An Analytic Study of Energy Eigenstates of Piecewise-constant Potentials Using the Wigner Quasi-probability Distribution

In the study of classical oscillating systems, a phase-space description is often useful in determining the long-term properties of a system’s motion. Wigner, over 70 years ago, was one of the first to introduce a phase-space description of quantum mechanics with a quasi-probability density joint in $x$ and $p$. The Wigner function is considered a ``quasi"- probability density because it can be negative for states which lack a classical analog and because of obvious problems raised by the Heisenberg uncertainty principle which restrict the ability to make simultaneous measurements of both $x$ and $p$. While many standard potentials have been analyzed using the Wigner function, including that of the free and accelerating particle and the harmonic oscillator, other familiar bound-state problems have not. We calculate the Wigner quasi-probability distribution for the energy eigenstates of several standard piecewise-constant one-dimensional potentials---attractive Dirac delta function, infinite well, finite well, asymmetric infinite well---as well as visualize the results.