The physical properties of linear and action-angle coordinates in classical and quantum mechanics

Abstract

The quantum harmonic oscillator is described in terms of two basic sets of coordinates: linear coordinates x, p_x and angular coordinates e^iφ, P_φ (action-angle variables). The angular “coordinate” e^iφ is assumed unitary, the conjugate momentum p_φ is assumed Hermitian, and e^iφ and p_φ are assumed to be a canonical pair. Two transformations are defined connecting the angular coordinates to the linear coordinates. It is found that x, p_x can be physical, i.e., Hermitian and canonical, only under constraints on the p_φ eigenvalue spectrum. The conclusion is that e^iφ can be a unitary operator. A parallel analysis of the classical harmonic oscillator is done with equivalent results.