The Wigner function negative value domains and Energy function poles of the polynomial oscillator

Perepelkin E.E., Polyakova R.V.1, Burlakov E.V., Afonin P.V.

Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia

1Joint Institute for Nuclear Research, Moscow region, Moscow,141980 Russia

With the development of quantum computing technique, quantum communication, cryptography and quantum information science, the mathematical apparatus of the Wigner function is becoming in demand. The Wigner function is used as a function of quasi-probabilities when describing a quantum system in a phase space. A quantum feature of the Wigner function is the presence of negative values in the phase domain. For the simplest system, a quantum harmonic oscillator, the expression for the Wigner function is known explicitly. The purpose of this study is to construct expressions for average energies using the method of finding the density matrix, and to show the presence of poles of the energy function in areas where the Wigner function takes negative values, thereby generalizing the result obtained for a harmonic oscillator. To achieve this goal, the following tasks are solved: theorems on the form of explicit expressions for the average kinematic quantities are proved, expressions for the average energies are constructed using the representations for the Wigner function in terms of the Weyl operator in the basis of the harmonic oscillator obtained in [1].


1. Perepelkin E.E., Sadovnikov B.I., Inozemtseva N.G., Burlakov E.V., Explicit form for the kernel operator matrix elements in eigenfunction basis of harmonic oscillator, Journal of Statistical Mechanics: Theory and Experiment, 2020, №. 023109


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