van Leeuwen, Hans; Maassen, Hans A \(q\) deformation of the Gauss distribution. (English) Zbl 0841.60089 J. Math. Phys. 36, No. 9, 4743-4756 (1995). Summary: The \(q\) deformed commutation relation \(aa^* - qa^* a = 1\) for the harmonic oscillator is considered with \(q \in [-1,1]\). An explicit representation generalizing the Bargmann representation of analytic functions on the complex plane is constructed. In this representation the distribution of \(a + a^*\) in the vacuum state is explicitly calculated. This distribution is to be regarded as the natural \(q\) deformation of the Gaussian. Cited in 24 Documents MSC: 60K40 Other physical applications of random processes 81S05 Commutation relations and statistics as related to quantum mechanics (general) Keywords:commutation relation; harmonic oscillator; Bargmann representation PDFBibTeX XMLCite \textit{H. van Leeuwen} and \textit{H. Maassen}, J. Math. Phys. 36, No. 9, 4743--4756 (1995; Zbl 0841.60089) Full Text: DOI References: [1] DOI: 10.1007/BF01197843 · Zbl 0671.60109 · doi:10.1007/BF01197843 [2] DOI: 10.1016/0022-1236(92)90055-N · Zbl 0784.46047 · doi:10.1016/0022-1236(92)90055-N [3] DOI: 10.1007/BF02100275 · Zbl 0722.60033 · doi:10.1007/BF02100275 [4] DOI: 10.1142/9789814354783_0005 · doi:10.1142/9789814354783_0005 [5] DOI: 10.2140/pjm.1994.165.131 · Zbl 0808.46094 · doi:10.2140/pjm.1994.165.131 [6] DOI: 10.1007/BF02099136 · Zbl 0734.60048 · doi:10.1007/BF02099136 [7] DOI: 10.1002/cpa.3160140303 · Zbl 0107.09102 · doi:10.1002/cpa.3160140303 [8] DOI: 10.1063/1.522937 · Zbl 0941.81549 · doi:10.1063/1.522937 [9] DOI: 10.4153/CJM-1980-053-8 · Zbl 0411.33009 · doi:10.4153/CJM-1980-053-8 [10] Jackson F., Q. J. Pure Appl. Math. 41 pp 193– (1910) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.