Wigner Quasiprobability with an Application to Coherent Phase States

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 ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.

Common Properties of Riemann Zeta Function, Bessel Functions and Gauss Function Concerning Their Zeros

“…with respect to the zeros in its finite Taylor series approximations. This function plays a role for the calculation of the properties of coherent phase states [19].…”

“…Taking separately the even and odd powers of z and applying the duplication formula for the factorials one may represent (9.1) in the form In Figure 12 we see the first three pairs of zeros as some accumulation points. Up to now we calculated only the first 4 pairs of zeros with sufficient accuracy [19] but S. Skorokhodov from the "Computing Centre of the Russian Academy of Sciences" calculated much more pairs of zeros with an essentially higher accuracy as he informed me in a nice email with the calculated zeros in the appended file [20] (see also [19]). These zeros agreed with my few in lower accuracy calculated zeros 5 .…”