We derive explicit expressions for the Wigner function of wave functions in
D dimensions which depend on the hyperradius---that is, of
s waves. They are based either on the position or the momentum representation of the
s wave. The corresponding Wigner function depends on three variables: the absolute value of the
D-dimensional position and momentum vectors and the angle between them. We illustrate these expressions by calculating and discussing the Wigner functions of an elementary
s wave and the energy eigenfunction of a free particle.
Abstract In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.
Bath-induced correlations in an infinite dimension Hilbert space has been shown. Quantum correlations between two free spinless dissipative distinguishable particles (initially uncorrelated) are studied analytically using the quantum master equation. We show that for temperature
T>0 the time-evolution of the reduced density matrix cannot be written as the direct product of two independent particles. We found a time-scale that characterizes the time when the bath-induced correlation is maximum before being wipe out by dissipation (purity, spatial, and mirror correlations are studied). We introduce a Wigner function associated to the Wannier lattice where the dissipative quantum walks move. Thus we discuss the peculiarities of the scenario of enlarging the lattice to the semi-integers in order to fulfilll all the fundamental properties of the Wigner function. We have supported the quantum character of the correlations by analyzing the negativity of the Wigner function.