Classical paths and quantum mechanics

ANNALS

OF PHYSICS

164,

411462

Classical

(1985)

Paths and Quantum

Mechanics

ROBERT D. CARLITZ Department

qf Physics and Astronomy, University Pittsburgh. Pennsylvania 15260

of Pittsburgh,

AND A. NICOLE*

DENS

Department of Physics, Santa Barbara, Received

University Cakfornia

March

of California, 93106

21, 1984

In the semiclassical limit, the path integral description of quantum mechanics is dominated by classical paths. A classical path method is developed to extract energy levels and wavefunctions for a one-dimensional quantum system and it is shown that this method reproduces the results of the WKB approach. The classical path method generalizes the instanton method and provides new insights into instanton interactions. It provides a convenient and intuitive approach to many problems in quantum mechanics and field theory. 0 1985 Academic

Press. inc.

I. INTRODUCTION The path integral approach to quantum mechanics describes the evolution of a system in terms of a sum over all possible paths of motion. Each path is weighted by a factor exp(iS/h), where S is the classical action evaluated along the given path. In the semiclassical limit (A-+ 0), the dominant paths are those close to the stationary points of S, i.e., the classical paths of motion. In this limit, then, the path integral approach is simple, intuitively clear and convenient for practical calculations [l-3]. Many physical situations call for this limit, particularly systems involving the decay of a metastable state or the mixing of degenerate states through quantum mechanical tunneling. Such tunneling phenomena are ubiquitous in quantum physics. The realization that such phenomena are important for quantum field theories has led to a considerable amount of work in this area in recent years [4-71. In the field theory case the relevant limit is that of small couplings (g + 0),

*Present

address:

Department

of Physics,

University

of Southhampton,

U.K.

411 0003-4916/S

$7.50

Copynghl CL’ 1985 by Academic Press. Inc All nghts of reproduction in any form reserved

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CARLITZ

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but the mathematical situation is virtually the same as in the semiclassical limit of quantum mechanics. Here the path integral formulation is a natural starting point for the whole theory, so it is clear that classical solutions of the theory play a rather important role. The instanton method has been developed to elucidate this role. In the language of quantum mechanics, instantons are paths which are confined to the classically forbidden region and which evolve purely imaginary time. Such paths are obviously relevant for the description of quantum tunneling, which involves the passage of a system through the classically forbidden region. Less obvious is the relation of these paths to the full set of classical paths (which evolve both real and imaginary time) and the relation of the instanton method to other approximation schemes (notably the WKB method [8], to which instanton results are often compared). In this paper we extract energy levels and wavefunctions from the semiclassical limit of the path integral for quantum mechanics. Classical paths will enter our discussion as saddle points of the full functional integral. There are typically a large number of such saddle points, only a small fraction of which are physically relevant. We will show how boundary conditions on the functional integral select the appropriate saddle points, much as boundary conditions on an ordinary complex integral [9] select a particular set of saddle points for the steepest descent approximation to the integral. The classical path description which results from our analysis provides a generalization of the instanton method, and our results are shown to include those of the WKB method. The reason that our results reproduce those of the WKB method is simply that we are exploring the same quantities about the same mathematical limit. As a practical matter our method may be easier to apply in many cases, since classical paths have a direct and intuitively clear interpretation. As a technical matter the equivalence of the classical path method to the WKB approximation helps explain a number of subtleties which arise in the path integral approach. Both techniques are in principle asymptotic expansions and, as such, exhibit a number of features common to asymptotic expansions. In particular, the analytic structure of an asymptotic expansion is typically not [9] the same as the analytic structure of the function being approximated. In the WKB method this fact is apparent in factors of the square root of the classical momentum which enter the WKB wavefunctions. In the classical path method this incorrect analytic structure makes it dangerous to analytically continue results from one region of parameter space to another. What happens in any asymptotic approximation is that as parameters are varied the approximation undergoes discontinuous changes. These “Stokes discontinuities” are present to compensate for the incorrect analytic structure of the asymptotic expression. It is this phenomenon which underlies the WKB connection formulae [8], which relate the form of the WKB wavefunction on either side of a classical turning point (where the classical momentum vanishes). The WKB approximation is itself invalid in the immediate neighborhood of a turning point, and one should distinguish the case in which the turning points are so well separated that between any two turning points there is a region in which the

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approximation is valid from the case where there are two (or more) nearby turning points. The form of the WKB connection formulae is different in the two cases, and we find a parallel difference in the classical path method as applied to these two cases. In the first case, the steepest descent method applies to all integrals in the problem. We thus establish an exact equivalence between the steepest descent evaluation of the quantum mechanical path integral and the WKB method with the connection formulae for isolated (or “linear”) turning points (where the classical momentum has a linear zero). If parameters of the problem are altered so that two turning points approach each other, the classical momentum will develop a quadratic zero. Correspondingly, the linear connection formulae of the WKB method must now be replaced by more complicated quadratic connection formulae. In terms of the path integral, what happens is that certain fluctuations about the dominant classical path are not small in this limit; parts of the functional integral are no longer Gaussian; and the steepest descent approximation cannot be used for these parts of the calculation. This situation is, however, still not too difficult to handle. If we choose the endpoints of the path appropriately, we can isolate the non-Gaussian pieces of the problem in a single ordinary (one-dimensional) integral. This is possible because the important non-Gaussian fluctuations correspond to classical trajectories with different energies and not to non-classical trajectories. Evaluating the ordinary integral, we are able to establish the equivalence of the classical path method with the WKB method even when quadratic turning points are present. Non-Gaussian fluctuations also arise in the instanton method [46], and a comparison with our treatment serves to highlight some features of our approach. Instanton solutions involve certain boundary conditions needed to restrict their evolution to purely imaginary time. These boundary conditions, it turns out, conflict with the boundary conditions which we would impose to restrict non-Gaussian fluctuations to a single one-dimensional integral. Thus in the instanton method one must introduce collective coordinates in the functional integral to isolate the nonGaussian modes of functional variation. No collective coordinates are needed in our approach, although the physical nature of the non-Gaussian components is just the same as in the instanton approach. If one is discussing the decay by tunneling of the lowest lying of a set of metastable states or the splitting by tunneling of a set of degenerate ground states, the technical differences between our approach and the instanton approach are not too important. For such applications the collective coordinates (which specify the instantons’ positions in imaginary time) are widely separated, and one can safely neglect the effect of interactions among the various instantons. These interactions cannot be ignored if one wishes to examine more highly excited states of the system. These states are of physical relevance for many quantum systems and are particularly important for the study of thermally assisted tunneling processes [lo]. Our approach incorporates instanton interactions automatically, and the treatment of arbitrary excited states is even simpler than the treatment of the lowest-lying levels.

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In field theories a complete classical path analysis is normally impossible, but there often exist classical solutions which correspond to interacting instantons. We believe that the knowledge of how to treat interacting instantons in quantum mechanics can lead to their proper treatment in field theory. This would enable one to extend the application of instantons in field theory beyond the dilute gas approximation which has been hitherto employed. We begin our analysis of classical paths in quantum mechanics in Section II. There we construct the path integral for a particle moving in a one-dimensional potential and evaluate this integral by the method of steepest descents. This establishes some general results and allows us to pose specific questions regarding the validity of the steepest descent approximation. To answer these questions we study in Sections III, IV and V a series of exactly solvable potentials. In Section III we analyse a linear potential, construct the corresponding wavefunction, examine its asymptotic behavior and make comparisons with the WKB method. The wavefunction here is an Airy function, and our analysis reproduces the usual features of the asymptotic expansion of that function. Stokes discontinuities, in particular, arise in a simple and graphic manner from jumps in the steepest descent contour. In Section IV we turn to a quadratic potential well (a simple harmonic oscillator). Again we find Stokes discontinuities as we move endpoints of the steepest descent paths past the classical turning points. A new feature is the possibility of two turning points approaching each other. In this limit the classical path method reproduces the result of the WKB approximation with the quadratic connection formulae appropriate to this regime. Section V describes a quadratic barrier. Here we show how complex turning points are naturally incorporated in our formalism. Such contributions are required to build the appropriate asymptotic approximation, but their correct treatment is not completely obvious in the usual WKB framework [ 8, 111. Sections II-V provide a technical framework adequate to describe any system with isolated linear and quadratic turning points. We apply our method in Section VI to the double-well potential and generalize results obtained previously by instanton methods. Our results match those of the WKB method, and our development makes it clear that this should be a general feature of our approach. Section VII is devoted to a comparison of the classical path method with the instanton method. We show the type of new information that our method supplies and illustrate it with a discussion of our results for the double-well potential. The final section is devoted to a summary of the strong points of the classical path method and a comparison of our approach with others in the literature. II. THE PATH INTEGRAL Consider a one-dimensional quantum mechanical system described by a potential V(x). We will study in this paper the semiclassical limit of the Green function G(x,, xi; E), which describes the propagation through this potential from point Xi

CLASSICAL

to point or of a particle Laplace transform [Z]

PATHS

AND

QUANTUM

with energy E. The Green function

G(x,, x,; E)=$=J=

0

415

MECHANICS

dTeiET1hK(.x,, x,; T)

G is defined as the

(IT.1 )

of the Feynman [ 1] kernel K. The integration path in Eq. (11.1) is taken along the positive real axis, and G is defined for real positive E as the E + 0 limit of E + k. If the spectrum is discrete, then G can be written as a sum on eigenfunctions [2] :x

G(x,, xi; E) = 1 ‘k,,,‘E’-

‘)

(11.2)

k

k

of the Hamiltonian H = (m/2) 1’ + V(x).

(11.3)

Thus knowledge of G is sufficient to determine the eigenfunctions $k and energy eigenvalues E, of the system. Even if the spectrum of H is not discrete, one can extract wavefunctions from G via the relation H(x) G(.K,x, ; E) = EG(.u, x,; E)

(11.4)

valid for x # .Y,. The Feynman kernel can be written as an integral K(s,, x,; T) = i 1[x(r)]

erSCr”‘lih

(11.5)

over paths .u(t) such that .K(0) = x, and x(T) = xf.

(11.6)

Each path in the integral (11.5) has an associated phase of i/h times the action S[x(t)]

= jordt [$xt’(t)-

V(x(t))].

(11.7)

In the semiclassicallimit (h + 0) neighboring paths in the integral (11.5) will tend to yield canceling contributions on account of the rapid variation of the phase of the factor exp(iS/h). An exception to this rule occurs at the stationary points of S, that is, for classical paths x,,(t) which pass from x, to -ur in time T. There will in

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general be a number of such paths, and one must sum over them to construct an approximation for K. These paths contribute to K an amount [2,3] (11.8) where (11.9) and the phase of [a*s,,/axiax,] ‘I2 depends on the topology of the path. The factor tv2 [ a2s,,/ax, ax,] I’* is associated with the functional determinant describing fluctuations about the classical path. This determinant can be written [12] as /L,

’ &(O) u T) [ 1 aZscI

axi ax,

=

liaE,,laT

)

(11.10)

where EC, is the energy along the path x,,(t). Fluctuations about the classical path make small contributions to K provided that the determinant is large. Note that the contribution of any given path x,,(t) to Eq. (11.8) is a function only of the endpoints and topology of the path. Deformations of the path to complex values [ 133 of x,,(t) are permitted as long as no singularities of S,, or zeros of the determinant (11.10) are crossed in the process. Equation (11.10) indicates that fluctuations about x,i(t) are, in general, small as h-+ 0. An exception occurs when xi or xr are in the vinicity of classical turning points (where i-,, vanishes). In that case the Gaussian estimate (11.8) is unreliable. We can, however, choose Xi and xr to avoid these points so that Eq. (11.8) will be reliable. One notes, incidentically, the presence in Eq. (11.8) [via Eq. (II.lO)] of factors proportional to pa l/*, where pc, is the momentum along x,,. These factors are, of course, pieces of the WKB wavefunction, which we begin to see emerging from our treatment. More of the WKB wavefunctions emerge when we examine the Laplace transform, Eq. (ILl), of the contributions (11.8) to K(x,xi; T). The presence of factors exp [i(S,, + ET)/h] invites a steepest descent approximation [2] for the integral (11.1) in the limit fi+ 0. Saddle points of this integral are given by the HamiltonJacobi equation

The saddle points are thus those T, for which a classical trajectory of energy EC, = E can pass from xi to xr. If the potential admits any sort of periodic motion, there will be a infinite number of saddles T,. One may construct a steepest descent approximation about any saddle point of Eq. (II. 1) by expanding the phase (S,, + ET)/h in a Taylor series about T,. Changing the integration variable to T/f?‘*, and integrating the Taylor series term by

CLASSICAL

PATHS

term, one obtains an asymptotic series will dominate’ unless

AND

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MECHANICS

417

series in powers of G”*. The leading term of this

(11.12) is of order h. Let us ignore this case for the moment and evaluate Eq. (II. 1) in the form G(,x,, xi; E) = ;C C [e’“i,,(O) rs w

ic,( T,)] -‘~‘eiwcl’rs’,

(11.13)

where (11.14) The sign of the square root in Eq. (II.1 3) will be fixed in our more detailed analyses given in Sections III-V. The precise set of saddle points to be included in Eq. (11.13) is determined by deforming the original contour (11.1) to pass by a series of steepest descent paths from T= 0 to Re T= co, Im T= 0. Note, in particular, that Eq. (II. 13) does not involve a sum over all possible saddle points. Note also that it is in principle a straightforward matter to obtain the full asymptotic series which generalizes Eq. (II. 13). Since E,, = + mi-f, + V(x)

is a constant of motion,

(11.15)

we can write WC, [with the aid of Eq. (11.7)] as

(11.16) Thus we recognize WC, as the usual WKB phase. This identifies more of the WKB wavefunction in Eq. (11.13). The expression (11.16) is not immediately useful, however, since the path x,,(t) has yet to be precisely specified. Such a specification will emerge as we discuss which saddle points are relevant for examples treated in the next few sections. These examples will also show the conditions under which Eq. (11.13) is valid. Briefly stated, if the classical turning points are well separated, then the expression (11.12) is non-vanishing, and it is legitimate to evaluate Eq. (II.1 ) by a steepest descent approximation. As two turning points approach each other, i3E,,/aT 1 A similar

discussion

applies

to the functional

integral

leading

to Eq. (11.8) (see Ref. [?I).

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CARLITZ AND NICOLE

approaches zero and Eq. (11.1) must be evaluated exactly. This, it turns out, is not too difficult and we can extend our formalism to cover this case as well. Note that as long as x, and xr lie far from the classical turning points, a Gaussian approximation to K [Eq. (11.8)] remains valid. Non-Gaussian fluctuations enter only in the one-dimensional Laplace transform integral, Eq. (II. 1). Indeed, in some of our examples there is only one classical path, and Eq. (11.8) is exact. Note that a clear separation of Gaussian and non-Gaussian pieces disappears if one evaluates [ 143 a trace of the Green function over initial and final positions.

III.

LINEAR POTENTIAL

In this section we apply the formalism example, a linear potential

of Section II to the simplest possible

V(x) = --Ax.

(III. 1)

There is only one classical path connecting xi to xr in time T: AT t--t++t2.

The corresponding

1

action is (111.3)

in terms of which K(xr, Xi; T) may be written [2]

K(x,, xi;T)= . [& 1e&llfi l/2

(111.4)

Note that this expression is exact; we will make no approximations until we apply the steepest descent method to the integral of Eq. (11.1). To that end let us examine the saddle points of this integral. It suffices to consider the case E = 0, since for a linear potential a shift in energy is equivalent to a translation in X. Applying Eq. (II. 11) we find T, = [~L@]“~(x;‘~

+ x,9’).

(111.5)

There are four saddles, corresponding to all possible signs of the square roots in Eq. (111.5). The value of the classical action (11.3) at each saddle point is S,] = $(2mn)1’2[x;‘2

+ xyq,

(111.6)

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419

with signs of $I2 and .y3/* chosen as in Eq. (111.5). The variation of the action about the saddles is governed’ by (111.7) It follows that the steepest descent approximation is valid provided neither x, nor xr is too close to the classical turning point I =O. The contribution to G from any given saddle is (111.8) This result can be compared with our general expression, Eq. (11.13). The product of the initial and final velocities for the path at the saddle point T, is i(O) A?(T,) = - (2+2)

x,“~.+‘~.

(111.9)

Since we have chosen E to be zero, Eqs. (II. 11) and (11.14) provide that WC, = S,,Jh.

(111.10)

It follows that Eqs. (11.13) and (111.8) are identical. The sign ambiguity in Eq. (11.14) is resolved by consistently maintaining the signs of (x,)“’ and (xr)l’* in Eqs. (IIIS)-(111.9). It remains only to specify which saddles are to be included in the sum of Eq. (11.13). This will depend on the specific choice of endpoints, for as xi or xr moves past the turning point (x =O), the steepest descent path for the integral (11.1) will jump discontinuously on or off one or another of the saddle points. These discontinuities are of course nothing but the Stokes jumps mentioned in Section I. To identify which saddles contribute to the integral (11.1) in any given case, one must see how the original contour along the positive real axis can be deformed to follow a series of steepest descent paths over ail the contributing saddles. “Steepest descent” implies a maximal variation of the real part of iS,,/h as the path leaves the saddle point; hence the steepest descent paths are specified by the condition Re S,,(T) = Re &A T,)

(111.11)

with the understanding that Im S,, should typically increase as one leaves the saddle point. These rules enable one to draw the appropriate contours for any choice of xi or xr. Consider first the case where xi and xr both lie in the classically permitted region x>O (with xf>+ui). This situation is illustrated in Fig. la. Figure lb shows the position and orientation of the four saddle points and specifies the steepest descent path for this case. Only the saddle points with Re T, > 0 contribute to Eq. (II.1 ), a

420

CARLITZ

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VIX)

X.

+

Xf

x

a

b

FIG. 1. (a) Linear descent contour.

potential

with

x, and xf in the allowed

region.

(b) Saddle

points

reflection of the causal structure of G. It is a simple matter to interpret relevant saddle points. One, with

T, = (2m/~)'i2[~~12-xXf/2],

and steepest

the two (111.12)

corresponds to the direct path from xi to xr. The other, with

T, = (2m/A)"2[x~'2 + x,"'],

(111.13)

corresponds to a path which moves left from Xi, reflects at x=0 and then moves right to xr. The different orientations of these saddles are related to the different signs of ii(O) in the two different cases. Since the two paths both lie in the classically allowed region, both saddle points involve purely real evolved time. The reflected path obviously requires more time to complete and hence lies farther out along the real axis. Since from Eq. (111.6) Re S,, differs for the two contributing saddles, the steepest descent contour must be built out of two pieces. Figure lb illustrates this and shows that the second saddle is crossed at an angle n/2 radians less than that at which the first saddle is crossed. This introduces a relative phase of --i in the contribution of the second saddle, consistent with the phase rule of Eq. (111.8) and removing the sign ambiguity inherent in the general expression, Eq. (11.13). [Note that a simple rule for the absolute phase comes from observing that

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the phase of the shortest path is the same as that of the free particle Green function, namely, -n/2.] Adding the contributions of the two saddle points we obtain

W,,x,;O)=-;[21(;i)L”]1’2~

in/4,(i/h)(2/3)(2mj.)“2.?lr

x 2cos

1

&2n,l.y

x’J2 - n/4 .

(111.14)

Suppose now that x, lies in the classically forbidden region to the left of the turning point as indicated in Fig. 2a. The saddle points and steepest descent path for this case are given in Fig. 2b. There is now only one contributing saddle, at T, = (2m/E,)“* [x;‘*-

(111.15)

i( -xi)“‘].

Physically this results because there is now only one possible path from xi to xr. The path necessarily passes through a portion of the classically forbidden region so T, necessarily has an imaginary part. In general (111.16) with p(x) = [2m(E-

(111.17)

v(x))]“2.

A V(X) \

D XL

Xf

X

a

FIG. 2. (a) Linear potential with points and steepest descent contour.

x, in the forbidden

region

and x( in the allowed

region.

(b) Saddle

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If we choose intermediate points of each path to pass through real values of x, then in the allowed region (E> I’) the path will evolve real time while in the forbidden region (EC V) it will evolve imaginary time. Comparing Eqs. (II.16) and (III.I6), we see that a negative imaginary contribution to T corresponds to a positive imaginary contribution to W. This portion of the path is therefore exponentially damped in the appropriate manner. The contribution of the saddle (111.15) gives G(x,, xi; 0) = -:

112

m

1

h [ 2A( -xixJi’2 x exp

-if

ei”/4e(i/Ll(2/3)(2mi)‘/2.~~/2

(2ml)1/2(-xi)3/2].

(111.18)

One can trace the manner in which the saddle of Eq. (111.13) drops off the steepest descent contour by setting xi = [Xi1 t?+

(111.19)

and examining what happens as 4 varies from 0 (as in Fig. 1) to rc (as in Fig, 2). As 4 reaches the value n/3, Re S,, for the saddles (111.12) and (111.13) become the same. The steepest descent contours for the two saddles merge at this point, as illustrated in Fig. 3. Beyond this point (4 > 1r/3) the first steepest descent contour drops below the second and the path from T=O to T= co (with Im T=O, Re T>O) no longer passes over the second saddle at all. This jump in the asymptotic form of G(x,, xi; E) under an infinitesimal change in xi is precisely the Stokes phenomenon [9] that we have been emphasizing. Actually we need not go to complex xi to find an instance where the steepest descent contours merge. Consider the case where xi and xr are both in the forbidden region (with xi < xt as in all our previous examples). This situation is illustrated in Fig. 4a. The physical reasoning developed in our previous examples makes it easy to understand the saddle points and steepest descent path of the corresponding Fig. 4b. A direct path through the forbidden region from xi to xf evolves purely imaginary time, as does a path which reflects from the turning point at x = 0. The second path is obviously longer. From Eqs. (II.1 5) and (111.17) it is clear that

FIG.

3.

Saddle

points

with equal

values

of ReS,,.

Steepest

descent

contours

have

merged.

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MECHANICS

Re IV,, (and hence Re S,,) vanish for both these paths so that the steepest descent path over the first saddle arrives at the second saddle as illustrated in Fig. 4b. Note that the subdominant saddle (corresponding to a longer trajectory through the forbidden region) is approached by a descending ridge, and that the steepest descent path from T=O takes a 90” bend at this point to drop off toward T= cc. Associated with this feature is a factor 4 for the contribution of the subdominant saddle. This factor arises because if xr were moved just above or just below the real axis, the subdominant saddle would lie either wholly on or wholly off the steepest descent path. The difference represents a Stokes discontinuity, related to the failure of our asymptotic approximation to reproduce the analytic properties of the exact Green function. One way to understand the factor of 4 is to note that since V’(x) is real, a wavefunction real for x > 0 must continue to be real for x < 0. In the present description this property is enforced by averaging the values obtained for x,+ k. This will be verified explicitly when we extract the connection formulas for the linear potential in the paragraphs below. For further discussion see the book by Dingle [ 151. The net contribution to G is thus found to be

,(1//;)(2/3)(2ml)'!*(-.,l)3'2

+

f. ~~(1/1;)(2/31(2mi.)"?(

-x[)"~

2

FIG. 4. (a) Linear descent contour.

potential

with

x, and xI in the forbidden

region.

(b) Saddle

points

1

(111

20)

and steepest

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The relative phases of the two bracketed terms follow from the sense in which the steepest descent contour crosses the two saddle points; the overall phase is specified by Eq. (111.8). We have argued previously that the Stokes discontinuities, which distinguish the various cases that we have considered, provide the underlying basis of the WKB connection formulae, which relate asymptotic wavefunctions on either side of a classical turning point. We can, in fact, compare Eqs. (III. 14), (111.18) and (111.20) and thereby extract the connection formulae from our work. When xi is varied [Eqs. (111.14) and (111.18)] we deduce the formulae -(l/li)(2/3)(2ml)‘12(,i’/2

&1x1-““e

-IxI-“4cos

[(l/h)$(2mA)1’21x13’2-7r/4],

(111.21)

and when xr is varied [Eqs. (III. 18) and (111.20)] we find f 1x1-‘14e ++e

-(1/6)(2/3)(2mn)‘~21~J’~* --in/41Xl

-

ilxl-

l/4

e

+ (1/1;)(2/3)(2m~)‘~zlx~~~2

- 1/4e(i/li)(2/3)(2m1)‘~21x~‘~2

(111.22)

These are precisely the standard results of the WKB formalism. To see this let us work out the explicit form of the momentum [Eq. (111.17)] for the potential (11.1) at E=O: p(x) = [2&x] The WKB

“2.

(111.23)

phase is defined by

w(x,=;,‘dxp(x), WJ where x0 is the turning point nearest x (here x,, = 0). Using Eq. (111.23) we obtain W(x) = (2/3@(2~1)“~x~‘~,

(111.25)

in terms of Eqs. (111.21) and (111.22) take the standard forms - in/4

-jeilW”” -I

W(x)1

+e-‘IW(XII

(111.26)

*

IPb)l l/2

jf&=’

and (i/2)

e-Iw”‘I

+

IP(X)l ‘I2

elw~”

ein/4

-Pol’!Ze

i,w(x),

.

(111.27)

Equation (111.26) refers to a wave incident from the allowed region (x > 0), which is reflected with an amplitude -i and transmitted into the forbidden region with an

CLASSICALPATHSANDQ~ANTUMMECHANICS

425

amplitude e -‘in’4.Similarly Eq. (111.27)refers to a wave incident from the forbidden region (x < 0) and decaying as x approaches 0. This wave is reflected back into the forbidden region with amplitude i/2 and transmitted into the allowed region with amplitude erni4.These factors are summarized in Table I. Note that since V is real, the complex conjugates of Eqs. (111.26) and (111.27)must also describe valid connection formulae. Thus if we multiply Eq. (111.27)by -i and add its complex conjugate, we find consistency with Eq. (111.26).This verities the claim that the factor of l/2 in Eq. (111.20) [which appears in Eq. (111.27)as i/2] is a consequenceof the reality of V. The expressions of Eqs. (111.14), (111.18)and (111.20)have all been obtained by a saddle point approximation to Eq. (11.1). Since the form of K in Eq. (111.4)is exact, it should be possible to avoid all approximations and deduce the full wavefunctions from Eq. (11.1). Those wavefunctions are Airy functions, and the asymptotic analysis that we have just computed must correspond to an analysis of the asymptotic behavior of the Airy functions. This is indeed the case, as we will now demonstrate. Let us start with the integral (11.1) with K defined by Eqs. (111.4)and (111.3).Let t = (/l/m)“‘T

(111.28)

ci = (Lm)“‘/h

(111.29)

and

in terms of which l/2 G(x,yq;O)=ar,-

dt

(x,-.Xi)*t-l+ 2

(Xl+Xi) 2

t

(111.30)

We will examine this expression as a function of X, in the limit X+ points of Eq. (111.30)are located at t = 29x;”

TABLE Connection forbidden forbidden

+ allowed -+ forbidden

allowed

--* allowed

allowed

--t forbidden

Formulae

+ .uy2),

(III.3 1)

I

for a Linear

Turning

CCI.The saddle

Point pi4 i/2 -i c -m/4

426

CARLITZ

AND

NICOLE

and while we are not planning to make a saddle point approximation, reasonable to shift the integration variable to 24= t - (2xr)“2,

it is still (111.32)

so that the dominant contributions come from finite exponent of Eq. (111.30) now takes the form

u even as xI-+ co. The

icr[2 I/2$ x:/2+ ux; - 2 u3 + o(x;“2)].

(111.33)

Thus in the limit xr+ ODwe obtain

X

s

m due ia[ux,

The last term provides an integral representation we have an expression for G G(x,, Xi; 0) e(i/1;)(2/3)(2ml)‘/?.~?/z[2

- (l/6)

u3]

--m

-1 - _ xf+ cc h 1/3~2/3]

m W2

for an Airy function of xi, Thus 112

[ 1 1/4nl1/2Ai

(111.34)

ern/4

( _ 21/3~2/3~,),

(111.35)

This equation describes the exact wavefunction for all x,-including xi in the vicinity of the classical turning point. In subsequent sections we will indicate other places where one can go beyond the Gaussian saddle point approximation to Eq. (11.1) and obtain exact wavefunctions in place of the asymptotic expansions. Note that it was necessary to go beyond the Gaussian approximation onl~j in the energy integral (11.1). Thus we have isolated the non-Gaussian fluctuations induced by proximity to the classical turning point in a single integral. We would like to emphasize several features of this simple analysis. First, exponentially small terms such as the last term in Eq. (111.20) may be swamped numerically by as yet uncalculated terms in the asymptotic series constructed about the dominant saddle point. Nonetheless careful treatment of the subdominant saddle point is essential to obtain the correct reality and factorization properties of G(Xi, xr, E) [see Eq. (11.2)]. Second, not all classical paths contribute to the steepest descent approximation. In fact, the set of contributing paths can jump discontinuously under a smooth variation of the endpoints even when no classical turning point has been approached [see Eq. (111.19)]. Third, as a consequence of the previous point, the analytic structure of an asymptotic expansion is in general different from that of the function which it approximates. Explicit Stokes jumps must be inserted to make a correct analytic continuation of the asymptotic expansion (see Ref. [lS]). This can make the application of Zinn-Justin’s method [16] rather delicate and subtle.

CLASSICALPATHSANDQUANTUM

IV. QUADRATIC

POTENTIAL

MECHANICS WELL

In this section we study a slightly more complicated or harmonic oscillator, V(x) = t 1710*x2.

427

example, the quadratic well

(IV.1)

For any positive energy there are now two classical turning points. If these turning points are far apart, the saddle points of the integral (11.1) are far apart, and the analysis of this quantum system is quite similar to the analysis of the linear potential given in the previous section. The principal difference is that there are now an infinite number of saddle points along the steepest descent contour. The physical reason for this is that the quadratic potential exhibits periodic motion in the classically allowed region of space. Examining the contribution of each saddle, we verify the rule of Section III for reflection at a linear turning point. Summing the contributions of all relevant saddle points, we obtain poles in G at the positions of bound states of the potential. Pole positions and the associated eigenfunctions are accurately represented by the steepest descent approximation as long as

E~h.

(IV.2)

At low energies there are important non-Gaussian fluctuations around each saddle. The saddles effectively merge, and one must evaluate Eq. (11.1) without approximations on E. We do this and show how the WKB results for a quadratic turning point are reproduced by our method. For E < 0 the disposition of saddles becomes quite different from that for E > 0. Just as for the linear potential, the Gaussian approximation to K is exact for the potential (IV.1 ). The resulting expression is [2]

(IV.3) We will concentrate first on the case illustrated in Fig. 5a, where the endpoints x, and xr lie far outside the classically allowed region. In terms of the variables (IV.4 (IV.5

428

CARLITZ AND NICOLE 4v \

I

E

0 ‘i

Xf

x

a

II )(

)(

)(

)(

)(

)(

FIG. 5. (a) Particle of energy E in a quadratic potential well. Endpoints x, and x, lie in the forbidden region on opposite sides of the well. (b) Saddle points and steepest descent path.

and q=-!.-2X.Xf x:+x;’

the semiclassical limit makes

(IV.6)

involves c1--f co at fixed /I and q. Our choice of endpoints (IV.7)

The sign of 9 indicates whether Xi and xr are on the same or opposite sides of the potential well. Introducing a dimensionless coordinate

we can rewrite Eq. (IV.3) as c1 K(Xfp

Xi;

T/O)

=

z?c(xf+ x:) sinr

w

1 [ exp

COST-V])

. I

We seek to evaluate the Laplace transform of K [cf. Eqs. (11.1) and (IV.5)],

(IV.9)

CLASSICAL

PATHS

AND

QUANTUM

MECHANICS

429

in the steepest desent approximation. The integration contour in Eq. (IV.10) runs along the real T axis passing beneath the singularities of K. The integral (IV.10) has saddle points fixed by the condition

&

1

&COST-q)+/?T

=o,

i

(IV.11)

whence (rj cos T, - 1 )/sin%, + p = 0,

(IV.12)

or cos Ts = (1/2/I) {rj f [q2 + 4fl(P - 1 ,I’!‘:.

(IV.13)

This result is quite general, with no restrictions on the endpoints or energy. If in fact the endpoints are far outside the well, then Eq. (IV.7) applies and Eq. (IV.13) takes one of the approximate forms cosTs=rl/B-

(IV.14)

1/v+W)

or COS 5, =

l/q

(IV.15)

+ o(b).

Both of these cosines are larger than one, corresponding to the fact that classical paths connecting xi to xt must traverse the forbidden region and hence must evolve some imaginary time. If p > 0 the real part of z, is given by Re T, = hn

(IV.16)

(yl>O)

or ReT,=(2n+l)n

(rl
(IV.17)

for either class of saddle points, (IV.14) or (IV.15). It is clear that the integer n in Eqs. (IV.16) and (IV.17) counts the number of bounces in the allowed region of the potential well, since 271/w is the real time required to execute one period of motion. For rl< 0 [Eq. (IV.17)] the endpoints are on opposite sides of the well, and one more half period of real evolved time is required relative to the q > 0 case [Eq. (IV.16)]. Having enumerated the saddle points, we must now examine their orientation, their disposition with respect to the steepest descent path and, of course, the whole question of validity of the steepest descent approximation. These questions all relate to the quantity (IV.18)

430

CARLITZ

AND

NICOLE

evaluated at the saddle point 7,. Noting that the sign of Im sin r, is the same as the sign of q Im rs, we obtain

D,=D(z,)=

+a

tj co? 7, - 2cos 7, + tj (cos*7s - q3’* ’

(IV.19)

where the f symbol refers to the sign of q Im zs. For D, < 0 the steepest descent path crosses the saddle parallel to the real z axis, while for D, > 0 it is parallel to the imaginary r axis. The magnitude of D, controls the quality of the steepest descent approximation. From Eqs. (IV.19) (IV. 14) and (IV.15) we find D,= &afl

for cos r, = qf/l-

l/q

(IV.20)

and

Taq2 DS”(l +y/*

for cos 5, = l/q.

The upper and lower signs here correspond to Im z, > 0 and Im z, < 0, respectively. It follows from Eqs. (IV.20) and (IV.21) that as c(+ co, fluctuations around the saddles near l/q are always small, while the fluctuations around the saddles near q/p - l/q require that (IV.22)

laPI % 1

as well. Since we are working in the limit (IV.7), this is not assured, and we must demand that Eq. (IV.22) or, equivalently, the condition (IV.2) be explicitly satisfied. This, of course, is just what we expected, and we can now proceed to sum the saddle point contributions appropriate to this restricted regime. The saddle points and steepest descent contour for the case v < 0 (Fig. 5a) are illustrated in Fig. 5b. Each contributing saddle has a simple physical interpretation. The first saddle corresponds to a direct path from Xi to xf. Such a path tunnels from xi to the well evolving imaginary time, passes through the well evolving half a period of real time and then tunnels on to xf evolving more imaginary time. Other paths involve one or more bounces in the allowed region. Each bounce adds a full period to the evolved real time, but all paths evolve the same amount of imaginary time. Note that none of the saddles of Eq. (IV.15) are physically relevant for q < 0. Note also that sections of the steepest descent contour with different values of Re S,,(z) join at singularities of S,,(r). In the examples of the previous section, these singularities were always at r = co, while here they also occur for finite 7. There is no contribution to the integral from the singularities themselves. To compute the steepest descent approximation to G we must find the value of Wcl=a

cosz,-7 [ sin rs

+ PG

1

(IV.23)

CLASSICAL

PATHS

AND

QUANTUM

431

MECHANICS

at the physically relevant saddle points given by Eq. (IV.14): W,,=icr[(l

-fl)-/?ln(2~//?-27cinfl+O(f12)].

(IV.24)

The prefactor for the nth saddle is found from Eqs. (IV.20) and (IV.9)

y[;]“2[,x~yx;)

J2=q

[x,x,, ‘12.

to be

(IV.25)

This is to be compared with the prefactor h-’ [a,, ;i-r] ~ I” in Eq. (11.13). Since for large X, (IV.26)

+ mi2 E -+ mo2x2,

the two expressions are equivalent. The phase factor ( - 1)” in Eq. (IV.25) results from the variation of the factor (sinr) ~ ‘j2 in Eq. (IV.9). Its presence confirms the rule of Section III and Table I that associates a factor of (-i) with each reflection in the allowed region, since each successive saddle along the steepest descent contour involves one more pair of reflections (at the right and left turning points in Fig. 5). Summing all contributing saddles, we obtain a series G(x,, xi; E)=&(x,r,)-“‘e”+“” x

f

(-1)ne2ni(n+1/2)zB

II=0

=_ 1

1

e - (mru(.x:

+ .4)/2/;)

ho2 cos(7rE/h) (IV.27)

This expression allows us to quickly read off a number of features of the quantum eigenvalues and eigenfunctions of the potential (IV.l). The propagator G has poles for

E=(n++)h

(integer n),

(IV.28)

which is the familar exact result for this system. Actually Eq. (IV.27) is strictly valid only for E+th, and it is somewhat accidental that Eq. (IV.28) is valid for small positive energies. The accidental’ nature of this agreement becomes more apparent when we realize that (IV.28) allows negative energy eigenvalues as well. In fact, as *This oscillator

accident arises from (see Ref. [8]).

the simple

configuration

of Stokes

lines which

occurs

for the harmonic

432

CARLITZ

AND

NICOLE

we will see below, the wavefunctions are not properly represented in Eq. (IV.27) except when E is large. Pieces of the correct wavefunction are, however, in evidence. Since Xi and xr are by hypothesis quite large, we can expect to see only the tails of the wavefunctions. And indeed we find them-an exponential factor exp( -mw2x2/2h) multiplied by a power xn for the nth level of the system. Consider now the case q > 0, where xiand xr lie in the forbidden region on the same side of the potential well. This case is not very different from the one just considered, but a comparison of results for rl< 0 and q > 0 will permit us to extract connection formulae as we did for the linear potential in the previous Section. Figure 6a illustrates the physical situation, while Fig. 6b shows the saddle points and the steepest descent contour. The saddle points of Eq. (IV.14) have now moved to the right as indicated by Eqs. (IV.16) and (IV.17). There is also now a direct physical path from xi to xr, corresponding to the presence on the steepest descent path of one saddle of the class (IV.15). [The steepest descent contour bypasses all other saddles of this class.] The first saddle of the class (IV.14), which corresponds to a path proceeding from Xi to the turning point and then reflecting back to xr, has the same value of Re S,, (namely, zero) as the single relevant saddle from Eq. (IV.15). The situation here is precisely the same as for the case in the previous section when Xi and xr both lie in the forbidden region. Once again the path which corresponds to a reflection in the forbidden region is associated with a subdominant saddle point in T and contributes with a strength of i. The prefactor and the value of W,, for saddles of the class (IV.14) have already

FIG. 6. (a) Particle of energy E in a quadratic potential well. Endpoints region on the same side of the well. (b) Saddle points and steepest descent

xi and xI lie in the forbidden path.

CLASSICAL

PATHS

AND

QUANTUM

433

MECHANICS

been specified in Eqs. (IV.25) and (IV.23), respectively. It remains only to compute these quantities for the saddle point (IV.15) corresponding to the direct path from x, to xr. This is easily done, with the results

Eff 1 w

7c(x;+x;)(l

-$)l’2

1 =z[x,xf]

-“’

(IV.29)

and W,,=icr

i

l+(l-$)“1

(1 -~2)“2-~ln [

=i[mW(ii

- X2)/2 -

II

?

(E/W)

(IV.30)

lIl(XJXi)].

The net result for G is thus found to be 1 G(x,. x,; E) = - e ~ mu.v;/?j; ho> x

Ce

+c

xe

( .yr)

+m
E/h

I ,‘2

~ E/h~O ~ I/2

2emo 2 Eil;ru sin( nE/h;) E ! 2 cos(~Ejl;w) -mol-~~26

(x,)E//&

Ii2

.

(IV.31 )

1

Note that factors in these equations have a simple relation to the phase rules of Table I. Saddles of the class (IV.14) differ by the number n of bounces in the allowed region. Each round trip in the allowed region evolves a WKB phase

Here k.‘c, are the classical turning points, x0 = [2E/mo’]

I/‘.

(IV.33)

The n = 0 term involves a reflection in the forbidden from Table I. For n = 1 there are factors e

‘n/4(

_

i) e’“/4e2~~“

=

The net effect of all saddles of the class (IV.14) factor

_

region and hence a factor i/2

ie2i”‘.

is to introduce

; + i f ( _ 1)neZlnP, _ sin M’ 2 cos M” n=l

(IV.34) in Eq. (IV.3 1) the

(IV.35)

434

CARLITZ

AND

NICOLE

We will now show that this factor and the corresponding factor of l/(2 cos w) in Eq. (IV.27) are the essential ingredients for deducing connection formulae from our approach. Since the derivation of these factors follows precisely the rules for linear turning points, it will come as no surprise that our approach reproduces the results of the WKB method. To extract connection formulae we must compare the expressions (IV.27) and (IV.31). To appreciate the content of these equations note first the boundary conditions satisfied by G(x,, Xi; E). At xr there is an outgoing solution, exponentially damped for increasing (xrl. For Xi on the opposite side of the well from xr [Eq. (IV.27)] the solution at Xi is also outgoing. For Xi on the same side of the well as xr the solution at Xi has incoming and outgoing pieces corresponding to terms in Eq. (IV.31) which decrease with increasing or decreasing values of xi, respectively. With this interpretation we obtain from Eqs. (IV.27) and (IV.31 ) the connection formulae ~

IP(

1 Ii2

1 e - I Wx)l I (P(X)l lJ2 1 + IW-y)1. +2cos(Whw)IpolllZe

e-‘w(-x)l +-+sin(ltE/hO)

(IV.36)

The classical momentum p(x) and the WKB phase W(x) have been defined in Eqs. (III. 17) and (111.24), respectively. The result (IV.36) is precisely what one obtains via the linear connection formulae of the WKB method. It is valid when the turning points are indeed linear, that is, for E$-ha. For small E one must employ the quadratic connection formulae in the WKB approach; and in our approach one must abandon the steepest descent approximation and evaluate the integral for c1large but with any value of c$. This we will now do, to obtain another comparison of our approach with the WKB method and to extend our method to encompass this case as well. Consider first q < 0. The only saddle points along the steepest descent path for this case were those of class (IV.14). When E is not large, contributions to the steepest descent path are no longer localized at these saddle points. Important nonGaussian contributions effectively make each saddle spread out and merge with its neighbors. Evaluation of the integral (IV.10) with no restrictions on IX/? will thus reveal the net effect of all the merged saddles. This is significant for the present problem; and, as we will see in Section VI, it is a situation which recurs in other problems. Since we are no longer discussing an asymptotic expansion in crj?, it is now legitimate to make analytic continuations in that variable. To avoid the singularities of G let us take /I < 0 for the moment. This allows us to rotate the contour of the integral (IV.10) downward to run along the negative imaginary r axis. The derivation of Eq. (IV.14) shows that dominant contributions to Eq. (IV.10) come from ir=q/fl$l.

(IV.37)

CLASSICAL

In this region it is legitimate

PATHS

AND

QUANTUM

MECHANICS

435

to approximate (IV.38)

and rewrite

Eq. (IV.10)

as

(IV.39)

A change of variables [=e-‘,

(IV.40)

gives the expression

(IV.41) whence

Here y denotes an incomplete complete r function and write

r function,

but since a+ 1 we can replace it by a

(IV.43 ) The poles of G for E > 0 are now seen to arise from poles of the r function. In the asymptotic limit E9h we can apply Stirling’s formula and show that Eq. (IV.43) reduces to our previous result, Eq. (IV.27). For negative E, Eq. (IV.43) has none of the spurious poles found in the approximation (IV.27), and for small positive E the two expressions also differ. The residue of the poles at E = h42, for example, differs in the two cases by a factor (n/e) ‘I* . We will relate this factor in

436

CARLITZ

AND

NICOLE

what follows to the ratio of expressions obtained via the WKB method with quadratic and with linear turning point connection formulae. First, however, we must consider the evaluation of the integral (IV.10) for q > 0 (and with cc~ 1 but c$ arbitrary as for the case just considered). We cannot simply make an analytic continuation of the result (IV.43) from q < 0 to q > 0, since we know that Stokes lines must be crossed in such a continuation. Specifically, we know that for q > 0 the steepest descent path includes one saddle of the class (IV.15) as illustrated in Fig. 6b. Recall that the contribution of that saddle remains Gaussian for any value of LX/?.The problem when c$ is not large is that all contributing saddles of the class (IV.14) spread out and merge as they do for the case q < 0 discussed just previously. We are interested, therefore, in the contributions of saddles of the class (IV.14) to the propagation function G. What we will now argue is that an analytic continuation of Eq. (IV.43) can be used to describe these contributions. The full expression for G is therefore obtained by adding to Eq. (IV.43), suitably continued in 4, the first term of the expression (IV.31). Note that on account of the nonintegral power of Xi (and hence of v]) that appears in Eq. (IV.43) the phase of the result obtained for rl real and positive will depend upon the direction of analytic continuation. For G to remain real it will be necessary to average the results obtained by continuing clockwise or counterclockwise in the complex q plane. These remarks are rather reminiscent of our discussion in Section III of the factor

FIG. 7. Saddle points and steepest descent path for the potential of Fig. 6a, but with a small imaginary part for q. (a) For one sign of Imq the path flops left and then passes through the n = 0 saddle point of the class (IV.14). (b) For the other sign of Imq the path flops right and bypasses the n = 0 saddle point of the class (IV.14).

CLASSICAL

PATHS

AND

QUANTUM

MECHANICS

437

saddle point. Indeed, suppose we examine the i associated with a subdominant steepest descent path of Fig. 6b for Im q nonzero. For one sign of Im q the path flops left as indicated in Fig. 7a. The resulting path crosses the n = 0 saddle of class (IV.14) in the proper sense, and that saddle must contribute fully to the integral along that path. For the other sign of Im 9 the path hops right, and the n = 0 saddle of class (IV.14) is completely bypassed. Averaging these two contributions, we obtain the factor of t associated with the n =0 saddle in the steepest descent approximation. Here we would not like to employ the steepest descent approximation [except for the saddle of class (IV.l5)]. If, however, we denote by G* the contribution to G of the saddles of class (IV.14) alone, we see that a suitable expression for G* is the average G*(x,,

xi; E) =&

[ j’x’ dt + j-l h] eiEr’h’mx K(x,, x,; T/W). (IV.44) -?I H The treatment of these integrals for XiXr> 0 is almost identical with our treatment of Eq. (IV.10) for x,x,< 0. First rotate the contours downward to obtain the analog of Eq. (IV.39),

xexp [(g-i) Then, following

Eqs. (IV.40)-(IV.43)

r2xrlC’].

(IV.45)

obtain the result

This is the continuation of Eq. (IV.43) that we have described (IV.46) provides the following expression for G: 1 G(x,, xi ; E) = -=- e - rrrwu:/2k(Xf)E/h<“- I/? ho

above. Equation

438

CARLITZ

AND

NICOLE

Note that relative to the steepest descent approximation replacement

we have simply made the

1 + K(E) = (27c2 cos (nE/hco )

(IV.48)

in the second term of Eq. (IV.31 ), Comparison of Eqs. (IV.27) and (IV.43) shows that the same rule applies, and hence the connection formula derived in the steepest descent aproximation is modified in our computation for arbitrary c$ to be 1

pe-lw(x)l*sin

IP(X)1 l/2

ICE 1 e - I Y.x)I ( ho > IP(X)1 1/Z (27~)“~(he/E)~‘~~ + lJ1/2-E/&0)

1 IPol”Ze

+Iw(x)l .

(IV.49)

We see that the modification of the formula (IV.36) derived from the steepest descent approximation involves only the coefficient of the term which is exponentially increasing in the forbidden region. This accords with the modification to the WKB connection formula induced by using the proper quadratic formula in place of the linear formulae. The WKB result with the quadratic connection formula is in fact precisely that given in Eq. (IV.49). This establishes the equivalence of our approach with the WKB method for both linear and quadratic turning points [S]. The present approach is perhaps simpler and certainly more physical than the standard derivation. It is interesting that the difference between the results (IV.36) and (IV.49) is so small for energies on the order of h;. As noted previously the discrepancy between the coefficients of the exponentially growing terms of the two expressions amounts only to a factor (e/r~)l’~, a correction of only about 7 %. This discrepancy was noted years ago by Furry [17], but the difference between Eqs. (IV.36) and (IV.49) seems to be a perennial [6, IS-201 source of difliculty. For negative E the two expressions are completely inequivalent, and an analytic continuation of the E > 0 saddle point approximation, Eq. (IV.36), is obviously wrong. For E large and negative, a new saddle point approximation should be valid. We will discuss this case briefly with q < 0 as illustrated in Fig. 8a. The turning points are now well separated but lie at imaginary values of the coordinate x. The steepest descent path for the integral (IV.10) is shown in Fig. 8b. It crosses only one saddle point, corresponding to the direct path connecting Xi to xr. The contribution to G of this saddle point can be computed as for our previous examples with the result ,-“dxf+$M2~(

-xixf)E/~-

Comparison with Eq. (IV.43) shows that we have again obtained asymptotic formula for r( t - E/h;).

112.

(1v.50)

the appropriate

CLASSICAL

PATHS

AND

QUANTUM

439

MECHANICS

As a final example relating to the potential (IV.1 ), suppose endpoints Xi and xr are inside the allowed region, with

now

p+1.

that the

(IV.51 )

This situation is illustrated in Fig. 9a. The saddle points of the integral (IV.10) are still given by Eq. (IV.13), but the approximation which led to Eqs. (IV. 14) and (IV.15) is no longer valid. In place of these equations we have now COST,= 1 -(l-r/)/28

(IV.52)

cos z, = - 1 + (1 + Y/)/28.

(IV.53)

or

Both saddles correspond

to paths with real elapsed time Ts= f((l-r/)//?)“‘+2xn

(IV.54)

z,= r((l+?/)/fl)+(2n+l)rc.

(IV.55)

and

FIG. 8. (a) Negative E for a quadratic potential well. (b) Saddle points and steepest descent path.

well. Endpoints

I, and xf lie on opposite

sides of the

440

FIG. region.

CARLITZ

AND

NICOLE

9. (a) Particle of energy E in a quadratic potential (b) Saddle points and steepest descent path.

well. Endpoints

x, and xf lie in the allowed

The signs here correspond to the sign of sin rS. All four classes of saddles indicated in Eqs. (IV.54) and (IV.55) lie on the steepest descent contour of the integral (IV.10). As with our previous examples, a physical explanation is straightforward. There are paths which (i) go directly from x, to x,; (ii) go from xi to the left turning point and bounce back to x,; (iii) go from xi past xr to the right turning point and reflect back to x,; (iv) go from xi to the left turning point, reflect there and travel to the right turning point, reflect again and go to xr. Accounting for n additional round trips in the allowed region, we obtain the elapsed times of Eqs. (IV.54) and (IV.55). The steepest descent path shown in Fig. 9b passes over saddles belonging to each of the four classes enumerated above. The contour of Fig. 9b should be contrasted with that of Fig. 8b. Recall that the original integration contour in Eq. (IV.10) was defined to pass under the singularities of the kernel K(x,, Xi; r/o). That contour was deformed in Fig. 8b to pass downward along the negative imaginary r axis-a deformation which crosses none of the singularities of K. It follows that the steepest descent contour of Fig. 8b really ends at r = --ice. Figure 9b is quite different. Sections of the steepest contour pass to z = (2n + 1) rr + ice. These points lie above the singularities, and hence the contour must return to (or below) the real axis in order that the net deformation will have crossed none of the singularities. Thus for E < 0 there was not an infinite sum of saddle point contributions and hence there were no poles generated in G.

CLASSICAL PATHS ANDQUANTUM

MECHANICS

441

Here, with E > 0 there is an infinite sum, and poles of G are generated. We will not evaluate the sum explicitly. Rather we use this example to emphasize once more the enormous differences in the steepest descent paths induced by simple changes in .yi, xy or E.

V. QUADRATIC

POTENTIAL

BARRIER

Let us consider now what happens if in Eq. (IV.1 ) we make the substitution

to construct a quadratic barrier V(x) = - f m12x2.

(V.2)

We know of course that it would be invalid simply to continue the asymptotic results of the previous section. Nonetheless, the derivation of the steepest descent approximation for the potential (V.2) proceeds parallel to that for the potential (IV.l), and we can borrow many results of the previous section. The parameters tl and B now have the form iY.=

imA(xf + xf) 2h

w.31

and 07.4)

and the integration contour in Eq. (IV.10) now runs up the imaginary axis [see Eq. (IV.8)]. We are working in the asymptotic limit ICX-+ cc with I/? 6 1. If

I@I B 1,

(V.5)

then the steepest descent approximation to the integral (IV.10) should be valid. We will study this case first. The saddle points of the integral (IV.10) are specified by Eqs. (IV.14) and (IV.15). Fluctuations about these saddles are controlled by the factors D, in Eqs. (IV.20) and (IV.21), respectively. The condition (VS) and the asymptotic limit in 1~1guarantee that fluctuations are indeed small. Note that the quantity D, is purely imaginary, so that orientations of the saddles have been rotated by 45” relative to saddles for the potential (IV.1 ). The precise positions of the saddle points and the path of the steepest descent contour depend upon the values of q and fl. Suppose fl> 0 (i.e., E < 0) and q < 0.

442

CARLITZ

AND

NICOLE

This physical situation is illustrated in Fig. 10a. Figure 10b indicates the corresponding disposition of saddle points. Since the endpoints x, and xf are on opposite sides of the barrier, all paths must tunnel through the barrier and evolve imaginary as well as real time. The direct path evolves a minimal amount of imaginary time and provides the dominant contribution. Other paths involve multiple bounces in the forbidden region and provide subdominant saddle points in Fig. lob. Since imaginary time corresponds to real z, these subdominant saddles appear displaced sequentially along the real 5 axis. Only the lirst saddle is crossed from below as one moves along the steepest descent path from 5 = 0. Infinitesimal variations of xi, xy or E will cause the subdominant saddle points to slip fully off or onto the contour; and the strength of their contribution is ambiguous. We have encountered this situation previously in Sections III and IV, where we argued that the first subdominant saddle on a given contour contributes to G with strength f. The same argument holds for the first subdominant saddle here. We know of no straightforward prescription for the strength of lower-lying subdominant saddle points, although Dingle [ 151 has treated a number of interesting examples. The situation for E-C 0 and q > 0 is similar to the one just discussed. There are now two paths from xi to xf which lie in the allowed region of the potential (see Fig. lla). One goes directly from xi to xf, while the other reflects from the barrier to reach xy after some longer elapsed real time. These paths provide the saddle points shown in Fig. llb along the imaginary z (real T) axis. There are also sub-

FIG. path.

10.

(a) Tunneling

through

a quadratic

potential

barrier.

(b) Saddle

points

and steepest

descent

CLASSICAL

PATHS

AND

QUANTUM

443

MECHANICS

AV

‘f

D X

‘i

E

FIG. path.

1 I.

(a) Reflection

from

a quadratic

potential

barrier.

(b) Saddle

points

and

steepest

descent

dominant saddles corresponding to penetration into the forbidden region, and the first of these saddles again contributes with the characteristic factor 4. Since the orientations of the two real time saddles in Fig. 1 lb differ by 90”, their contributions to G differ by a phase factor -i. Combining our results for Figs. lob and 11b, we are led to the connection formula 1

4 WV)1_ i

IPole

, p --iI IP(X)l ‘I1 F(E)

W(x)l

c-* F(E)

enE’M

IPCX)l “2

,il

WI\-)I

’

V.6)

with a flux factor F(E)

=

1 _

4 e2nE/fii.

+

0(~4nE/~~),

W.7)

The higher-order terms in Eq. (V.7) correspond to the undetermined contributions of low-lying subdominant saddle points. The usual WKB connection formula for this system is also of the form (V.6) but with a flux factor F= [ 1 + $wlii.]

~ l/2

(V.8)

specified by the constraint of unitarity. In the limit IEl $hA the expressions (V.7) and (V.8) are equivalent; indeed they coincide even at order ezrrElfi’. This agreement is physically significant in that it verifies that the reflection coeffkient R= -iF

W.9)

444

CARLITZ

and the transmission

AND

NICOLE

coefficient T = &“E’fiA

(V.10)

that can be deduced from Eq. (V.6) satisfy the unitarity [RI*+

ITI*=

constraint

1

(V.11)

at the leading non-trivial order. We can improve on Eq. (V.6) considerably. The restriction of Eq. (V.5) was introduced only to simplify evaluation of the Laplace transform integral [Eq. (II.l)] which defines G. If this constraint is relaxed, that integral can be evaluated exactly, and exact expressions for R and T can be obtained. We will do this below. First we study the case where Ic$I $1 but with /I < 0. Here E > 0, and the system is above the potential barrier. If q < 0 we have the physical situation illustrated in Fig. 12a. The corresponding steepest descent path is shown in Fig. 12b. The dominant saddle point evolves real T (or imaginary r) and corresponds to the direct path from xi to xr. Subdominant saddles correspond to paths which tunnel to and bounce off the (complex) classical turning points. The situation with E > 0 and n > 0 is shown in Figs. 13a and 13b. The steepest descent contour crosses two saddles from below and then threads its way down a sequence of subdominant saddles. The dominant saddles correspond to a direct path from xi to xr (which evolves real time or imaginary time) and a path involving

FIG. path.

12.

(a) Transmission

above

a quadratic

potential

barrier.

(b) Saddle points

and steepest descent

CLASSICAL

PATHS

AND

QUANTUM

445

MECHANICS

*V

E

XL

D X

Xf

a

‘f Jr

J/ Jr

Jr

Jr

ib FIG. path.

13.

(a) Reflection

above

a quadratic

potential

barrier.

(b) Saddle

points

and steepest

descent

a reflection from one of the complex turning points (which evolves complex r). Subdominant saddles involve multiple reflections from the complex turning points. Combining our results for Figs. 12b and 13b, we obtain a connection formula of the form F(E) e - nE/kl 1 ,ilW.)l _ j e ~ I WX)l ( ; F(E) ,~l~Cx)l IP(X)1 II2 IP(X)1 II2 IP( “2 ’

(V.12)

with a flux factor F(E)=

1 _~,-2~EIfij.+o(,-4n~l~~)~

(V.13)

Application of the WKB method to this example is not completely straightforward. A careful treatment including the constraint of unitarity gives a result of the form (V.12) but with f’= [ 1 + ,-=/fii] Again the dominant We will (V.6) and

595/164/l-15

~ j/2,

(V.14)

two procedures agree at the order of the contribution from the first subsaddle point. now show that our procedure is really correct, and that the results, Eqs. (V.12), specify the true behavior of the asymptotic wavefunctions. To do

446

CARLITZ

AND

NICOLE

this we will examine the domain where /E/&J = la/?l is not large. This will involve evaluating Eq. (IV.10) without recourse to the steepest descent approximation. This discussion follows the lines of that given for the quadratic well in the previous section. For the quadratic barrier, the integration contour in Eq. (IV.10) runs directly up the imaginary axis. If that contour is deformed toward the steepest descent contour of Fig. lob, then significant contributions to the integral come only for Imz>> 1.

(V.15)

Since fluctuations about each of the contributing saddles in Fig. 10b are not small, we cannot meaningfully separate the contributions of individual saddle points. Rather we must in effect sum the contributions of all these saddle points. Equation (V.15) makes it appropriate to make the approximations -isinr2:cosrNe-iT/2

(V.16)

in the integrand of Eq. (IV.10). The substitution c=e”

(V.17)

yields the expression

(V.18) which may be compared with our Eq. (IV.41) for the quadratic well. Proceeding along the lines of the argument following Eq. (IV.41), one is led to the exact connection formula 1

4 WV)1

GGPe

j

e

fcE)

I P(X)1 1’2

--iI

W()I

++ftE)

enE’M

eil W(.x)l

(V.19)

r(& - iE/&).

(V.20)

IP(X)1 Ii2

with f(E) = (27c)“2exp

{ (iE/&)[ln

I E/hAJ + in/2 - l]}

This equation correctly reproduces our previous results, Eqs. (V.6) and (V.12) (with unit flux factors), in the respective limits E6 -h; and EBhl. It also helps to clarify our discussion on the role of subdominant saddle points in the integral which defines G. Suppose that for E % h3, one employs Stirling’s formula for r( 4 - iE/hA) in Eq. (V.20). There results an expression which coincides with Eq. (V.13) at the leading order in E/Q with a series of correction terms in ascending powers of l&/E.

CLASSICALPATHSANDQUANTUM

MECHANICS

Interestingly, these power corrections contribute mission and reflection coefficients

447

only to the phase of the trans-

T= f(E) &=EIKA

(V.21)

R = - ij-(E).

(V.22)

and

This is not to say that T has modulus unity. Indeed with no approximations / TI 2= enW~~[2 cosh(rrE/fiL)] - ‘,

(V.23)

which corresponds with the result of Eq. (V.12) with the flux factor (V.14). This illustrates that there are significant exponential corrections to the asymptotic series of the r function, and that Stirling’s formula (which neglects these terms) is inadequate for the present problem. Indeed, Stirling’s formula corresponds to retaining in Fig. lob only the contribution of the first (dominant) saddle. Our steepest descent treatment, by contrast, gives unambiguously the correct contribution of the next (first subdominant) saddle.

VI. DOUBLE-WELL

POTENTIAL

In previous sections we have developed a method for evaluating the contributions of classical paths in one-dimensional quantum mechnical problems. Examining several exactly solvable systems, we have learned how to identify the paths which are relevant, the phases with which these paths contribute to the propagation function, the conditions under which fluctuations about these paths are small and the procedure to employ when fluctuations are not small. In this section we will apply this knowledge to a system which is not exactly solvable, the quantum mechanical double well, which is described by the potential V(x) = i(x2 - 2)’ illustrated

in Fig. 14. In the vicinity of x = fa,

(VII)

V(s) has the form

V(x) 2;md(xfay

(VI.2)

with w = 2a[2A/m] ‘12.

(VI.3)

The quantity w represents the fundamental frequency of each of the separate potential wells in the absence of any quantum tunneling.

448

CARLITZ

AND

NICOLE

v

c E I

& x. I

-x0

-x,

x,

x0

FIG. 14. Double-well potential with km < E< E,. Turning lie in the forbidden region on opposite sides of the wells.

Xf

points

x

are k x0. &x,

Endpoints

xi and x,

A classical path which travels from xi at t = 0 to xf at t = T is described by an elliptic function x(t): w x(r) t= m - aq2]-1’2. 02 J3 dx[E-1(x2

Possible values of the energy E in this expression are specified by the endpoint condition x(T) = xf. For a given E the classical turning points are located at the points +x, and fx, (illustrated in Fig. 14 for real E), where x($= a2 + (E/A)“’

(VI.5)

xf = a2 - (E/l)‘12.

(VI.6)

and

In terms of these variables an explicit expression for x(t) is x(t)=x,

sn

iox,( (

t - t’)

2a

x,

‘x0’ 1

(VI.7)

The constant t’ denotes the time at which x(t’) = 0. An explicit expression for t’ will be given below. The simplest applications of the classical path method are for systems with wellseparated real turning points. If the energy E is far below the top of the barrier between the two wells (E, in Fig. 14) but far above 0, the double-well system satisfies these conditions. It follows that a good approximation to G(xr, xi; E) can be obtained simply by summing contributions of the form (11.13) from the classical paths which link Xi to xf. These are easily enumerated. There is a direct path which evolves both real and imaginary time, and an infinity of other paths involving bounces in the regions between the turning points. Each bounce in the permitted region involves an additional elapsed real time dx tR =

(2m)1’2

lx;

[E-

J/(~),I/Z

=G

4a K(k)

(V1.8)

CLASSICAL

and an additional

WKB

PATHS

AND

QUANTUM

phase

2(2m)“’ 2WR = h

‘0 1r,

d.x[E-

v(x)]“2

= 2

The functions K(k) and E(k) in Eqs. (VI.8) functions of modulus k, where k2

Each bounce in the forbidden

=

[a2E(k)

- xfK(k)].

(VI.9)

and (V1.9) denote complete

2(El~)“2 uz + (E/n)“”

region evolves an additional

f, = -i(2m)1’2 and an additional

449

MECHANICS

elliptic

(VI.10) imaginary

time

dx -8ia I’ s -r, [V(x-E,1~2=zqK(k’)

(VI.1 1 )

phase 2w = 2i(2m)“’ I 6

rl y,

dx[ V(x) - E]‘12 = 2iA,

(VI.12)

with A= 2 Here k’ denotes the complementary

[a’E(k’)

- (E/l)“‘K(k’)].

(VI.13)

modulus, (VI.14)

The imaginary value of wr corresponds to the fact that tunneling bidden region is suppressed. The requirement that Im w,zO

through

the for-

(VI.15)

fixes the sense of the path in the complex I plane and hence determines the sign of Im t, [Eq. (VI.1 l)]. The set of paths involving all possible bounces leads to a lattice of saddle points contributing to the Lapiace transform integral (Il. 1). This lattice is shown in Fig. 15. Since for the potential (VI.l) there are an infinity of paths contributing to G(x+ xi; E), each of the saddle points depicted in Fig. 15 may be associated with several of the contributing paths. This means that-in contrast with our previous examples-there is now a non-trivial multiplicity factor associated with each saddle point. Furthermore, as we have seen previously, some saddle points are approached from above along the steepest descent path; and these subdominant saddles con-

450

CARLITZ AND NICOLE

(

l

l

l

FIG. 15. Saddle points and weight factors for the double-well potential.

tribute to G(X,, Xi; E) with only half strength. Finally, there is a phase in Eq. (11.13) which must be specified for each contributing saddle point. The net weight for each saddle point is easily found either by application of the connection rules of Table I or by direct enumeration of the contributing paths x(t). Consider, for example, the lifst row of saddle points in Fig. 15, corresponding to (VI.16)

T,=t,+~t*+(n+l)t,. Here t, denotes the (imaginary)

time evolved in the region outside the wells, dx

(VI.17)

[V(x)-E11’2

The explicit form of t, involves elliptic integrals of the first kind F(d, k’): l0

= -(2iu/wxO)

CFt4(xi),

k’)

+

F(Q)(xf),

k’)

1,

(VI.18)

where #(x)=sin-‘[(x2-x:)/(x*-x:)].

(VI.19)

The constant t’ which appears in Eq. (VI.7) can be written similarly: t’ = (-2iU/WXO)

F(~(Xi),

k’) + a t, + f t,.

(VI.20)

The integer n in Eq. (VI.16) counts the number of bounces in the allowed region. Paths with n = 0, 1 or 2 are illustrated in Fig. 16a along with the associated weight factors, as deduced from the rules of Table I. Figure 16b similarly illustrates the weight factors for saddles at to + $ fi + t, and t, + $ t, + 2t,. Each of the paths described above corresponds to one of the appropriately directed, topologically distinct classical paths which evolves from xi at t =0 to xr at t = T,. Two paths are topologically distinct only if they are separated by singularities of the classical action S,,. For the potential (VI.l) this corresponds to singularities of the function x(t) [Eq. (VI.7)]. The direction of any path is specified by the requirement that for real x the path must evolve positive real t in classically

4.51

-x0

-x, b

FIG. 16. (a) Weight factors (b) Weight factors for paths with

for paths with one extra round

X,

X0

no reflection into the forbidden trip through the forbidden region.

region

1x1 C-Y,.

allowed regions and positive imaginary W in classically forbidden regions. In Fig. 17 we indicate with open and solid circles the zeros and poles, respectively, of the function x(t). This function assumes real values along the dashed lines shown in the figure. Horizontal lines correspond to motion in the allowed region and vertical lines to motion in the forbidden region. Figure 17a also shows a path associated with the saddle point t, + t,/2 + 1,. The path is indicated (as in Fig. 16a) to evolve purely real values of x. Deformations of the path which do not cross the singularities of x(t) are topologically equivalent and need not be illustrated separately. Any deformation which does cross one of the singularities would produce a path which is misdirected, i.e., a path topologically equivalent to one which evolves negative real time or negative imaginary W along some portion of its development. Since there is a unique path for the saddle we are considering, its weight in Fig. 15 is 1. Figure 17b shows representative paths for the saddle point t, + t,/2 + 2t,. There are two topologically distinct paths relevant for this saddle point. For clarity the paths illustrated in Fig. 17b have been deformed away from the real x contour of Fig. 16a. It is a simple matter to compute the contribution to G(xr, xi ; E) from all the saddle points of Eq. (VI.16). Each of the contributing paths involves a common phase w(x,, Xi) = w, + 2w, + WI,

(VI.21)

452

CARLITZ

AND

NICOLE

-+---,---,---k--+I b . ’ ox -+---I-;-----;I I 0 : I -+---,-I I

1

I P

.

I 0

I

+--+--+--&c-l--

I

I

: I -.--+---l

4

.

0 I

I bl

. I

-?---I--

FIG. 17. Structures of the classical solution x(r) for the potential (VIA). values of x. Open and solid circles indicate zeros and poles, respectively. topologically unique path from f = 0 to the saddle point T, = to + 1,/2 + I,. topologically distinct paths to the saddle point f, + 1,/2 + 21,.

Dashed lines denote real (a) Solid line shows the (b) Solid lines show two

where W, = id, = $2rr~)‘~~

-~Y-ix[V(x)-Ell~‘+

j~xfdx[v(x)-E]1~2 %I

describes the phase accumulated outside the two wells and 2w, + wi [Eqs. (VI.9) and (VI.12)] describes the phase accumulated on the obligatory passage from -x0 to x0. The explicit form of A, is A,= (g~a3x0/3fi~) where

Cf(Xi) +f(xf)I,

(VI.23)

f(x) = E(&x),k’)- (E/la4)1’2F(#(X), k’) (“x;~y +X “‘-xy:) ( x0>(

(VI.24)

The argument C$of the elliptic integrals E and F has been specified in Eq. (VI.19). Paths with n bounces in the allowed regions evolve a phase of 2nw, in addition to the phase w(x,, xi) of Eq. (VI.21). Summing all such paths, we obtain a contribution to G, G”‘(x,,

xi; E) = (i/~)(aiaf)-“2e~“a~d(1/(2

COSW~)‘).

(VI.25)

CLASSICAL

PATHS

AND

QUANTUM

453

MECHANICS

Comparison with Eq. (IV.27) makes it clear that we have simply counted all paths which proceed first from x, to some point inside the barrier (say, x = 0) and then from that point on to xr. On the passage through the left and right hand wells of Fig. 14 any number of bounces within the allowed region are included. Suppose now we consider saddle points along the second row of Fig. 15. These correspond to paths with three transits through the forbidden region between the two wells. Before and after each transit there are allowed an arbitrary number of bounces in one or more of the allowed regions. Thus there results an expression G” ‘(xi, x, ; E) = G”‘(x,,

’ P23 e ’

x, ; E)

since relative to Eq. (VI.25) there are two more transits of the forbidden region (yielding a suppression factor of e-*‘) preceded by path segments which start and end in the forbidden region and reflect an arbitrary number of times in one of the allowed regions. These latter path segments are precisely what entered in Eq. (IV.35), and hence a factor (sin wR/2 cos wR) results from each of them. The arguments just given are readily extended to the case with 2n + 1 transits of the forbidden region to give 2n

(VI.27) and hence

G(x,, -xi; E) = T G’“‘(x,,

x,: E)

,I= 0

(VI.28 It follows

that energy levels of the double-well

quantum

cos wlR = +$ eP”sin Equation (VI.29) i.e., only when

is valid only when the turning E$hJ

M’~.

system are given by (VI.29

points of Fig. 14 are far apart,

(VI.30)

and E, - E$iico,

(VI.3 1)

where E, = l,a4.

(VI.32)

454

CARLITZ

AND

NICOLE

It is not hard to extend our results to the case where one of the constraints (VI.30) or (VI.31) is violated. Suppose first that E is small. Then x,, is close to x, , and fluctuations about each of the saddle points in Fig. 15 become large. The contribution around each saddle is spread out over a region in t of order 6t N (h/Ew)”

(VI.33)

[see Eq. (IV.20) 3 while the separation of adjacent saddles along any row of Fig. 15 approaches a constant value t, NW-‘. (VI.34) It follows that individual saddles in any given row cannot be distinguished, and that the contribution of each row must be considered in toto. Different rows remain distinct since (VI.35)

ti - - (2i/o) ln(64Aa4/E) $61

in the limit we are considering. This means that while it is not meaningful to identify paths in the neighborhood of the turning points, it is still meaningful to distinguish paths according to the number of transits they make through the forbidden region and to discuss separately their individual contributions [G(“)] to G. Consider, for example, the term G(O), which results from paths which pass only once through the forbidden region. Any such path will comprise segments as illustrated in Fig. 18. The points x2 and xj are chosen to be close enough to a that quartic terms in V(x) are negligible at these points, i.e.,

and (VI.36)

(x3 - a) 6 (wz/A)‘/~~.

FIG. 18. Particle of energy E in the double-well potential. Intermediate to lie in the forbidden region away from the turning points x, and x0.

points

x2 and x3 are chosen

CLASSICAL

PATHS

AND

QUANTUM

MECHANICS

455

We require that .x2 and .x3 be far enough from the turning points that mw(x: +x:)/b

(VI.37)

1

[cf. Eq. (IV.4)]. Equations (VI.36) and (VI.37) guarantee that G(x,, x,; E) [and G( --x2, -x3 ; E)] can be treated by the methods of Section IV. In terms of the path segments of Fig. 18, G”’ may be written as a product G’O’(xu,, Xi; E)= G(x,,

E)

~3;

ih.t,G(~,,

x G(x2, -x2;

E)( -ih.t,)

~2;

E) it%, G( --x2, xi; E).

(VI.38)

Here .CJ, .t2, etc., denote velocities at the points x3, x2, etc., along the classical paths which build each factor of G in Eq. (VI.38). If each factor of G were evaluated in the steepest descent approximation, the product representation (VI.38) would follow trivially from Eq. (11.13). It is true more generally, provided the intermediate points -yj, x2, etc., are not too close to the turning point [and this is guaranteed by Eq. (VI.36)]. To prove Eq. (VI.38) or the equivalent statement that the semiclassical kernel K(x,, xi; T) can be written as a series of time convolutions [Zl], one need only substitute for K(x,, xi; T), K(x,, s3, t) etc., the expressions from Eqs. (11.8) and (11.10) and evaluate the convolution integral by steepest descents. Once again Eq. (VI.37) validates the steepest descent procedure. Saddle points of the convolution integral occur at such points as to conserve E,, along the various path segments. The Gaussian factors (aE,,/at)-“’ that result from the steepest descent evaluation then serve to cancel extra factors of (~?E,,/~t)i’~ coming from Eq. (11.10) and provide the result (VI.38). If E$>h; we could use the steepest descent approximation to evaluate the Laplace transform integral defining each of the factors of G in Eq. (VI.38). When E/h; is not small, then G(x,, x2; E) [and G( -x2, -x,; E)] cannot be evaluated by the steepest descent procedure. Since, however, V(x) is essentially quadratic between x2 and xg [as guaranteed by Eq. (VI.36)], the results from Section IV for this range of E/h; are immediately applicable. Thus in Eq. (VI.25) we simply make the substitution indicated by Eq. (IV.48) to obtain G”‘(xf,

x,; E) = (~~.~~)~“2e~do-dK2(E).

(VI.39)

A similar discussion for the expression G(“) arising from 2n + 1 transits of the forbidden region leads to a modification of Eq. (VI.27) to read G(“)(x~, x;; E) = G”‘(x,,

Xi; E)[K”(E)

sin2wR]‘“eP2”“.

(VI.40)

The final result for G(x,, xi; E) is thus (~~l~$-IIZe-4-d

G(x,, x, ; E) =

K-‘(E)

- eP2dsin2w,’

(VI.41)

456

CARLITZ

AND

NICOLE

In the language of the WKB method one would say that Eq. (VI.41) corrects Eq. (VI.28) to accommodate a quadratic connection formula for the neighboring turning points x,, and xi. Indeed the result (VI.41) is identical with that prescribed by the WKB approach. For low-lying levels the level splitting indicated by Eq. (VI.41) also coincides with that obtained by instanton methods. The instanton approach is rather different from ours; we will explore these contrasts in Section VII. The instanton method is inapplicable for E9>h;. Our method remains applicable even for EN E,, the case which we now consider. As E approaches E, the turning points fx, approach each other and generate large fluctuations for classical paths near x = 0. Since near x = 0 the double-well system reduces to the quadratic barrier, we can use the results of Section V to correct the steepest descent approximation (VI.28). In terms of the saddle points of Fig. 15 we find that as (E, - E)/h decreases there are growing fluctuations about each T, but a finite limit for t,. This means that vertical bands of saddles merge together. Since t, + co as E + E,, the columns of Fig. 15 remain distinct. Thus, as expected, the situation in the complex T plane resembles that of Fig. lob. It is convenient to rewrite the saddle point approximation (VI.28) in a manner which groups multiple transits through the forbidden region: G(x,, Xi; E) =

(VI.42)

The bracketed factors in the numerator and denominator correspond to estimates of the transmission and reflection coefficients deduced from the linear turning point rules of Table I. A discussion parallel to that given above for EN /ic~ shows that a proper evaluation of the merged vertical bands of saddle points in Fig. 15 converts the factors of T and R in Eq. (VI.42) to the values given in Eqs. (V.21) and (V.22). This then leads to an expression for G, (VI.43) which is valid for E-E,. The function f(E) is defined in Eq. (V.20). Equation (VI.43) is valid for E > EC as well as E < EC. Since, however, it was assumed in the derivation of (VI.43) that lx,1 E, and the problem is not too different from the quadratic barrier with E > 0, which we discussed in Section V.

CLASSICALPATHSANDQUANTUMMECHANICS

VII.

INSTANTON

457

METHODS

Many of the results obtained by classical path techniques in the previous section have been obtained previously via the instanton approach. Instantons also provide a natural framework for the discussion of tunneling phenomena in field theories. There are, however, limitations on the applicability of instanton methods which severely restrict their practical utility in many problems. These limitations arise when instantons and anti-instantons overlap extensively and interact appreciably. We will argue that classical path techniques remain applicable in this regime, that they provide answers not previously accessible via the instanton approach and that they provide useful clues toward extending instanton methods to encompass this regime. The instanton approach concentrates on the quantity K(x,, Xi; T) for large imaginary time, T= -iz.

(VII.1 )

The trace of K, Tr K(z) = j dx K(x, x; - iz),

(VII.2)

Tr K(z) = c epEk‘,

(VII.3)

has the behavior

k

Therefore an analysis of the large r behavior of K(s) is sufficient to determine the energy eigenvalues of the system in question. There are several obvious differences between the instanton method and the classical path method. One method concentrates upon the Feynman kernel K(x,, xi; T) and the other on its Laplace transform G(x,, xi; E). One takes xr= xi and averages over these coordinates while the other explicitly separates xr from x, and moves both far from any classical turning point. This restriction is imposed in the classical path method to ensure that the functional integral (11.5) be Gaussian. Since this restriction is dropped in the instanton approach, the functional integral in this approach is not always Gaussian. There are soft modes of fluctuation about some classical field conligurations and these modes must be separated for special treatment [4-61 in the evaluation of Eq. (11.5). This is actually a virtue of the instanton method. The soft modes of fluctuation are replaced by collective coordinates associated with the motion of individual instantons and anti-instantons. If the mutual interactions of widely separated instantons and anti-instantons are insignificant, there results in the large r limit a simple dilute instanton gas. For a given large r the contribution to Tr K(z) from the N-instanton sector of this gas is a quantity of order e - N3zN/N!

(VII.4)

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times a normalizing factor involving no instantons or anti-instantons. Here as in Eq. (VI.27) the quantity e-’ refers to the suppression induced by tunneling through the forbidden region of the potential. There is thus a direct correspondence between the N-instanton (and anti-instanton) term arising from the dilute instanton gas and the classical path contribution resulting from N transits of the forbidden region. A coefficient arises in the instanton method from the Jacobian of the transformation to collective coordinates [i.e., from the normalization of the soft modes removed from the integral (IIS)] and multiplies e -’ in Eq. (VII.4). This corresponds to the factors deduced from the analysis of the quadratic turning point formula in Section IV and corrects for the difference in energy between the instanton path (which has E = 0) and the paths used in Section IV (which are centered around E = ho/2). Therefore the result of the instanton method coincides with that of the classical path analysis for the splitting by quantum mechanical tunneling of the ground state of the double-well potential. What about excited states? From the classical path analysis there is certainly a non-trivial variation of the magnitude of the level splitting as one raises E above &0/2. There is, in particular, a transition from the region where the quadratic connection formula is de rigueur [i.e., where saddle points of the Laplace transform integral (11.1) are non-Gaussian] to a region where linear connection formulae are adequate (and the saddle points are all Gaussian). How is this transition achieved in the instanton approach? From Eq. (VII.3) it is clear that an analysis of the excited state energy splitting is equivalent to analysis of subdominant contributions to the asymptotic behavior of Tr K(t). This means that one must examine Tr K(z) for large but finite values of r and include in Eq. (VII.3) terms that were neglected in the ground state analysis. At finite z there is of course a finite separation (namely, r/N) for the instantons and anti-instantons of an N-kink solution. The non-vanishing interactions among the N kinks provide corrections to the dilute gas approximation and thereby permit an extension of the instanton method to evaluate the splitting by tunneling of excited states of the system [ 161. There are drawbacks to this approach which have limited the application of instanton methods for the double-well problem to energies EaE,

(VIIS)

and which have similarly limited applications of instantons in field theoretic systems to the domain where the instanton gas is still dilute. The relation of Eq. (VIM) to the condition of diluteness is found in Eq. (VI.35), which states that the imaginary period t, is large only when E is small. The drawback concerns first the proper identification of the instanton interaction and then the proper incorporation of the interaction into the gas of instantons and anti-instantons. The classical path method provides an answer to the first question and gives it in a form which may mitigate the difficulty of the second. A classical path contributes to K(x,, xi, -ir) an amount proportional to - ir)l/h]. For any given N, an N-kink solution contributes to S,, exp[:-l&h

xi;

CLASSICAL

PATHS

AND

QUANTUM

459

MECHANICS

a constant amount (proportional to N) in the limit r -+ 03. For finite z we can define the interaction between neighboring kinks as Sint(rrel) = (l/N)[S,r(.X,

X;

-ir)

- lim SCl(X, X; - ir)], r+*

where S,, in this expression refers to a classical solution kinks evolving a total time -it (with z real) and Trel = z/N.

with

N kinks

(VII.6) and anti(VII.7)

There is another definition [22], more typically employed in instanton interactions and equivalent to Eq. (VII.6) in the limit z/N+ co. This definition involves superposing kink and anti-kink solutions at a separation z,,, and computing the difference between the action S which results for this configuration and the action similarly evaluated in the limit r,,, + co. Note that for a potential of the form (VI.1 ) there is no superposition law and that the field configurations used in this definition are not actual classical paths of the system. There do exist classical paths which correspond to kinks and anti-kinks. In Fig. 17 these paths would be represented by vertical lines through the zeros of x(t). Just such periodic solutions have been invoked in the definition (VII.6). In this sense our definition is closer to the original aim of evaluating K(x,, Xi; T) by a steepest descent approximation than is the method of superposition. This distinction is irrelevant for large trel where either approach gives the same expression for Si”t. For smaller r,,, where the mutual deformation of neighboring kinks is large the definition (VII.6) remains valid while the definition by superposition is meaningless. For the double-well problem the new domain made accessible by the definition (VII.6) is the one where

EhE,,

(VII.8)

i.e., energy levels near the top of the hump in the double-well potential. Before specifying the form of S,,, that results from Eq. (VII.6) for instantons of the separation relevant for Eq. (VII.8), we must address the second problem engendered by instanton interactions. This concerns the sign of Sint(Trel) for large rre,, which is negative. [This result refers to the region where Eq. (VII.6) and the superposition method agree, so there is no question of inappropriate approximations here.] A negative value of S,,, means that instantons and anti-instantons attract each other at large distances [16]. If this attraction grows stronger at short distances, then the entire gas of kinks may condense. Just such a problem would seem to arise if we evaluate Sint(~re,) from Eq. (VII.6) for large z,,, and analytically continue the result toward z,,, = 0. This, however, is wrong, because in such an analytic continuation the paths x,, which build S,, in Eq. (VII.6) undergo a Stokes jump and spoil the analytic structure of S,,,. It is not hard to understand the nature of this Stokes discontinuity if we examine Fig. 18 once more. As E approaches EC1from below, the turning points +x, coalesce and large

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fluctuations result in the path x,,. Simultaneously, the contribution wr [Eq. (VI.lZ)] to the WKB phase, which results from passage through the classically forbidden region, decreases to zero. Above EC, the turning points fx, move into the complex plane. Paths which circle these turning points still evolve purely imaginary time, so one can still associate these paths with multi-kink solutions. The sign of d = -iwr, which was positive for E < E,, must remain positive for E > EC. Indeed it is the sign of A which determines the direction of passage along a given path through the classically forbidden region. Analytic continuation of wr from E < EC to E > EC would specify the wrong sign for w, and hence the wrong direction along the various classical paths. The discontinuity in i3w,/iYE which results at E = EC is evidence of the Stokes phenomenon cited above. In Fig. 19 we sketch the form of Sint(Zrel) which follows from Eq. (VII.6). For large positive rre, , Sint is negative but small, corresponding to the dilute gas regime when E/E,4 1. As E is raised, r,,, decreases and the attractive interaction becomes stronger. Stokes lines radiate from E = EC, corresponding to a rrel of (VII.9)

zc=23bc/w.

For E > EC, the relevant values of rre, are negative, and the corresponding Sint is positive. This indicates that tunneling to complex turning points involves a passage into the classically forbidden region and is an exponentially suppressed quantum process. One concludes that for E > E, the standard instanton description is no longer appropriate even if one incorporates instanton interactions. One might have expected that it would be impossible to generate instanton-anti-instanton configurations with arbitrarily small positive values of ~~~~~since in the double-well system instantons and anti-instantons must always alternate along a path in T = - i7. To understand the exact mechanism by which instantons and anti-instantons with 0 5 z,,~ 5 t, disappear from the system, one should probably study the instanton method for r 1: r, and extract the structure of the zero mode fluctuations which result in the limit r = r,. A

'in!

) _-.I Tc

-___----Tc

j

/-----T I

rel

/' I' : 1 I I I

FIG. 19. of T,, which

Instanton contribute

interaction as a function of spacing in imaginary time. Solid in the classical path method: for 0 < E < E,, T > zC, while for

curve

shows values

E > EC, -5, < T < 0.

CLASSICAL PATHSANDQUANTUM

VIII.

SUMMARY

MECHANICS

461

AND CONCLUSIONS

In previous sections we have described how classical paths build the functional integral for one-dimensional quantum mechanical systems in the limit /& 0. The results of the classical path method coincide with those of the WKB method for systems with isolated linear and quadratic turning points. These results go beyond those of previous instanton calculations and show directions in which the instanton approach may be generalized. More specifically, the classical path approach provides a description of quantum mechanical systems involving overlapping and strongly interacting instanton configurations. Field theories often exhibit classical paths of this sort and thus demand an extension of the instanton method beyond the customary dilute gas regime. Our discussion of complex classical paths is not without precedent, and many of the points raised in this paper appear in some form in other related work [21]. In particular, subtleties of the WKB method associated with the Stokes phenomenon have been known for some time. For readers unfamiliar with the subject we recommend the paper of Berry and Mount [8] cited many times in previous sections of this paper. The relationship of Stokes phenomena to instanton methods has not been emphasized explicitly in previous papers with which we are familiar, and this omission has led to a certain amount of confusion in the literature. One finds factors of f missing or inserted surreptitiously in the contributions of subdominant saddle points. Connection formulae for linear turning points have been misapplied to quadratic systems and mysterious factors of (e/~)‘!* have been thereby engendered. Non-contributing saddle points have been occasionally included in sums analogous to Eq. (11.8). We will not attempt to enumerate specific instances of these errors but will only warn our readers that they exist. To avoid them one must not routinely treat the integral (11.1) as if it were Gaussian; one must treat carefully all paths with endpoints in the neighborhood of classical turning points; and one must avoid the analytic continuation of asymptotic expansions past Stokes discontinuities. Our own work owes much to that which precedesit. Recent interest in applying semiclassical methods to field theory was stimulated by the work of Dashen, Hasslacher and Neveu [12]. Extensions of these methods to quantum tunneling phenomena were made by Lapedes and Mottola [14]. And the idea that classical paths describe instanton-anti-instanton field configurations was forcefully stated in papers by Richard and Rouet [23,24]. It is typically not possible to categorize all the classical paths of a given field theory. Nonetheless significant paths corresponding to an instanton or to multiinstantons often exist. We believe that paths corresponding to mixed instanton-anti-instanton configurations should also exist. Quantum mechanical examples suggestthat a key feature of such paths is periodicity. These solutions do not have finite action but only a finite action per unit volume (or unit time in the quantum mechanics case). Consequently they have typically been ignored in past searchesfor instanton solutions. They are also unstable, because there are negative modes of

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fluctuation about these solutions, which indicate that at large distances instantons and anti-instantons attract. In our quantum mechanical example this attraction does not persist to arbitrarily short separation distances, since instantons and antiinstantons disappear below a critical separation r,. An analogous phenomenon would be very welcome in certain field theoretic systems, which are marred by divergent instanton size integrals. We advocate a search for this phenomenon in systems which exhibit a dilute instanton gas regime. In this regime a symmetric and periodic configuration of instantons and anti-instantons will be stabilized by the mutual attraction of each neighboring pair. This provides the metastable solutions that we seek. As the instanton density is raised, the attractive forces will increase; but if our quantum mechanical example is a good analogy, there will be a critical density above which the instantons and anti-instantons will disappear. If this happens the instanton gas will be stable. Dense configurations will be excluded, and a well-defined model of interacting instantons will be available for computation.

REFERENCES AND A. R. HIBB~, “Q uan t urn Mechanics and Path Integrals,” McGraw-Hill, New York, 1965. 2. L. S. SCHULMAN, “Techniques and Applications of Path Integration”, Wiley-Interscience, New York, 1981. 3. M. P. GUTZWILLER, J. Math. Phys. 8 (1967), 1979; 10 (1969), 1004; 11 (1970), 1791; 12 (1971), 343. 4. S. COLEMAN, Whys.Rev. D 15 (1977), 2929; erratum 16 (1977), 1248. 5. C. CALLAN AND S. COLEMAN, Phys. Rev. D 16 (1977) 1762. 6. E. GILDENER AND A. PATRASCIOIU, Phys. Rev. D 16 (1977), 423; erratum 16 (1977), 3616. 7. N. J. GUNTHER, D. A. NICOLE, AND D. J. WALLACE, J. Phys. A 13 (1980), 1755. 8. M. V. BERRY AND K. E. MOUNT, Rep. Progr. Phys. 35 (1972), 315. 9. H. JEFFREYSAND B. SWIRLES, “Methods of Mathematical Physics,” Chap. 17, Cambridge Univ. Press, Cambridge, 1972. 10. I. AFFLECK, Phys. Rev. Left. 46 (1981), 388. 11. R. BALIAN, G. PARISI, AND A. VOROS, Phys. Rev. Lets. 41 (1978), 1141. 12. R. F. DASHEN, B. HASSLACHER, AND A. NEVEU, Phys. Rev. D 10 (1974), 4114. 13. D. W. MCLAUGHLIN, J. Math. Phys. 13 (1972) 1099. 14. A. LAPEDE~ AND E. MOITOLA, Nucl. Phys. B 203 (1982), 58. 15. R. B. DINGLE, “Asymptotic Expansions: Their Derivation and Interpretation,” Academic Press, London/New York, 1973. 16. J. ZINN-JUSTIN, Nucl. Phys. B 192 (1981), 125; 218 (1983), 333; in “Recent Advances in Field Theory and Statistical Mechanics” (J.-B. Zuber and R. Stora, Eds.), North-Holland, Amsterdam, 1984. 17. W. H. FURRY, Phys. Rev. 71 (1947), 360. 18. S. LEVIT, J. W. NEGELE, AND Z. PALTIEL, Phys. Rev. C 22 (1980), 1979. 19. A. PATRASCIOIU,Phys. Rev. D 24 (1981), 496. 20. H. K. SHEPARD, Phys. Rev. D 27 (1983), 1288. 21. U. WEISS AND W. HAEFFNER,Phys. Rev. D 27 (1983), 2916. 22. C. G. CALLAN, R. F. DASHEN, AND D. J. GROSS,Phys. Rev. D 17 (1978), 2717. 23. J. L. RICHARD AND A. ROIJET, Nucl. Phys. B 183 (1981) 251. 24. J. L. RICHARD AND A. ROLJET,Nucl. Phys. B 185 (1981), 47. 1. R. P. FEYNMAN