Presentations

Properties of quasi-probability density functions for quantum systems with electromagnetic interaction

Perepelkin E.E., Polyakova R.V.¹, Sadovnikov B.I., Inozemtseva N.G.², Afonin P.V.

Lomonosov Moscow State University, Faculty of Physics, Moscow, 119991, Russia, pevgeny@mail.ru, afonin.pv19@physics.msu.ru

¹Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980, Russia, polykovarv@mail.ru

²Dubna State University, Dubna, Moscow Region, 141980, Russia

In this work, we examine the properties of quasi-probability density functions for quantum systems with electromagnetic interactions of two types: the Wigner function and the gauge-invariant Weyl–Stratonovich function (1). Both functions are used to describe quantum systems in phase space. The term quasi-density reflects the possibility of negative values in these probability distributions.

A comparative analysis of the properties of these quasi-probability functions was carried out using exact model solutions of the electromagnetic Schrödinger equation. It was found that, unlike the Wigner function, the Weyl–Stratonovich function violates Hudson’s theorem and its three-dimensional generalizations for a Gaussian wavefunction. Within the PSI-model, we obtained an exact solution of the electromagnetic Schrödinger equation for which the Weyl–Stratonovich function is everywhere positive in phase space. It is shown that the coordinate-space probability densities derived from the Wigner and Weyl–Stratonovich functions coincide, whereas the momentum-space densities do not. A comparison of several mean kinematic quantities computed using both types of quasi-density functions is also presented.

Furthermore, we derived an evolution equation for the Wigner function of an electromagnetic system in a mathematical form closely resembling the second Vlasov equation, which led us to propose an electromagnetic Vlasov–Moyal approximation. This dynamical approximation makes it possible to cut the infinitely self-coupled Vlasov chain at the second equation and thereby to treat quasi-classical electromagnetic systems, as well as to assess the validity of the phenomenological Vlasov approximation widely used in plasma physics, accelerator physics, and astrophysics.

• abstract in Russian (PDF)