Classical representation of wave functions for integrable systems

Classical exact $\left(\mathrm{CE}\right)$ 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.

Figure 1

Semiclassical wave functions (solid curves) and quantum wave functions (dashed curves) for the four lowest states of the perturbed particle-in-box system. The potential energy function is shown in heavy lines. Energy and coordinate units are consistent with values of unity for Planck’s constant, the particle mass, and the well length. Each wave function is translated upwards in the figure by a distance equal to the corresponding energy eigenvalue. The differences between the semiclassical and quantum wave functions are too small to be resolved on the scale shown for $n=3,4$.