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Journal of Physics B Atomic Molecular and Optical Physics

The phase formalism for the one-dimensional eigenvalue problem and its relation with the quantum Bohr-Sommerfeld rule

A. V. Abarenov, A. V. Stolyarov

Published Aug 14, 1990• 29 citations
Publisher

Abstract

A quantum phase formalism is offered for searching for eigenvalues of the one-dimensional Schrodinger equation. Its application in the shooting method appreciably sped up convergence and made it absolute. The introduced quantum phase functions are close to their semiclassical analogues, unlike those used before, and transform into them in the semiclassical limit. A form of quantum analogue of the Bohr-Sommerfeld quantisation rule, allowing it to operate with non-integer quantum numbers within the framework of precise quantum consideration is obtained.

DOI
10.1088/0953-4075/23/15/010

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A quantum phase formalism is offered for searching for eigenvalues of the one-dimensional Schrodinger equation. Its application in the shooting method appreciably sped up convergence and made it absolute. The introduced quantum phase functions are close to their semiclassical analogues, unlike those used before, and transform into them in the semiclassical limit. A form of quantum analogue of the Bohr-Sommerfeld quantisation rule, allowing it to operate with non-integer quantum numbers within the framework of precise quantum consideration is obtained.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
Classical mechanics and the quantum conditions of Planck, Bohr, Sommerfeld, Wilson and Einstein are presented. The virtues and defects of this “old quantum theory” are pointed out. Its replacement by quantum mechanics is described, leading to the Schrödinger equation for the wave function and the corresponding energy eigenvalues. For separable systems, the reduction of this equation to ordinary differential equations and their asymptotic solution by the WKB method are described, as well as the resulting corrected quantum conditions with integer or half-integer quantum numbers. For nonseparable systems, the analogous asymptotic solution constructed by the author is described, together with the corrected quantum conditions to which it leads. Examples of the use of these conditions in the solution of eigenvalue problems are presented. It is explained that difficulties arise in using this method when the classical motion is stochastic or chaotic. Suggestions for overcoming these difficulties are mentioned.
The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order ℏ2\hbar^2ℏ2 (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in ℏ\hbarℏ, starting with ℏ0\hbar^0ℏ0, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by ℏ\hbarℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of 2π2π2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.