The quantum normal form and its equivalents

Abstract

A quantum analog, called the quantum normal form, of the classical Birkhoff–Gustavson normal form is presented. The algebraic relationship between the quantum and Birkhoff–Gustavson normal forms has been established by developing the latter using Lie transforms. It is shown that the Birkhoff–Gustavson normal form can be obtained from the quantum normal form. Using an anharmonic oscillator and a Henon–Heiles system as test cases, the equivalence between the quantum normal form and the Rayleigh–Schrödinger perturbation method is shown. This equivalence provides an algebraic connection between the Birkhoff–Gustavson normal form and the Rayleigh–Schrödinger perturbation approach. The question of Weyl and torus quantizations of the Birkhoff–Gustavson normal form is discussed in the light of the quantum normal form.