Quantum and classical probability distributions for position and momentum

Abstract

The classical and quantum probability distributions for both position and momentum are compared for several model systems admitting bound states including the harmonic oscillator, the infinite well, and the linear confining potential (V(x)=F‖x‖). Examples corresponding to unbound systems, including the uniformly accelerating particle and the motion of a particle moving away from a point of unstable equilibrium, i.e., the ‘‘unstable oscillator’’ defined by V(x)=−kx2/2, are also considered. The quantum and classical distribution of kinetic and potential energy for the harmonic oscillator is briefly discussed.