37 research outputs found
Wigner function properties for electromagnetic systems
Using the Wigner-Vlasov formalism, an exact 3D solution of the Schr\"odinger
equation for a scalar particle in an electromagnetic field is constructed.
Electric and magnetic fields are non-uniform. According to the exact expression
for the wave function, the search for two types of the Wigner functions is
conducted. The first function is the usual Wigner function with a modified
momentum. The second Wigner function is constructed on the basis of the
Weyl-Stratonovich transform in papers [Phys. Rev. A 35 2791 (1987)] or [Phys.
Rev. B 99 014423 (2019)]. It turns out that the second function, unlike the
first one, has areas of negative values for wave functions with the Gaussian
distribution (Hudson's theorem).
On the one hand, knowing the Wigner functions allows one to find the
distribution of the mean momentum vector field and the energy spectrum of the
quantum system. On the other hand, within the framework of the Wigner-Vlasov
formalism, the mean momentum distribution and the magnitude of the energy are
initially known. Consequently, the mean momentum distributions and energy
values obtained according to the Wigner functions can be compared with the
exact momentum distribution and energy values. This paper presents this
comparison and describes the differences. For the first Wigner function, an
analog of the Moyal equation with an electromagnetic part and the
Hamilton-Jacobi operator equation are obtained. An operator analogue of the
{\guillemotleft}motion equation{\guillemotright} with electromagnetic
interaction is constructed. For the second Vlasov equation, an operator
expression for the Vlasov-Moyal approximation for systems with electromagnetic
interaction is obtained.Comment: 26 pages, 2 figure
The Wigner function negative value domains and energy function poles of the polynomial oscillator
For a quantum oscillator with the polynomial potential an explicit expression
that describes the energy distribution as a coordinate (and momentum) function
is obtained. The presence of the energy function poles is shown for the quantum
system in the domains where the Wigner function has negative values.Comment: 21 pages, 4 figure