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首页> 外文期刊>Advances in Pure Mathematics >Wigner Quasiprobability with an Application to Coherent Phase States
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Wigner Quasiprobability with an Application to Coherent Phase States

机译：Wigner拟概率及其在相干态中的应用

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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 qua siprobability 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.
机译：从维格纳（Wigner）对现在以他命名的函数的定义开始，我们系统地开发了该拟概率性的不同表示形式，重点是与相空间的规范变量（q，p）有关的对称表示形式，并使用与奇偶运算符的已知关系。表示之一是借助于Laguerre 2D多项式，该多项式在量子光学中特别有效。对于相干态，我们表明它们的傅立叶变换再次是相干态。我们计算了方形孔中无限高不可穿透壁面的粒子的本征态的Wigner拟拟概率，该空间在空间坐标上不平滑并且在壁面边界之外消失。它不太适合用于期望值的计算。对于量子光学中相干相态的Wigner拟概率性的计算非常重要，这在本质上是新的。我们表明，非常规的整个函数在大多数公式中都发挥了作用，这使所有计算变得困难。计算并以图形方式表示了相干态的维格纳拟似性，但是由于所涉及的非正统函数，该维格纳拟似性仅作为示例，并不适合于期望值的计算。通过状态数表示的另一种方法，以及使用最近开发的借助于广义欧拉数的求和公式，可以近似收敛地近似计算基本期望值，尤其是基本不确定性，这些不确定性在图形中另外表示。所考虑的两个例子，即方阱和相干相态，都属于具有SU（1,1）对称性的系统，具有统一的不可约表示的索引K = 1/2。

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