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Journal of Physics A Mathematical Nuclear and General

Functional integral representation of operators and the Weyl quantization rule

T. Lukes

Published Jul 1, 1973• 3 citations
Publisher

Abstract

The Weyl quantization rule is shown to follow from a semiclassical expansion of an exact expression for a functional of an operator given by Feynman.
DOI
10.1088/0305-4470/6/7/001

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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.