TITLE:
Wigner Quasiprobability with an Application to Coherent Phase States
AUTHORS:
Alfred Wünsche
KEYWORDS:
Parity Operator, Quantum Square Well, Coherent States, SU (1, 1) Group and Realizations, Glauber-Sudarshan and Husimi-Kano Quasiprobability, London Phase States, Phase Distribution, Unorthodox Entire Function, Laguerre 2D Polynomials, Generalized Eulerian Numbers
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.8 No.6,
June
28,
2018
ABSTRACT: Starting from Wigner’s definition of the
function named now after him we systematically develop different representation
of this quasiprobability with emphasis on symmetric representations concerning
the canonical variables (q,p) of phase space and
using the known relation to the parity operator. One of the representations is
by means of the Laguerre 2D polynomials which is particularly effective in
quantum optics. For the coherent states we show that their Fourier transforms
are again coherent states. We calculate the Wigner quasiprobability to the
eigenstates of a particle in a square well with infinitely high impenetrable
walls which is not smooth in the spatial coordinate and vanishes outside the
wall boundaries. It is not well suited for the calculation of expectation
values. A great place takes on the calculation of the Wigner quasiprobability
for coherent phase states in quantum optics which is essentially new. We show
that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wigner
quasiprobability for coherent phase states is calculated and graphically
represented but due to the involved unorthodox function it may be considered
only as illustration and is not suited for the calculation of expectation
values. By another approach via the number representation of the states and
using the recently developed summation formula by means of Generalized Eulerian
numbers it becomes possible to calculate in approximations with good
convergence the basic expectation values, in particular, the basic
uncertainties which are additionally represented in graphics. Both considered
examples, the square well and the coherent phase states, belong to systems with SU (1,1) symmetry with the same
index K=1/2 of unitary irreducible
representations.