Phase and Angle Variables in Quantum Mechanics

摘要: The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical these coordinates. relations among operators state vectors under consideration are clarified in context Heisenberg uncertainty relations. familiar case azimuthal variable $\ensuremath{\phi}$ its "conjugate" angular momentum ${L}_{z}$ discussed. Various pitfalls associated periodicity problem avoided by employing periodic ($sin\ensuremath{\phi}$ $cos\ensuremath{\phi}$ to describe variable. Well-defined derived A detailed analysis three-dimensional harmonic oscillator excited coherent states given. simple usual assumption that a (Hermitian) operator $\ensuremath{\varphi}$ (conjugate number $N$) exists shown be erroneous. However, cosine sine $C$ $S$ exist appr\'opriate variables. Poisson bracket argument using action-angle (rather $J$, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) used deduce $S$. spectra eigenfunctions investigated, along important "phase-difference" properties various types analyzed special attention transition classical limit. utility as basis for evolution density matrix emphasized. In this it easy identify Liouville equation "corrections." Mention made possible physical applications superfluid systems.