Visualizing classical and quantum probabilitydensities for momentum using variations on familiarone-dimensional potentials

Abstract

After briefly reviewing the definitions of classical probabilitydensities for position, PCL(x), and for momentum,PCL(p), we present several examples of classicalmechanical potential systems, mostly variations on such familiarcases as the infinite well and the uniformly acceleratedparticle for which the classical distributions can be easilyderived and visualized. We focus especially on a simplepotential which interpolates between the symmetric linearpotential, V(x) = F|x|, and the infinite well, which canillustrate, in a mathematically straightforward way, how thedivergent δ-function classical probability density formomentum for the infinite well can be seen to arise. Such examples can help students understand the quantummechanical momentum-space wavefunction (and its correspondingprobability density) in much the same way that othersemiclassical techniques, such as the WKB approximation, can beused to visualize position-space wavefunctions.