Moyal star product approach to the Bohr-Sommerfeld approximation

TL;DR

This paper derives the Bohr-Sommerfeld eigenvalue approximation to order ar^2 using the Moyal star product, applicable to general Hamiltonians with a Weyl symbol, and introduces diagrammatic techniques for higher-order terms.

Contribution

It introduces a method to derive the Bohr-Sommerfeld approximation to order ar^2 using the Moyal star product for general Hamiltonians, including a normal form transformation.

Findings

Derived eigenvalue approximation to order ar^2

Applicable to Hamiltonians with a Weyl symbol as a power series in ar

Presented diagrammatic techniques for higher-order Moyal series manipulation

Abstract

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