Generalized Transformation to a Rotating Coordinate System in Quantum Mechanics

Abstract

A generalized transformation to a rotating coordinate system for the quantum-mechanical equation of motion for the density matrix is presented. The transformation is of value when the time dependence of the perturbation is harmonic in time and contains resonant and nonresonant terms. We show that after the transformation, the resonant terms are slowly varying in time and are mostly responsible for the physical behavior of the system in the neighborhood of the resonance. The nonresonant terms give a small contribution and have a rapidly varying phase. A formal procedure is given which yields an equation containing only the low-frequency part of interest. A general result of this transformation is the appearance of the Bloch-Siegert shift of the resonance frequency due to the influence of the nonresonant terms. As an example, a straight-forward application of the method is carried out in order to compute the Bloch-Siegert shift of the resonance for a spin-\textonehalf{} system.