Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets

Abstract

We calculate the Wigner quasi-probability distribution for position and momentum, P (n) W (x, p), for the energy eigenstates of the standard infinite well potential, using both x-and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P W (x, p; t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like 'mini-packets' which appear at fractional multiples of the exact revival time.

Analytic Results for Gaussian Wave Packets in Four Model Systems: I. Visualization of the Kinetic Energy

“…Gaussian solutions such as those considered here can find use as examples of model systems in discussing other theoretical constructs, such as the Wigner quasi-probability distribution [32] - [42]. In that case, one defines P W (x, p; t) via which can then be evaluated explicitly in closed form, in either position-or momentumspace, for all of the solutions discussed above.…”

Phase-Space Position-Momentum Correlation and Potentials

“…The numerical procedure was checked by numerically calculating Wigner functions of the PIAB model and comparing them to their known analytical expressions [31]. The normalization of the Wigner function was also verified.…”

Two-particle Wigner functions in a one-dimensional Calogero-Sutherland potential

“…(24), rather than Eq. (25), to better visualize the phase space behaviors of WFs presented in Figs. 1-2., i.e., how the localization in q happens when the strength of spatial confinement ω is increased.…”

“…The method, however, is not easy to handle when the potential contains higher order powers of coordinates, since this case comprises a differential equation for the Wigner function with terms as much as the number of the order . In recent years there has been a number of works to calculate the WF for various type of potentials: Infinite square well [25], a double well potential [26], the Pösch-Teller potential [27], the Morse oscillator [28], a quantum damped oscillator [29], the hydrogen atom [30], the rotational motion of a spherical top [31] are notable applications. A discrete WF for non-relativistic quantum systems with one degree of freedom has been developed in finite dimensional phase space and applied to a few simple system [32].…”