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American Journal of Physics

Bohr–Sommerfeld, WKB, and modified semiclassical quantization rules

D. J. W. Geldart, David Kiang

Published Feb 1, 1986• 11 citations
Publisher

Abstract

The WKB approximation for energy eigenvalues can be advantageously introduced to students when the Bohr–Sommerfeld quantization rule is discussed. A simple heuristic derivation of these quantization rules is given. Exceptional cases occur when conditions necessary for the validity of semiclassical approximations are violated, but appropriate modifications can still provide useful semiclassical approximations even in such cases. Various situations are illustrated by examples.

DOI
10.1119/1.14707

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The WKB approximation for energy eigenvalues can be advantageously introduced to students when the Bohr–Sommerfeld quantization rule is discussed. A simple heuristic derivation of these quantization rules is given. Exceptional cases occur when conditions necessary for the validity of semiclassical approximations are violated, but appropriate modifications can still provide useful semiclassical approximations even in such cases. Various situations are illustrated by examples.
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.
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.
A fresh look at the origin of the Wilson–Sommerfeld quantization rule has been pursued to gain new insight. The rule is shown to provide states that satisfy several well-known theorems of standard quantum mechanics. A few other useful results and scaling relations are also derived. They emerge to act as nice guiding rules of thumb in the course of rigorous computations. Certain features of true excited-state densities can be understood. Goodness of approximate densities can be assessed. Compressed systems can be studied profitably. A route is also sketched that allows one to retrieve classical trajectories from near-exact energy eigenfunctions for both bound and resonant states by exploiting this rule. Additionally, a discussion on semiclassical perturbation theory is presented emphasizing the asymptotic behavior. Pilot calculations demonstrate the success of the present endeavor under various circumstances. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 113–125, 2001
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.