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.
Highlights
Most representations of probability theory begin with the discussion of some examples where probabilities play a main role and introduce axiomatically the probability as a positively semi-definite and “normalized” function over a set of events
Some expectation values for the coherent phase states ε can be effectively calculated in other way that we demonstrate
We derived in this article formulae for the calculation of the Wigner quasiprobability with emphasis of relations which are symmetric in the canonical operators (Q, P)
Summary
Most representations of probability theory begin with the discussion of some examples where probabilities play a main role and introduce axiomatically the probability as a positively semi-definite and “normalized” function over a set of events. The main purpose of this function is to allow us to calculate mean values and their variances or more generally expectation values for arbitrary functions over the set of events when the initial conditions or the prehistory of the events are not fully under control of the experimenter or observer The results of such calculations are true in the mean for great ensembles of “equal”. In Hamilton dynamics of a system of one degree of freedom a trajectory is fully determined by a pair (q, p) of canonical coordinate and momentum in two-dimensional phase space as initial condition and the probability function is given by a positive semi-definite function.
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