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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Bohr-Sommerfeld quantization with the effective action variable

Edward R. Floyd

Published Mar 15, 1982• 40 citations
Publisher

Abstract

The effective action variable with its attendant modified potential is applied herein to the bound state, for which it is hypothesized that the Bohr-Sommerfeld whole-integer quantization is correct. Results for various one-dimensional potentials support this hypothesis. Finite turning points for bound-state effective action integrands do not exist. Also, it is shown that the modified potential for bound states is not unique.

DOI
10.1103/physrevd.25.1547

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The effective action variable with its attendant modified potential is applied herein to the bound state, for which it is hypothesized that the Bohr-Sommerfeld whole-integer quantization is correct. Results for various one-dimensional potentials support this hypothesis. Finite turning points for bound-state effective action integrands do not exist. Also, it is shown that the modified potential for bound states is not unique.
This investigation concerns some model potentials with very steep walls, for which the quantal Bohr–Sommerfeld half-integer quantization condition, if necessary generalized to correspond to modified phase-integral approximations of arbitrary order, and used without or with higher-order corrections included, can be used for obtaining, very accurately, the energy eigenvalues of the bound states, apart possibly from the lowest ones. One of the cases treated is the potential proportional to cot2x, for which a modified Bohr–Sommerfeld half-integer quantization condition yields the energy eigenvalues exactly. That the Bohr–Sommerfeld half-integer quantization condition is applicable to potentials with very steep walls may at first sight seem surprising in view of the well-known fact that the energy eigenvalues of a square-well potential with infinitely high walls are obtained exactly from the Bohr–Sommerfeld integer quantization condition, i.e., the quantization condition obtained by replacing (s+1/2) π by (s+1) π in the right-hand member of the Bohr–Sommerfeld half-integer quantization condition. From the study of a potential with horizontal bottom and linearly rising walls, which goes over into a square well when the inclination of the walls tends to zero, it can easily be understood, why the half-integer quantization condition is appropriate when the steep walls have a finite slope.
An algebraic reformulation of the Bohr - Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrödinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrödinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl - Teller potential and several anharmonic oscillator potentials, which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.