Since the Wigner function (WF) is related to a Lindard-constant type linear dielectric function derived in the symmetric gauge,7it is expected to show de Haas-van Alphen (dHvA) oscillations. Starting with the symmetric eigenfunctions, we derived the pure-state WF in a magnetic field, whose plots in phase space and in term of B-1for increasing n are consistent with the dHvA effect. Furthermore the asymptotic expansion of WF at large n show periodic oscillations with a period related to the Fermi energy. The phase space plots of WF also show that dHvA and similar oscillations could be a consequence of Nature's strategy for increasing the effective spatial range without violating the uncertainty principle. Properties of the symmetric eigenfunctions were derived. The dynamics of WF can be obtained from the solution of the time-dependent Schrödinger equation (SE). A new method to solve the SE in a magnetic field in the interaction picture based on expansion in term of symmetric eigenfunctions has been developed. The matrix element for a Gaussian potential were derived explicitly, plotted against B-1, and showed oscillations. The total WF was shown to be a linear combination of the diagonal pure-state WF's by using the orthogonality for symmetric eigenfunctions. The no-special-point property for WF was confirmed, which is important for the construction of a numerical algorithm to solve the SE in a magnetic field.
Since the Wigner function (WF) is related to a Lindard-constant type linear dielectric function derived in the symmetric gauge,7it is expected to show de Haas-van Alphen (dHvA) oscillations. Starting with the symmetric eigenfunctions, we derived the pure-state WF in a magnetic field, whose plots in phase space and in term of B-1for increasing n are consistent with the dHvA effect. Furthermore the asymptotic expansion of WF at large n show periodic oscillations with a period related to the Fermi energy. The phase space plots of WF also show that dHvA and similar oscillations could be a consequence of Nature's strategy for increasing the effective spatial range without violating the uncertainty principle. Properties of the symmetric eigenfunctions were derived. The dynamics of WF can be obtained from the solution of the time-dependent Schrödinger equation (SE). A new method to solve the SE in a magnetic field in the interaction picture based on expansion in term of symmetric eigenfunctions has been developed. The matrix element for a Gaussian potential were derived explicitly, plotted against B-1, and showed oscillations. The total WF was shown to be a linear combination of the diagonal pure-state WF's by using the orthogonality for symmetric eigenfunctions. The no-special-point property for WF was confirmed, which is important for the construction of a numerical algorithm to solve the SE in a magnetic field.
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. The approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy. The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well.
Using the Wigner-Vlasov formalism, an exact three-dimensional solution of the Schr\"odinger equation for a scalar particle in an electromagnetic field is constructed. Electric and magnetic fields are nonuniform. According to the exact expression for the wave function, the search for two types of Wigner function is conducted. The first function is the usual Wigner function with a kinetic momentum. The second Wigner function is constructed on the basis of the Weyl-Stratonovich transform [Javanainen et al., Phys. Rev. A 35, 2791 (1987); Nedjalkov et al., 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). An example of an electromagnetic quantum system described by a non-Gaussian wave function has successfully been found. The second Wigner function is positive over the whole phase space for the non-Gaussian wave function. This result is analogous to the Hudson theorem for the gauge invariant Wigner function. 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. The Vlasov-Moyal approximation of average acceleration flow has been built in phase space for a quantum system with an electromagnetic field. The obtained approximation makes it possible to cut the Vlasov chain off at the second equation and also to analyze the Boltzmann H-function evolution. By averaging the Vlasov-Moyal approximation over momentum space we can derive an expression for the electromagnetic force of the classical system. This averaging makes the high-order quantum terms disappear from the ``motion equation.''
The two-dimensional hydrogen atom in an external magnetic field is considered in the context of phase space. Using the solution of the Schrödinger equation in phase space, the Wigner function related to the Zeeman effect is calculated. For this purpose, the Bohlin mapping is used to transform the Coulomb potential into a harmonic oscillator problem. Then, it is possible to solve the Schrödinger equation easier by using the perturbation theory. The negativity parameter for this system is realised.
By using the technique of integration within an ordered product (IWOP) of operators, we derive the normal-product form of the gauge-invariant Wigner operator of an electron in a uniform magnetic field and its expression in the newly constructed |λ> representation. The virtue of working in the |λ> representation lies in the fact that it is expressed in terms of the ladder operators Π±, K±(see Eq. (10)), thus the magnetic vector potential can naturally be included. On these bases we can easily obtain Wigner functions of some important electron states.
We found analytically the exact joint Wigner function (WF) of an atom-field system described by the Jaynes-Cummings model in the dispersive limit for a given initial state. Using the solution of a master equation with a dissipative term calculated by Peixoto de Faria and Nemes (Peixoto de Faria J G and Nemes M C 1999 Phys. Rev. A 59 3918) and a general expression for the joint WF proposed by Czirják and Benedict (Czirják A and Benedict M G 1996 Quantum Semiclass. Opt. 8 975), we obtained the exact reduced atom and field WFs and analysed their time evolution. Our results show that, at the maximum entanglement times, the atom tends to a statistical mixture state represented by a spherical polar plot. Also, the atomic WF shows how fast the negative-valued part disappears as a function of dissipation constant and the strength of the field coherent state.
Abstract This paper presents a comprehensive investigation of the problem of a harmonic oscillator with time-depending frequencies in the framework of the Vlasov theory and the Wigner function apparatus for quantum systems in the phase space. A new method is proposed to find an exact solution of this problem using a relation of the Vlasov equation chain with the Schrödinger equation and with the Moyal equation for the Wigner function. A method of averaging the energy function over the Wigner function in the phase space can be used to obtain time-dependent energy spectrum for a quantum system. The Vlasov equation solution can be represented in the form of characteristics satisfying the Hill equation. A particular case of the Hill equation, namely the Mathieu equation with unstable solutions, has been considered in details. An analysis of the dynamics of an unstable quantum system shows that the phase space square bounded with the Wigner function level line conserves in time, but the phase space square bounded with the energy function line increases. In this case the Vlasov equation characteristic is situated on the crosspoint of the Wigner function level line and the energy function line. This crosspoint moves in time with a trajectory that represents the unstable system dynamics. Each such trajectory has its own energy, and the averaging of these energies by the Wigner function results in time-dependent discreet energy spectrum for the whole system. An explicit expression has been obtained for the Wigner function of the 4th rank in the generalized phase space x , p , p ̇ , p ̈ .