Wigner Quantum Density Function in the Classical Limit. Development of a Three-Dimensional WKBJ-Type Solution

The one-dimensional Wigner quasiprobability function is considered rigorously in the classical limit ($\hslash \to 0$). With the implicit $\hslash$-dependence of the wave function $\psi$ ignored in the integral representation, this limit differs from ${\left|\psi \mathrm{}\right(q\left)\right|}^2\delta \mathrm{}\left(p\right)\mathrm{}$ by terms $\frac{o\left(\hslash \right)}{\hslash }$. With the implicit $\hslash$ dependence of $\psi$ accounted for (as it should be), rigorous asymptotic methods are applied, with the results that: (1) in the "classically allowed" region, the limit is an asymptotic exponential function of $q$, multiplied by an energy-conserving $\delta$ function; (2) in the "nonclassically allowed" region, this limit is zero. The stationary-phase approach leads nowhere in this regard. A WKBJ-type solution for the radial Schrödinger equation is developed, with rigorous error bounds in the different situations. The Wigner function is computed for an ensemble of one-dimensional harmonic oscillators, and an associated phase-space "density matrix" is constructed therefrom. This is compared with the Slater sum for the ensemble—in general, and in the particular cases of high temperature and of $\hslash \to 0$; where $\hslash \to 0$, the asymptotic behavior of the Wigner function is explicitly manifest.