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

A normal form transformation is carried out on one-dimensional quantum Hamiltonians that transforms them into functions of the quantum harmonic oscillator. The method works with the Weyl transform (or 'symbol') of the Hamiltonian. The Moyal star product is used to carry out the normal form transformation at the level of symbols. Diagrammatic techniques are developed for handling the expressions that result from higher order terms in the Moyal series. Once the normal form is achieved, the Bohr–Sommerfeld formula for the eigenvalues, including higher order corrections, follows easily.