Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials

Abstract

After briefly reviewing the definitions of classical probability densities for position, $P_{CL}(x)$, and for momentum, $P_{CL}(p)$, we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, $V(x) = F|x|$, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent, $δ$-function classical probability density for momentum for the infinite well can be easily seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semi-classical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.