Phase and Angle Variables in Quantum Mechanics
Abstract
The quantummechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable ϕ and its "conjugate" angular momentum L_{z} is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables (ϕ and ϕ to describe the phase variable. Welldefined uncertainty relations are derived and discussed. A detailed analysis of the threedimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator φ (conjugate to the number operator N) exists is shown to be erroneous. However, cosine and sine operators C and S exist and are the apprópriate phase variables. A Poisson bracket argument using actionangle (rather J, φ, φ) variables is used to deduce C and S. The spectra and eigenfunctions of these operators are investigated, along with the important "phasedifference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in actionangle variables along with quantummechanical "corrections." Mention is made of possible physical applications to superfluid systems.
 Publication:

Reviews of Modern Physics
 Pub Date:
 April 1968
 DOI:
 10.1103/RevModPhys.40.411
 Bibcode:
 1968RvMP...40..411C