# Effective numerical algorithm for constructing the Wigner function of a quantum system with a polynomial potential in the phase space

Seminars

### Seminar of the Scientific Department of Computational Physics

**Date and Time**: Thursday, 1 October 2020, at 11:00 AM

**Venue**: Online seminar on Webex, Laboratory of Information Technologies

**Seminar topic**: *«Effective numerical algorithm for constructing the Wigner function of a quantum system with a polynomial potential in the phase space»*

**Authors**: E. E. Perepelkin^{abcd}, B. I. Sadovnikov^{b}, N. G. Inozemtseva^{cd}, E. V. Burlakov^{bd}, R. V. Polyakova^{a}

^{a} – Joint Institute for Nuclear Research

^{b} – Lomonosov Moscow State University

^{c} – Dubna State University

^{d} – Moscow Technical University of Communications and Informatics

**Speaker**: E. V. Burlakov

**Abstract**:

When considering quantum systems in the phase space, the Wigner function is used as a function of the quasi-probability density. Finding the Wigner function is related to the calculation of the Fourier transform of a certain composition of wave functions of the corresponding quantum system. As a rule, the knowledge of the Wigner function is not the ultimate goal, and computations of the average values of different quantum characteristics of a system are required.

An explicit solution of the Schrödinger equation can be obtained only for a narrow class of potentials; therefore, numerical methods to find wave functions are used in most cases. Consequently, finding the Wigner function is associated with the numerical integration of grid wave functions. When considering a one-dimensional system, it is obligatory to calculate N2 Fourier integrals of the grid wave function. To provide the required accuracy for the wave functions corresponding to the higher states of a quantum system, a larger number of grid nodes is needed.

The goal of the given work was to construct a numerical-analytical method for finding the Wigner function, which would significantly reduce the number of computational operations. Quantum systems with polynomial potentials, for which the Wigner function is represented as a series in some known functions, was considered.
The work was supported by the RFBR grant No. 18-29-10014.

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