAection in this case and leads to much better agreement with the exact result. 6. Conclusion Although the semiclassical WKB approximation is generally expected to be most useful near the semiclassical limit, where quantum mechanical e?ects are small, semiclassical WKB wave functions can be used to advantage far from the semiclassical limit, even near the anticlassical, the extreme quantum limit of the Schr4odinger equation. This is because the conditions for the accuracy of the WKB wave functions are inherently local. Under conditions which are far from the semiclassical limit for the Schr4odinger equation as a whole, there may still be large regions of coordinate space where WKB wave functions are highly accurate approximations of the exact quantum mechanical wave functions; they may even be asymptotically exact. For example, for homogeneous potentials proportional to 1=r & with & ¿ 2, the threshold E = 0 represents the anticlassical limit of the Schr4odinger equation, but WKB wave functions become exact for r → 0 at all energies, in particular at threshold and in the near-threshold region. The (local) condition for the accuracy of WKB wave functions such as (25) is conveniently expressed via the quantality function (36): |Q(r)|1. The regions of coordinate space, where |Q(r)| is not negligibly small are quantal regions where quantum e?ects such as classically forbidden tunnelling or reAection can be generated. In many physically important situations, exact or highly

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accurate quantum mechanical wave functions are available in the quantal regions and globally accurate wave functions can be constructed by matching these to WKB wave functions in the semiclassical regions. In this way, several highly accurate, sometimes asymptotically exact results have been derived via WKB waves far from the semiclassical limit of the Schr4odinger equation. Numerous examples have been given in the preceding sections. WKB wave functions are singular at a classical turning point and the connection formulas relating the oscillating WKB waves on the classically allowed side to the exponential waves on the forbidden side are usually formulated under restrictive assumptions for the potential (linearity), which are not related to whether or not WKB wave functions may be accurate away from the turning point. Allowing more general connection formulas (44), (45) greatly widens the range of applicability of WKB wave functions. Correct choice of the phase of the WKB wave (56) on the allowed side of a classical turning point via an appropriate de>nition of the reAection phase is a vital ingredient for the construction of accurate WKB wave functions in the classically allowed region. This leads, e.g. to a formula for the scattering phase shifts for repulsive inverse-power potentials which is highly accurate at all energies, and to an exact expression (74) for the scattering lengths, which determine the behaviour of the phase shifts in the anticlassical limit, see Section 3.3. For bound state problems, a generalization (83) of the conventional WKB quantization rule to allow an appropriate de>nition of the reAection phases at the classical turning points leads to greatly improved accuracy without complicating the procedure, see e.g. Figs. 3, 4 and 7, in Section 3.4. The generalized connection formulas (44), (45) also lead to more precise expressions for tunnelling probabilities, in particular near the base of a barrier, where conventional WKB expressions fail when the potential tail falls o? faster than 1=r 2 , see Section 3.5. For deep potential wells, where WKB wave functions are accurate in some region of small or moderate r values, the properties of bound states for E ¡ 0 and continuum states for E ¿ 0 are largely determined by the tail of the potential beyond this WKB region. For potentials falling o? faster than 1=r 2 , the threshold E = 0 represents the anticlassical limit of the Schr4odinger equation and the near-threshold properties of the wave functions are determined by three independent tail parameters, which can be derived by matching zero-energy solutions of the Schr4odinger equation to WKB waves in the WKB region. These three parameters are the characteristic length b, Eq. (150), the mean scattering length aU0 , Eq. (151), and the zero-energy reAection phase 0 , which is the phase loss of the WKB wave due to reAection at the outer classical turning point at threshold, see Eq. (136) in Section 4.1. Immediately below threshold, the quantization rule acquires a universal form (174) which becomes exact for E → 0 and contains the tail parameter b as well as the threshold quantum number nth , which depends on 0 and also on the threshold value of the action integral over the whole of the classically allowed region, see Eq. (148). The characteristic length b determines the leading singular contribution to the near-threshold level density according to Eq. (175), and also the near-threshold behaviour of the quantum reAectivity of the potential tail according to Eq. (241) in Section 5.1. The mean scattering length aU0 determines the near-threshold behaviour of the phase of the amplitude for quantum reAection according to Eq. (242), and hence also the time and space shifts involved in the quantum reAection process. With the correct tail parameters, Eqs. (174), (175), (241) and (242) are asymptotically exact relations for the near-threshold behaviour of the bound and continuum states. When the zero-energy solutions of the Schr4odinger equation for the potential tail beyond the WKB region are known analytically, analytical expressions for the tail parameters b, aU0 and 0 can

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be derived. A summary for various potential tails is given in Table 4 in Section 4.3. If analytical solutions of the Schr4odinger equation are not available, the tail parameters can be obtained from numerical zero-energy solutions of the Schr4odinger equation. Knowing the asymptotic (E → 0) behaviour of the bound and continuum states near threshold is of considerable practical value, because the direct numerical integration of the Schr4odinger equation for small >nite energies is an increasingly nontrivial exercise as the energy approaches zero. For potential tails falling o? more slowly than 1=r 2 , the threshold E = 0 represents the semiclassical limit of the Schr4odinger equation, and semiclassical methods are expected to work well in the near-threshold region, e.g. in the derivation of the near-threshold quantization rule (164) for the energies of the bound states in the limit n → ∞. Potential tails vanishing as 1=r 2 represent the boundary between long-ranged potentials, which support in>nitely many bound states, and shorter-ranged potentials which can support at most a >nite number of bound states. The behaviour of potentials with inverse-square tails (191) depends crucially on the strength parameter (, as discussed in detail in Section 4.4. For ( ¡ − 14 , i.e. for potential tails more attractive than the s-wave centrifugal potential in two dimensions (see Eq. (59) in Section 3.3), the potential supports an in>nite dipole series of bound states (193), in which the energy depends exponentially on the quantum number near threshold. Potentials with weak inverse-square tails, − 41 6 ( ¡ 34 , support at most a >nite number of bound states, but the leading energy dependence in the near-threshold quantization rule is still of order less than O(E), so the near-threshold level density is still singular at E =0. The properties of short-ranged potential tails falling o? faster than 1=r 2 appear as a special case of weak inverse-square tails with ( = 0. A summary of quantization rules is given in Table 5 in Section 4.4. Probabilities for transmission through the quantal region of an inverse-square tail with ( ¿ − 14
1

are proportional to E (+ 4 , which is a generalization of Wigner’s threshold law (218). For weak inverse-square tails, − 14 6 ( ¡ 34 , the transmission probability above threshold is related to the leading energy dependence in the near-threshold quantization rule below threshold, Eq. (227). For all potentials which are asymptotically (r → ∞) less repulsive than the p-wave centrifugal potential in two dimensions ((= 34 ), the leading near-threshold energy dependence in the quantization rule is of order less than O(E) and can be derived from the tail of the potential. For potentials with more repulsive tails, i.e. for inverse-square tails with ( ¿ 34 or for repulsive tails falling o? more slowly than 1=r 2 , the leading energy dependent terms in the near-threshold quantization rule are of order O(E) and include e?ects of the potential for smaller r values; they cannot be derived from the properties of the potential tail alone. These results have been derived using WKB waves to approximate the quantum mechanical wave functions only in regions where such an approximation is highly accurate or asymptotically exact. The results do not depend on the conditions of the semiclassical limit being ful>lled for the Schr4odinger equation as a whole. Indeed, the comprehensive results for the near-threshold region refer to the immediate vicinity of the anticlassical or extreme quantum limit in the case of potential tails falling o? faster than 1=r 2 . Many of the results summarized in this article are of direct practical importance in various >elds, e.g. in atomic and molecular physics, where the intense current interest in ultra-cold atoms and molecules has drawn attention to quantum e?ects speci>c to small velocities and low energies. The near-threshold phenomena studied in Sections 4 and 5 are explicit examples of such quantum e?ects. The quantum reAection of atoms moving as slowly as a few centimetres per second towards

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a surface occurs several hundreds or thousands of atomic units from the surface and can be observed in present-day experiments. Understanding this and similar phenomena is important for technical developments such as the construction of atom-optical devices. The theoretical considerations and practical applications discussed in this review refer mostly to the Schr4odinger equation for one degree of freedom. Matching the exact or highly accurate solutions of the Schr4odinger equation in the “quantal region” of coordinate space to WKB waves which are accurate in the “WKB region” can occur at any point where these regions overlap. An extension to systems with more than one degree of freedom does not seem straightforward, because matching between quantal and WKB regions would have to occur on a subspace of dimension one or more, and it is not clear whether this is easy to do in general. A more promising >eld for generalising the techniques reviewed in this article is that based on coupled ordinary Schr4odinger equations, such as the coupled channel equations used in the description of scattering and reactions in nuclear, atomic and molecular physics. Multicomponent WKB waves have been used in the treatment of coupled wave equations with the individual equations referring to spin components of Pauli or Dirac particles or di?erent Born-Oppenheimer energy surfaces in a molecular system [154–158]. Generalizing such theories to allow for signi>cant deviations from the semiclassical limit may greatly enhance their range of applicability. A simple example is the inAuence of quantum reAection on Landau-Zener curve crossing probabilities as described in Section 5.4. Acknowledgements The authors express their gratitude to the current and former collaboraters who have contributed to the results presented in this review, namely Kenneth G.H. Baldwin, Robin CˆotPe, Christopher Eltschka, Stephen T. Gibson, Xavier W. Halliwell, Georg Jacoby, Alexander Jurisch, Carlo G. Meister and Michael J. Moritz. Harald Freidrich also wishes to thank the members of the Department of Theoretical Physics and of the Atomic and Molecular Physics Laboratories, in particular Brian Robson, Steve Buckman, Bob McEachran and Erich Weigold, for hospitality and enlightening discussions during his stay at the Australian National University during spring and summer 2002/2003. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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