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Phase and Angle Variables in Quantum Mechanics

P. CARRUTHERS and MICHAEL MARTIN NIETO

P. CARRUTHERS

  • Laboratory of Nuclear Studies and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York

MICHAEL MARTIN NIETO

  • Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York

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Rev. Mod. Phys. 40, 411 – Published 1 April, 1968

DOI: https://doi.org/10.1103/RevModPhys.40.411

Abstract

The quantum-mechanical 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 Lz is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables (sinϕ and cosϕ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional 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 action-angle (rather J, cosφ, sinφ) variables is used to deduce C and S. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" 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 action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.

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Rev. Mod. Phys. 40, 411– Published 1 April, 1968

    © 1968 American Physical Society

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