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Physical Review A

Classical representation of wave functions for integrable systems

Kenneth G. Kay

Published Jun 15, 2004• 8 citations
Publisher

Abstract

Classical exact $(\mathrm{CE})$ wave functions are certain integral representations of energy eigenfunctions that are parameterized in terms of the motion of the corresponding classical system in a semiclassically relevant way. When applied to systems for which they are not exact, such expressions serve as semiclassical approximations. Previous work identified $\mathrm{CE}$ wave functions for a number of specific systems and established their semiclassical usefulness. This paper explores the degree to which such representations can be found for more general systems. It is shown that $\mathrm{CE}$ wave functions exist, in principle, for bound states of an arbitrary integrable system that are confined to a single classically allowed region. Evidence is presented that $\mathrm{CE}$ representations also exist for more general states of such a system that are unbound, or that extend over more than one allowed region. The $\mathrm{CE}$ expressions are not unique: an innumerable variety exists for each such system. The existence proof provides a formal method for constructing $\mathrm{CE}$ expressions by Fourier transforming certain superpositions of energy eigenstates. The parameterization in terms of the classical motion is achieved by identifying certain quantities in these superpositions as classical action and angle variables. The semiclassical relevance of this identification is ensured by imposing some mild conditions on the coefficients in the superposition. This procedure for parameterizing exact wave functions in terms of classical variables indicates a basic relationship between the quantum and classical descriptions of states. The method of constructing $\mathrm{CE}$ wave functions introduced in the proof is shown to be consistent with a number of previously obtained $\mathrm{CE}$ formulas and is used to derive two new, closed-form, $\mathrm{CE}$ expressions. A simple numerical example is presented to illustrate the semiclassical application of one of these expressions and to further verify the physical significance of the classical parameterization.

DOI
10.1103/physreva.69.062106

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Classical exact (CE)(\mathrm{CE})(CE) wave functions are certain integral representations of energy eigenfunctions that are parameterized in terms of the motion of the corresponding classical system in a semiclassically relevant way. When applied to systems for which they are not exact, such expressions serve as semiclassical approximations. Previous work identified CE\mathrm{CE}CE wave functions for a number of specific systems and established their semiclassical usefulness. This paper explores the degree to which such representations can be found for more general systems. It is shown that CE\mathrm{CE}CE wave functions exist, in principle, for bound states of an arbitrary integrable system that are confined to a single classically allowed region. Evidence is presented that CE\mathrm{CE}CE representations also exist for more general states of such a system that are unbound, or that extend over more than one allowed region. The CE\mathrm{CE}CE expressions are not unique: an innumerable variety exists for each such system. The existence proof provides a formal method for constructing CE\mathrm{CE}CE expressions by Fourier transforming certain superpositions of energy eigenstates. The parameterization in terms of the classical motion is achieved by identifying certain quantities in these superpositions as classical action and angle variables. The semiclassical relevance of this identification is ensured by imposing some mild conditions on the coefficients in the superposition. This procedure for parameterizing exact wave functions in terms of classical variables indicates a basic relationship between the quantum and classical descriptions of states. The method of constructing CE\mathrm{CE}CE wave functions introduced in the proof is shown to be consistent with a number of previously obtained CE\mathrm{CE}CE formulas and is used to derive two new, closed-form, CE\mathrm{CE}CE expressions. A simple numerical example is presented to illustrate the semiclassical application of one of these expressions and to further verify the physical significance of the classical parameterization.
The wave function for a state of definite angular momentum of the three-dimensional isotropic harmonic oscillator is expressed exactly in terms of the corresponding classical trajectories. In particular, the three-dimensional wave function as well as the radial wave function and the spherical harmonics are obtained as integrals over quantities determined entirely by the classical motion. The expression for the harmonic-oscillator radial wave function is shown also to yield the exact radial wave function for the free particle. The expressions are cast in forms suitable for use as uniform semiclassical approximations for wave functions of other systems. Numerical examples confirm that such wave functions obey boundary conditions appropriate for spherical coordinates and that they are free of caustic singularities. The wave functions obtained by this technique can be quite accurate even for low-energy states where semiclassical approximations are expected to be poor.
It is shown that application of the time-dependent variational principle to trial wave functions of a particular form results in a description of the quantum system as a classical system consisting of two mutually interacting parts: the ordinary classical analog of the quantum system and a (generally finite) number of additional degrees of freedom. This holds out the prospect of simulating quantum dynamics by coupling the corresponding classical system to additional degrees of freedom, and it is pointed out that such calculations have, in effect, already been performed. Since the trial wave functions that qualify for this treatment may be arbitrarily accurate, there is no limitation to the accuracy of the resulting classical description. In a particular case, the canonical coordinates and momenta for all degrees of freedom of the overall classical system are identified and investigated. In general, certain parameters appearing in the trial wave functions treated in this work may become redundant at certain times, leading to singularities in the Hamiltonian equations of motion. It is shown, however, that in certain such cases, a classical formalism for the quantum dynamics remains valid if the Hamiltonian function is suitably modified. Calculations applying the present Hamiltonian formalism are carried out that demonstrate that it has better convergence properties than an alternative, nonvariational, Hamiltonian technique suggested by previous work. These numerical studies also suggest that the observed spreading of wave packets and development of complicated structure in wave functions may be interpreted classically in terms of energy transfer among the various degrees of freedom in the overall classical system.
A calculation of the classical analogue for the quantum wave function and local denity of states, in energy representation, is presented for simple Hamiltonian systems. Sucha analogous were proposed by M. V. Berry and A. voros considering the intersection of energy shells of two systems as the only semiclassical object which can give support to eigenfunctions. One of them is the system unser study and the other is the "unperturbed system" used to express the wave functions, even in the case that both systems are not close. For simple systems and as for scalable ones analytical expressions are obtainable. In the present work we offer examples of both.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
Wave functions for the Coulomb, Morse, Rosen-Morse, and Eckart systems are expressed exactly and explicitly in terms of the corresponding classical trajectories. This is accomplished by a technique that maps previously obtained classical representations for wave functions of the three-dimensional isotropic harmonic oscillator to the wave functions for the present systems. By design, the classical behavior enters these expressions in a manner that allows them to be used as uniform semiclassical approximations for other systems. The construction of a hydrogen orbital from the corresponding classical orbits is illustrated for a particular case. The classical representation of the Eckart wave function is analyzed to understand how ordinary, real, classical trajectories that are confined to one side of a barrier are able to describe tunneling through the barrier.
Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger equation. The following are proved: The “quantum-mechanical momentum” which is nearly equal to the corresponding classical momentum in “classical region” can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrödinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to “quantum-mechanical quantization condition”, equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the “classical region” of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrödinger equation always exists the corresponding classical system.