TY - JOUR

T1 - The action-angle Wigner function

T2 - A discrete, finite and algebraic phase space formalism

AU - Hakioǧlu, T.

AU - Tepedelenlioǧlu, E.

PY - 2000/9/15

Y1 - 2000/9/15

N2 - The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigner function is shown to exist in the semi-discrete limit of this quantization scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an action-angle phase space distribution respects are derived. The dynamical properties of the action-angle Wigner function are analysed for discrete and finite-dimensional model Hamiltonians. The limit of the discrete and finite-dimensional formalism including a discrete analogue of the Gaussian wavefunction spread, viz. the binomial wavepacket, is examined and shown by examples that standard (continuum) quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity.

AB - The action-angle representation in quantum mechanics is conceptually quite different from its classical counterpart and motivates a canonical discretization of the phase space. In this work, a discrete and finite-dimensional phase space formalism, in which the phase space variables are discrete and the time is continuous, is developed and the fundamental properties of the discrete Weyl-Wigner-Moyal quantization are derived. The action-angle Wigner function is shown to exist in the semi-discrete limit of this quantization scheme. A comparison with other formalisms which are not explicitly based on canonical discretization is made. Fundamental properties that an action-angle phase space distribution respects are derived. The dynamical properties of the action-angle Wigner function are analysed for discrete and finite-dimensional model Hamiltonians. The limit of the discrete and finite-dimensional formalism including a discrete analogue of the Gaussian wavefunction spread, viz. the binomial wavepacket, is examined and shown by examples that standard (continuum) quantum mechanical results can be obtained as the dimension of the discrete phase space is extended to infinity.

UR - http://www.scopus.com/inward/record.url?scp=0037908608&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/33/36/307

DO - 10.1088/0305-4470/33/36/307

M3 - Article

AN - SCOPUS:0037908608

SN - 0305-4470

VL - 33

SP - 6357

EP - 6383

JO - Journal of Physics A: Mathematical and General

JF - Journal of Physics A: Mathematical and General

IS - 36

ER -