Quantum normal forms, Moyal star product and Bohr–Sommerfeld approximation

Abstract

The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order 2 (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 , starting with 0 , 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.

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“…The extension of the Bohr-Sommerfeld quantization, namely the calculation of eigenvalues for discrete simple spectra has been treated over the past 25 years by many authors in different approaches; we refer to [11] for an extensive discussion and references.…”

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