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Wigner Quasiprobability with an Application to Coherent Phase States

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Document pages: 51 pages

Abstract: Starting from Wigner’s definition of thefunction named now after him we systematically develop different representationof this quasiprobability with emphasis on symmetric representations concerningthe canonical variables (q,p) of phase space andusing the known relation to the parity operator. One of the representations isby means of the Laguerre 2D polynomials which is particularly effective inquantum optics. For the coherent states we show that their Fourier transformsare again coherent states. We calculate the Wigner quasiprobability to theeigenstates of a particle in a square well with infinitely high impenetrablewalls which is not smooth in the spatial coordinate and vanishes outside thewall boundaries. It is not well suited for the calculation of expectationvalues. A great place takes on the calculation of the Wigner quasiprobabilityfor coherent phase states in quantum optics which is essentially new. We showthat an unorthodox entire function plays there a role in most formulae which makes all calculations difficult. The Wignerquasiprobability for coherent phase states is calculated and graphicallyrepresented but due to the involved unorthodox function it may be consideredonly as illustration and is not suited for the calculation of expectationvalues. By another approach via the number representation of the states andusing the recently developed summation formula by means of Generalized Euleriannumbers it becomes possible to calculate in approximations with goodconvergence the basic expectation values, in particular, the basicuncertainties which are additionally represented in graphics. Both consideredexamples, the square well and the coherent phase states, belong to systems with SU (1,1) symmetry with the sameindex K=1 2 of unitary irreduciblerepresentations.

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