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On the global nature of the semiclassical wavefunction describing elastic scattering in several coordinate dimensions
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
ON THE GLOBAL NATURE OF THE SEMICLASSICAL WAVEFUNCTION DESCRIBING ELASTIC SCATTERING IN SEVERAL COORDINATE DIMENSIONS * B.J.B. CROWLEY University of Oxford, Department of Theoretical Physics, Oxford OXI 3NP, UK Received 16 October 1978 Revised manuscript received 16 February 1979
The short-wavelength (semiclassical) limit of the quantal wavefunction in a multidimensional eudidean configuration space is discussed. Attention is drawn to the shortcomings of some previous approaches with regard to the treatment of multivaluedness and caustics. A global representation of the wavefunction based on generalised Kirchoff diffraction integrals is proposed. Application of the saddle-point approximation leads to a local semiclassical representation of the wavefunction as a series of JWKB approximants.
In their recently published letter in Phys. Lett. A, Korsch and Möhlenkamp [1] raise a number of interesting questions regarding the nature of the semiclassical limit of the quantal wavefunction in a multidimensional euclidean configuration space. However, their analysis deserves further comment concerning the implications of multivaluedness associated with solutions of the classical Hamilton—Jacobi problem. Multivaluedness merits very careful consideration, particularly when discussing global semiclassical wavefunctions, in view of the fact that several of the usual approaches to the semiclassical approximation fall to yield a selfconsistent and well-posed mathematical description when multivaluedness is taken into account. For example, following Korsch and Mohlenkamp, substitution of the trial function ~L’(r)‘g(r)exp[(i/Fl)W(r)]
(1)
into the Schrodinger equation describing a particle of mass m and energy E moving in a potential V(r) leads to the following equations: 2
=
(1/2m)(VW)
+
v•’
0
2vrAr~— —
V+ ‘
r’~u
E,
(2) (3\ ~
~ Work supported by research grant from Science Research Council (UK).
186
‘
in which the so-called quantum potential is given by (4 = ~2 ‘2 ~ V2 qu k / m,g g. For given g(r), eq. (2) is the classical Hamilton—Jacobi equation for the characteristic function W describing motion in the potential V + Vqu. The solutions of this equation are, in general, many-valued functions of r and this in turn leads to correspondingly many-valued solutions of eq. (3). The quantum potential (4) is therefore itself a many-valued function of the coordinates, and consequently the nature of any solutions to eqs. (2) and (3) becomes complicated and difficult to understand. A treatment of the many-valuedness of the solutions of the Hamilton—Jacobi problem is possible if eqs. (2) and (3) can be decoupled. This involves the prior assumption that Vqu 0 in the semiclassical limit (“11 -+0” [2]). Eqs. (2) and (3) then describe the corresponding classical problem and no longer present any formal difficulties. However, one has sacrificed the description relating the many-valued local JWKB approximant(s) (1) to the global semi—~
—
-~
classical wavefunction. Even if it were true (and in general it is not) that the wavefunction is locally dominated by one single-valued JWKB term (of the form of eq. (1)), one still has the problem of connecting the wavefunction in different regions where the solution is characterised by different dominant terms.
Volume 71 A, number 2,3
PHYSICS LEUERS
This is the nature of the connection problem in many dimensions and the change in the analytical structure of the semiclassical limit of the wavefunction in passing from one such (complex) coordinate domain to another is a generalisation of the Stokes phenomenon [2—4]. At this point we would like to stress that, in the treatment of eqs. (l)—(4) and also in what follows, it is incorrect to impose global conditions to the effect that either or both ofg and W be real and/or singlevalued. No advantage would seem to be offered by this procedure. Moreover, the mathematics is considerably (if not impossibly in the case of complex trajectories) by unnecessary realitycondiconditions and complicated lack of analyticity, while imposing tions of single-valuedness can only lead to inconsistencies. The other approach, which received critical mention in ref. [I], involves the ubiquitous expansion of the quantum phase in powers of it. This depends upon the unjustifiable assumption that such an expansion exists, and leaves unresolved difficulties associated with multivaluedness and the connection problem. In the following we suggest an alternative approach that avoids some of the difficulties mentioned above, and, in particular, permits a proper treatment of the role of multivaluedness of solutions of the classical HJ problem. The method possesses the attribute of mathematical rigour. However, space restrictions prevent a comprehensive account here and only an outline omitting much of the mathematical detail is possible. Further information will be found in the cited references as well as in future publications. Let 9((p, q), which is independent of time, be the hamiltonian of an integrable system whose motion is describable in an n-dimensional eucidean space of n independent coordinates. These are represented by q whose (cartesian) components are conjugate to the n components of the canonical momentum p. We assume only that (i) the components of q are independent and that any holonomic constraints have already been applied in the removal of redundant coordinates; (ii) ~ (p, q) V(q) is a quadratic form in the cartesian components ofp A(q), whereA(q) is a vector potential; and (iii) ~( is a regular function of q in a domain which includes all points for which q is real. For ifiustration we consider a generalised scattering problem in which we introduce the additional assump—
—
30 April 1979
tions (iv) that 9~has a well defined limit (K(p)), and vanishing derivativeswith respect to q in the “asymptotic region” q oo• The time-independent Schrödinger equation for the system is: -+
-
~ (—lit a/aq, q)1i(q) = E~/i(q), in which a/aq = (a/aq1, a/aq2,
(5)
a/aq~)= V is the generalised n-dimensional gradient operator. The notation q, is used to denote the ith cartesian component of q. Substituting the trial solution, ...,
~i(q) ag(q,P)exp[(i/h)W(q, F)] (6) and insisting that W and g 2) satisfy 0 the equations (7 8) 9C(V ~ q) = E V (ug (where u a9C lop is, by hamilton’s equations [6], the velocity, i’), leads to the sufficient requirement that a satisfies 2guVa=ihV (ag).
(9)
In the above, P denotes the n constant momentum components used in specifying the scatteringboundary conditions (eg. “incoming” momenta). Eq. (7) is the classical HJ equation for the characteristic function W, while eq. (8) is a continuity equation in which g2 appears as the density. The advantage of this approach is that eqs. (7) and (8) are decoupled prior to taking the limit “it -÷ 0”. The construction of a, global semiclassical wavefunction from the solutions of eqs. (7) and (8) involves studying the behaviour of the appropriate solutions of eq. (9) in the semiclassical limit. Thus, even without taking the limit “it 0”, this approach has apparently succeeded in separating the problem into specifically classical and quantal parts. The solutions of eqs. (7) and (8) describe the classical microcanonical ensemble (for fixed P)as families of dynamical trajectories. These solutions are manyvalued in coordinate space wherever different trajectories, or different parts of the same trajectory, pass through the same point. Associated with this is the problem of singularities of g2 where l/g2 vanishes and where a may be expected to exhibit rapid or discontinuous behaviour. For these reasons, the local JWKB approximant (1) does not, except in trivial cases, provide a valid global approximation to the quantal wavefunction. The occurrence of singularities in manyvalued solutions of this type is directly associated with the presence of caustics [7] which are the dominant -~
187
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
generic features of the wavefield in the short-wavelength (semiclassical) limit. Thus we are not concerned here with difficulties of a purely formal nature. The direct connection with physical reality precludes any possibility of their avoidance. The characteristic function W, which is the particular integral of eq. (7), is the generating function of a canonical transformation [6], (p, q) (F, Q), whereby g((p, q) becomes K(F) and is cyclic in all the Q~• The transformation is:
side of eq. (11) is a local representation of the wavefunction and yields a local semiclassical wavefunction in the limit “it -÷ 0” when the coefficients a~become constants. Thus far the approach has been heuristic. We now introduce an element of mathematical rigour by noting that a uniformly valid global approximation, that is equivalent to eq. (11) in the semiclassical limit, is provided by a representation of the wavefunction in the form of a generalised Kirchoff diffraction integral.
P = 0 W(q;P)/Oq, Q = 0 W(q; P)/OP. (10) Parameterising the motion in terms of time t yields that q = Q(t) represents unperturbed motion of the system (when C?( has its asymptotic form K(p) everywhere). The mapping Q(t) q(t) therefore provides
That is
-~
-+
a complete representation of the particle’s motion, and is in general many-to-one. Its representation in terms of a generating function provides a logical description of multivaluedness of solutions of the HJ problem. The required particular integral of eq. (8) is the Van Vieck determinant [8,5], 2 = det[02W/Oq-OP-] = [O(Q)/O(q)] g I which is the jacobian of the mapping q -÷ Q. Points where 1 /g2 = 0 define the caustic surfaces in the systern of trajectories pararneterised by Q. A description of caustic structures is provided by catastrophe theory [9—11].The mapping Q q, which is continuous and single valued, defines a manifold CX which contains the dynamical trajectories as generators and on which W(q; F) possesses a single-valued representation. Unfortunately, it is now apparent that eq. (9) becomes undefined at caustics owing to the presence there of singularities in g, and to the inability to treat the different branches of a many-valued function such as u separately. However, in a coordinate domain in which there are no caustics, the function (6) may be an exact solution of the Schrodinger equation. The ‘
-+
quantal wavefunction,which must satisfy global conditions of regularity, may be represented in such a domain as a linear superposition of all valid solutions and may therefore be expressed in the form: i,1i(q) =
E a~g~(q, F)exp [(i/h)W~(q,F)] ,
(11)
where ~ labels a trajectory through q. The right-hand 188
=
2
(2irih)_hh/
x fMhI2f(Q;q;P)e(iI P(Q;q;P)dnQ , where M(Q; q; F)
=
(12)
det [02ct/0Q
1 8q1] and 1 is a generating function (sometimes called a “universal unfolding”) which is a single-valued extension of the action and which is defined so that: (i) the gradient map, O~/OQ= 0 ~ OW/OF Q,M ~ 0, and thus defmes CX as the critical manifold containing the saddle points of 4 with respect to Q. (ii) On CX , ~ is everywhere equal =toW(q;P); the characteristic function. Hence ~t(OW/OP; q;P) (iii) The functionMf contains only integrable singularities. In addition we defme f, for which ~,1i, given by eq. (12), satisfies eq. (5), so that: (iv) on, and in the neighbourhood of, CX ,f possesses a uniform asymptotic expansion in positive powers of it, for it 0. This last constraint may lead to the general wavefunction being expressible, not as a single term in the form of eq. (12), but as a sum of such terms. Integrals of the type (12) have been described in the work of Duistermaat [17], Berry [11,19] and others [10,18], and establish a direct connection with catastrophe theory [9—11]in cases when 4 may be locally mapped into a function involving the generator of an elementary catastrophe by means of a sequence of one—one uniform transformations. At a saddle point v where 04/OQ vanishes, g2 det [02 W/Oq~OF 1] = M~IH~. Therefore, evaluation of the integral (12) using the saddle-point approximation (e.g. ref. [171) yields —~
i,li
~~
,
(13)
2)~14v (where .z where c~,= 0 or e(hI 7, is an integer) according to the nature of the saddle point, and f~= f(Q~q;F).
Volume 71 A, number 2,3
30 April 1979
PHYSICS LETrERS
The result (13) establishes the connection between the representation (12) and a semiclassical wavefunction having the form of eq. (11). It is also clear that there are circumstances under which the saddle-point approximation fails, such as at caustics which are characterised by zeros of H = det [a2cF/0Q1 0Q1] on CX. Therefore, infinities associated with caustics arise through misapplication of the saddle-point approximation. The wavefunction (12) is regular at caustics and provides a basis for uniform and transitional approximations [11] valid in domains containing caustics. The representation of the wavefunction in the form of eq. (12) thus provides the means of copidg with difficulties associated with multivaluedness and caustics. We can now concern ourselves with the properties of an analytic function namely the integrand of eq. (12), and proceed to investigate the properties of the function f(Q; q; F) for which ,t’ is an exact or approximate solution of the Schrodinger equation. The property of this function that it possesses a uniform asymptotic expansion in positive powers of it, for it 0, gives rise to the possibility of generating an asymptotic expansion for R(q) = (~( E)~(i(q)in terms involving integrals of the type (12). In particular, it can be shown that, when f is independent of q and Q, R(q) = O(h 2) away from caustics, while at a caustic R(q) = O(h 20), where a (<2) is the singularity index of the caustic. This confirms the earlier suggestion that, away from caustics, the quantities a~= c~f~ behave, in the semiclassical limit, like constants. In this limit, the integral (12) may exhibit the Stokes phenomenon [2—4],in which case it is found that a~take the roleinof Stokes multipliers [2] This the is clear, for on example, the treatment [3,18—20] of the integral representation of the Airy function (in which n = 1, f 1, 4) = (q + P)Q ~Q3)by which the aforementioned procedure provides the exact quantal solution in the case of a linear potential. It will be noticed, however, that, when 4) is a “universal unfolding”, the Stokes phenomenon arises only in the treatment of the asymptotic evaluation of the integral (12) in the semiclassical limit. When a “universal unfolding” exists, the connection problem [2,4] is not an essential part ofthe semiclassical method, In general, it would appear [21] that there does not exist a single universal function 4) leading to an expression of the wavefunction in the form of a single diffraction integral (12). Thus, generically, a wave—
-+
—
-
—
function will be expressible as a sum of terms, each of which has the form of eq. (12). There remains the problem of how the functions 4) may be found. Practical approaches to solving the HJ equation include the method of separation of variables when components of Q appear as separation constants. This provides the solution on a series of one-dimensional coordinate submanifolds spanning a subspace of {q, Q}. These solutions in one dimension may exhibit Stokes phenomena (such as one finds in connection with the presence of linear and parabolic barriers) and may often be conveniently represented in terms of a multiple-reflection series or its equivalent wave-propagation matrix expression- [151 Such a representation may be obtained by means of local uniform approximations [12,14] in which the wavefunction near a turning point is locally expressible in terms of Airy or para. bolic cylinder functions. In this way a global representation of the wavefunction may involve a set of generating functions {4)(~~~} associated with which is a set 1~CX(ic)} of contiguous manifolds, CX (“) = {q, Q I a4)(K)/OQ 0). The complete critical manifold CX generated by all real and complex trajectories representing solutions of the classical HJ problem is the union of all the manifolds belonging to this set. An approximation to the wavefunction, uniformly valid in some open domain D{q} in which none of the CX have boundaries (other than Stokes lines) is as follows: -
(K)
= 7-’
/
X
(2mh\~ ‘
/
f E [M(”~jl/2f(K)e(iIli )
~n
(14)
K
A representation of the same wavefunction in terms of a universal generating function, as in eq. (12), is, by the saddle-point approximation, equivalent in D{q} to the above in the semiclassical limit. In another paper [21] we demonstrate that a representation of the form of eq. (14) is applicable in the treatment of scattering by a central field in three dimensions where it leads to accurate representations of the wavefunction and smatrix as partial-wave series. Justification is also given of the wavefunctions described in ref. [5] through use of the saddle-point approximation. The functions f(K) appearing in eq. (14) consist of multiple reflection coefficients such as arise in the wave-propagation matrix treatment [15,16] of the 189
Volume hA, number 2,3
PHYSICS LETTERS
one-dimensional radial problem, and may contain factors that depend on the form(s) of quantisation condition(s) through the value(s) of the appropriate Maslov indices [24]. Finally we observe that, as well as providing global representations of the quantal wavefunction in the semiclassical limit, this procedure also leads to a cornplete unification of dynamical [12,22] and diffractive [23]short-wavelength scattering theories. References
30 April 1979
[9] R. Thom, Structural stability and morphogenesis (Benjamin, Reading, MA, 1975). [10] T. Poston and I.M. Stewart, Catastrophe theory and its applications (Pitman, London, 1978). Eli] M.V. Berry, Adv. Phys. 25 (1976)1. [121J. Knoll and R. Schaeffer, Ann. Phys. (NY) 97 (1976) 307.
[13] J.B. Keller, Proc. Symp. Appl. Math. 8 (1958) 27. [14] D.M. Brink and N. Taidgawa, Nucl. Phys. A279 (1977) [15] 159. S.Y. Lee and N. Takigawa, Nuci. Phys. A308 (1978) 189. [16] N. Takigawa, talk presented at the second Louvain—
Krakow Seminar (Louvain-La-Neuve, Belgium, June 1978), to be published.
[1] H.J. Korsch and R. Mohlenkamp, Phys. Lett. 67A (1978) 110. [2] M.V. Berry and K.E. Mount, Rep. Prog. Phys. 35 (1972)
315. [3] G.G. Stokes, Mathematical and physical papers (Cambridge U.P., London) (1904) pp. 77—109, 283—98; (1905) pp. 221—5, 283—7. [4] E.C. Kemble, Fundamental principles of quantum mechanics (McGraw-Hill, New York, 1937). [5]BJ.B. Crowley, submitted to Phys. Rep. (1978). [6] H. Goldstein, Classical mechanics (Addison-Wesley, New York, 1950)Ch. 7—9. [7] O.N. Stravroudis, The optics of rays, wavefronts and caustics (Academic Press, New York, 1972). [8] J.H. Van Vleck, Proc. Nat. Acad. Sci. USA 14 (1928) 178.
190
[17] J.J. Duistermaat, Commun. Pure Appl. Math. 27 (1974) 207.
[18] J.N.L. Connor, Mol. Phys. 31(1976) 33. [19] M.V. Berry, Proc. Phys. Soc. 89 (1966) 479. [20] C. Chester, B. Friedman and F. Ursell, Proc. Camb. Phil. Soc. 53 (1957) 599. [211 B.J.B. Crowley, submitted to Phys. Lett. A (1979). [22] R. Schaeffer, in: Nuclear physics with heavy ions and mesons, Proc. session XXX (Les Houches, 1977), Vol. I, eds. R. Balian, M. Rho and G. Ripka (North-Holland, Amsterdam, 1978). [23] W.E. Frthn, in: Heavy-ion, high-spin states and nuclear structure, Proc. extended seminar (Trieste, 1973), Vol. I (IAEA, Vienna, 1975) p. 157. [24] I.C. Percival, Adv. Chem. Phys. 36 (1977) 1.
PHYSICS LETTERS
30 April 1979
ON THE GLOBAL NATURE OF THE SEMICLASSICAL WAVEFUNCTION DESCRIBING ELASTIC SCATTERING IN SEVERAL COORDINATE DIMENSIONS * B.J.B. CROWLEY University of Oxford, Department of Theoretical Physics, Oxford OXI 3NP, UK Received 16 October 1978 Revised manuscript received 16 February 1979
The short-wavelength (semiclassical) limit of the quantal wavefunction in a multidimensional eudidean configuration space is discussed. Attention is drawn to the shortcomings of some previous approaches with regard to the treatment of multivaluedness and caustics. A global representation of the wavefunction based on generalised Kirchoff diffraction integrals is proposed. Application of the saddle-point approximation leads to a local semiclassical representation of the wavefunction as a series of JWKB approximants.
In their recently published letter in Phys. Lett. A, Korsch and Möhlenkamp [1] raise a number of interesting questions regarding the nature of the semiclassical limit of the quantal wavefunction in a multidimensional euclidean configuration space. However, their analysis deserves further comment concerning the implications of multivaluedness associated with solutions of the classical Hamilton—Jacobi problem. Multivaluedness merits very careful consideration, particularly when discussing global semiclassical wavefunctions, in view of the fact that several of the usual approaches to the semiclassical approximation fall to yield a selfconsistent and well-posed mathematical description when multivaluedness is taken into account. For example, following Korsch and Mohlenkamp, substitution of the trial function ~L’(r)‘g(r)exp[(i/Fl)W(r)]
(1)
into the Schrodinger equation describing a particle of mass m and energy E moving in a potential V(r) leads to the following equations: 2
=
(1/2m)(VW)
+
v•’
0
2vrAr~— —
V+ ‘
r’~u
E,
(2) (3\ ~
~ Work supported by research grant from Science Research Council (UK).
186
‘
in which the so-called quantum potential is given by (4 = ~2 ‘2 ~ V2 qu k / m,g g. For given g(r), eq. (2) is the classical Hamilton—Jacobi equation for the characteristic function W describing motion in the potential V + Vqu. The solutions of this equation are, in general, many-valued functions of r and this in turn leads to correspondingly many-valued solutions of eq. (3). The quantum potential (4) is therefore itself a many-valued function of the coordinates, and consequently the nature of any solutions to eqs. (2) and (3) becomes complicated and difficult to understand. A treatment of the many-valuedness of the solutions of the Hamilton—Jacobi problem is possible if eqs. (2) and (3) can be decoupled. This involves the prior assumption that Vqu 0 in the semiclassical limit (“11 -+0” [2]). Eqs. (2) and (3) then describe the corresponding classical problem and no longer present any formal difficulties. However, one has sacrificed the description relating the many-valued local JWKB approximant(s) (1) to the global semi—~
—
-~
classical wavefunction. Even if it were true (and in general it is not) that the wavefunction is locally dominated by one single-valued JWKB term (of the form of eq. (1)), one still has the problem of connecting the wavefunction in different regions where the solution is characterised by different dominant terms.
Volume 71 A, number 2,3
PHYSICS LEUERS
This is the nature of the connection problem in many dimensions and the change in the analytical structure of the semiclassical limit of the wavefunction in passing from one such (complex) coordinate domain to another is a generalisation of the Stokes phenomenon [2—4]. At this point we would like to stress that, in the treatment of eqs. (l)—(4) and also in what follows, it is incorrect to impose global conditions to the effect that either or both ofg and W be real and/or singlevalued. No advantage would seem to be offered by this procedure. Moreover, the mathematics is considerably (if not impossibly in the case of complex trajectories) by unnecessary realitycondiconditions and complicated lack of analyticity, while imposing tions of single-valuedness can only lead to inconsistencies. The other approach, which received critical mention in ref. [I], involves the ubiquitous expansion of the quantum phase in powers of it. This depends upon the unjustifiable assumption that such an expansion exists, and leaves unresolved difficulties associated with multivaluedness and the connection problem. In the following we suggest an alternative approach that avoids some of the difficulties mentioned above, and, in particular, permits a proper treatment of the role of multivaluedness of solutions of the classical HJ problem. The method possesses the attribute of mathematical rigour. However, space restrictions prevent a comprehensive account here and only an outline omitting much of the mathematical detail is possible. Further information will be found in the cited references as well as in future publications. Let 9((p, q), which is independent of time, be the hamiltonian of an integrable system whose motion is describable in an n-dimensional eucidean space of n independent coordinates. These are represented by q whose (cartesian) components are conjugate to the n components of the canonical momentum p. We assume only that (i) the components of q are independent and that any holonomic constraints have already been applied in the removal of redundant coordinates; (ii) ~ (p, q) V(q) is a quadratic form in the cartesian components ofp A(q), whereA(q) is a vector potential; and (iii) ~( is a regular function of q in a domain which includes all points for which q is real. For ifiustration we consider a generalised scattering problem in which we introduce the additional assump—
—
30 April 1979
tions (iv) that 9~has a well defined limit (K(p)), and vanishing derivativeswith respect to q in the “asymptotic region” q oo• The time-independent Schrödinger equation for the system is: -+
-
~ (—lit a/aq, q)1i(q) = E~/i(q), in which a/aq = (a/aq1, a/aq2,
(5)
a/aq~)= V is the generalised n-dimensional gradient operator. The notation q, is used to denote the ith cartesian component of q. Substituting the trial solution, ...,
~i(q) ag(q,P)exp[(i/h)W(q, F)] (6) and insisting that W and g 2) satisfy 0 the equations (7 8) 9C(V ~ q) = E V (ug (where u a9C lop is, by hamilton’s equations [6], the velocity, i’), leads to the sufficient requirement that a satisfies 2guVa=ihV (ag).
(9)
In the above, P denotes the n constant momentum components used in specifying the scatteringboundary conditions (eg. “incoming” momenta). Eq. (7) is the classical HJ equation for the characteristic function W, while eq. (8) is a continuity equation in which g2 appears as the density. The advantage of this approach is that eqs. (7) and (8) are decoupled prior to taking the limit “it -÷ 0”. The construction of a, global semiclassical wavefunction from the solutions of eqs. (7) and (8) involves studying the behaviour of the appropriate solutions of eq. (9) in the semiclassical limit. Thus, even without taking the limit “it 0”, this approach has apparently succeeded in separating the problem into specifically classical and quantal parts. The solutions of eqs. (7) and (8) describe the classical microcanonical ensemble (for fixed P)as families of dynamical trajectories. These solutions are manyvalued in coordinate space wherever different trajectories, or different parts of the same trajectory, pass through the same point. Associated with this is the problem of singularities of g2 where l/g2 vanishes and where a may be expected to exhibit rapid or discontinuous behaviour. For these reasons, the local JWKB approximant (1) does not, except in trivial cases, provide a valid global approximation to the quantal wavefunction. The occurrence of singularities in manyvalued solutions of this type is directly associated with the presence of caustics [7] which are the dominant -~
187
Volume 71A, number 2,3
PHYSICS LETTERS
30 April 1979
generic features of the wavefield in the short-wavelength (semiclassical) limit. Thus we are not concerned here with difficulties of a purely formal nature. The direct connection with physical reality precludes any possibility of their avoidance. The characteristic function W, which is the particular integral of eq. (7), is the generating function of a canonical transformation [6], (p, q) (F, Q), whereby g((p, q) becomes K(F) and is cyclic in all the Q~• The transformation is:
side of eq. (11) is a local representation of the wavefunction and yields a local semiclassical wavefunction in the limit “it -÷ 0” when the coefficients a~become constants. Thus far the approach has been heuristic. We now introduce an element of mathematical rigour by noting that a uniformly valid global approximation, that is equivalent to eq. (11) in the semiclassical limit, is provided by a representation of the wavefunction in the form of a generalised Kirchoff diffraction integral.
P = 0 W(q;P)/Oq, Q = 0 W(q; P)/OP. (10) Parameterising the motion in terms of time t yields that q = Q(t) represents unperturbed motion of the system (when C?( has its asymptotic form K(p) everywhere). The mapping Q(t) q(t) therefore provides
That is
-~
-+
a complete representation of the particle’s motion, and is in general many-to-one. Its representation in terms of a generating function provides a logical description of multivaluedness of solutions of the HJ problem. The required particular integral of eq. (8) is the Van Vieck determinant [8,5], 2 = det[02W/Oq-OP-] = [O(Q)/O(q)] g I which is the jacobian of the mapping q -÷ Q. Points where 1 /g2 = 0 define the caustic surfaces in the systern of trajectories pararneterised by Q. A description of caustic structures is provided by catastrophe theory [9—11].The mapping Q q, which is continuous and single valued, defines a manifold CX which contains the dynamical trajectories as generators and on which W(q; F) possesses a single-valued representation. Unfortunately, it is now apparent that eq. (9) becomes undefined at caustics owing to the presence there of singularities in g, and to the inability to treat the different branches of a many-valued function such as u separately. However, in a coordinate domain in which there are no caustics, the function (6) may be an exact solution of the Schrodinger equation. The ‘
-+
quantal wavefunction,which must satisfy global conditions of regularity, may be represented in such a domain as a linear superposition of all valid solutions and may therefore be expressed in the form: i,1i(q) =
E a~g~(q, F)exp [(i/h)W~(q,F)] ,
(11)
where ~ labels a trajectory through q. The right-hand 188
=
2
(2irih)_hh/
x fMhI2f(Q;q;P)e(iI P(Q;q;P)dnQ , where M(Q; q; F)
=
(12)
det [02ct/0Q
1 8q1] and 1 is a generating function (sometimes called a “universal unfolding”) which is a single-valued extension of the action and which is defined so that: (i) the gradient map, O~/OQ= 0 ~ OW/OF Q,M ~ 0, and thus defmes CX as the critical manifold containing the saddle points of 4 with respect to Q. (ii) On CX , ~ is everywhere equal =toW(q;P); the characteristic function. Hence ~t(OW/OP; q;P) (iii) The functionMf contains only integrable singularities. In addition we defme f, for which ~,1i, given by eq. (12), satisfies eq. (5), so that: (iv) on, and in the neighbourhood of, CX ,f possesses a uniform asymptotic expansion in positive powers of it, for it 0. This last constraint may lead to the general wavefunction being expressible, not as a single term in the form of eq. (12), but as a sum of such terms. Integrals of the type (12) have been described in the work of Duistermaat [17], Berry [11,19] and others [10,18], and establish a direct connection with catastrophe theory [9—11]in cases when 4 may be locally mapped into a function involving the generator of an elementary catastrophe by means of a sequence of one—one uniform transformations. At a saddle point v where 04/OQ vanishes, g2 det [02 W/Oq~OF 1] = M~IH~. Therefore, evaluation of the integral (12) using the saddle-point approximation (e.g. ref. [171) yields —~
i,li
~~
,
(13)
2)~14v (where .z where c~,= 0 or e(hI 7, is an integer) according to the nature of the saddle point, and f~= f(Q~q;F).
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PHYSICS LETrERS
The result (13) establishes the connection between the representation (12) and a semiclassical wavefunction having the form of eq. (11). It is also clear that there are circumstances under which the saddle-point approximation fails, such as at caustics which are characterised by zeros of H = det [a2cF/0Q1 0Q1] on CX. Therefore, infinities associated with caustics arise through misapplication of the saddle-point approximation. The wavefunction (12) is regular at caustics and provides a basis for uniform and transitional approximations [11] valid in domains containing caustics. The representation of the wavefunction in the form of eq. (12) thus provides the means of copidg with difficulties associated with multivaluedness and caustics. We can now concern ourselves with the properties of an analytic function namely the integrand of eq. (12), and proceed to investigate the properties of the function f(Q; q; F) for which ,t’ is an exact or approximate solution of the Schrodinger equation. The property of this function that it possesses a uniform asymptotic expansion in positive powers of it, for it 0, gives rise to the possibility of generating an asymptotic expansion for R(q) = (~( E)~(i(q)in terms involving integrals of the type (12). In particular, it can be shown that, when f is independent of q and Q, R(q) = O(h 2) away from caustics, while at a caustic R(q) = O(h 20), where a (<2) is the singularity index of the caustic. This confirms the earlier suggestion that, away from caustics, the quantities a~= c~f~ behave, in the semiclassical limit, like constants. In this limit, the integral (12) may exhibit the Stokes phenomenon [2—4],in which case it is found that a~take the roleinof Stokes multipliers [2] This the is clear, for on example, the treatment [3,18—20] of the integral representation of the Airy function (in which n = 1, f 1, 4) = (q + P)Q ~Q3)by which the aforementioned procedure provides the exact quantal solution in the case of a linear potential. It will be noticed, however, that, when 4) is a “universal unfolding”, the Stokes phenomenon arises only in the treatment of the asymptotic evaluation of the integral (12) in the semiclassical limit. When a “universal unfolding” exists, the connection problem [2,4] is not an essential part ofthe semiclassical method, In general, it would appear [21] that there does not exist a single universal function 4) leading to an expression of the wavefunction in the form of a single diffraction integral (12). Thus, generically, a wave—
-+
—
-
—
function will be expressible as a sum of terms, each of which has the form of eq. (12). There remains the problem of how the functions 4) may be found. Practical approaches to solving the HJ equation include the method of separation of variables when components of Q appear as separation constants. This provides the solution on a series of one-dimensional coordinate submanifolds spanning a subspace of {q, Q}. These solutions in one dimension may exhibit Stokes phenomena (such as one finds in connection with the presence of linear and parabolic barriers) and may often be conveniently represented in terms of a multiple-reflection series or its equivalent wave-propagation matrix expression- [151 Such a representation may be obtained by means of local uniform approximations [12,14] in which the wavefunction near a turning point is locally expressible in terms of Airy or para. bolic cylinder functions. In this way a global representation of the wavefunction may involve a set of generating functions {4)(~~~} associated with which is a set 1~CX(ic)} of contiguous manifolds, CX (“) = {q, Q I a4)(K)/OQ 0). The complete critical manifold CX generated by all real and complex trajectories representing solutions of the classical HJ problem is the union of all the manifolds belonging to this set. An approximation to the wavefunction, uniformly valid in some open domain D{q} in which none of the CX have boundaries (other than Stokes lines) is as follows: -
(K)
= 7-’
/
X
(2mh\~ ‘
/
f E [M(”~jl/2f(K)e(iIli )
~n
(14)
K
A representation of the same wavefunction in terms of a universal generating function, as in eq. (12), is, by the saddle-point approximation, equivalent in D{q} to the above in the semiclassical limit. In another paper [21] we demonstrate that a representation of the form of eq. (14) is applicable in the treatment of scattering by a central field in three dimensions where it leads to accurate representations of the wavefunction and smatrix as partial-wave series. Justification is also given of the wavefunctions described in ref. [5] through use of the saddle-point approximation. The functions f(K) appearing in eq. (14) consist of multiple reflection coefficients such as arise in the wave-propagation matrix treatment [15,16] of the 189
Volume hA, number 2,3
PHYSICS LETTERS
one-dimensional radial problem, and may contain factors that depend on the form(s) of quantisation condition(s) through the value(s) of the appropriate Maslov indices [24]. Finally we observe that, as well as providing global representations of the quantal wavefunction in the semiclassical limit, this procedure also leads to a cornplete unification of dynamical [12,22] and diffractive [23]short-wavelength scattering theories. References
30 April 1979
[9] R. Thom, Structural stability and morphogenesis (Benjamin, Reading, MA, 1975). [10] T. Poston and I.M. Stewart, Catastrophe theory and its applications (Pitman, London, 1978). Eli] M.V. Berry, Adv. Phys. 25 (1976)1. [121J. Knoll and R. Schaeffer, Ann. Phys. (NY) 97 (1976) 307.
[13] J.B. Keller, Proc. Symp. Appl. Math. 8 (1958) 27. [14] D.M. Brink and N. Taidgawa, Nucl. Phys. A279 (1977) [15] 159. S.Y. Lee and N. Takigawa, Nuci. Phys. A308 (1978) 189. [16] N. Takigawa, talk presented at the second Louvain—
Krakow Seminar (Louvain-La-Neuve, Belgium, June 1978), to be published.
[1] H.J. Korsch and R. Mohlenkamp, Phys. Lett. 67A (1978) 110. [2] M.V. Berry and K.E. Mount, Rep. Prog. Phys. 35 (1972)
315. [3] G.G. Stokes, Mathematical and physical papers (Cambridge U.P., London) (1904) pp. 77—109, 283—98; (1905) pp. 221—5, 283—7. [4] E.C. Kemble, Fundamental principles of quantum mechanics (McGraw-Hill, New York, 1937). [5]BJ.B. Crowley, submitted to Phys. Rep. (1978). [6] H. Goldstein, Classical mechanics (Addison-Wesley, New York, 1950)Ch. 7—9. [7] O.N. Stravroudis, The optics of rays, wavefronts and caustics (Academic Press, New York, 1972). [8] J.H. Van Vleck, Proc. Nat. Acad. Sci. USA 14 (1928) 178.
190
[17] J.J. Duistermaat, Commun. Pure Appl. Math. 27 (1974) 207.
[18] J.N.L. Connor, Mol. Phys. 31(1976) 33. [19] M.V. Berry, Proc. Phys. Soc. 89 (1966) 479. [20] C. Chester, B. Friedman and F. Ursell, Proc. Camb. Phil. Soc. 53 (1957) 599. [211 B.J.B. Crowley, submitted to Phys. Lett. A (1979). [22] R. Schaeffer, in: Nuclear physics with heavy ions and mesons, Proc. session XXX (Les Houches, 1977), Vol. I, eds. R. Balian, M. Rho and G. Ripka (North-Holland, Amsterdam, 1978). [23] W.E. Frthn, in: Heavy-ion, high-spin states and nuclear structure, Proc. extended seminar (Trieste, 1973), Vol. I (IAEA, Vienna, 1975) p. 157. [24] I.C. Percival, Adv. Chem. Phys. 36 (1977) 1.