The distribution of r⋅p in quantum mechanical systems

Abstract

The position–momentum dot product (`posmom') s = r·p of a particle is a quantum mechanical observable. In principle, its density S(s) can be derived from the position or momentum wavefunction using Mellin transforms but this leads to complicated integrals and it has therefore been largely neglected by the molecular physics community. However, we show that S(s) can be obtained easily as the Fourier transform of the hyperbolic autocorrelation of the wavefunction. Our findings are illustrated using numerical results for various states of a harmonic oscillator, a hydrogenic ion and particles in a box.