Reality of linear and angular momentum expectation values in bound states

TL;DR

This paper explores the coordinate dependence of momentum operators in quantum mechanics, especially in spherical coordinates, and demonstrates that expectation values in bound states are zero, clarifying misconceptions and addressing interpretational issues.

Contribution

It explicitly calculates momentum expectation values in spherical coordinates and discusses the interpretation of angular variables in quantum mechanics.

Findings

Momentum expectation values in bound states are zero.

Coordinate dependence affects the form of the momentum operator.

Heisenberg's equation of motion for the radial momentum is derived.

Abstract

In quantum mechanics textbooks the momentum operator is defined in the Cartesian coordinates and rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work we introduce the difficulties one faces when the question of the momentum operator in spherical polar coordinate comes. We have tried to point out most of the elementary quantum mechanical results, related to the momentum operator, which has coordinate dependence. We explicitly calculate the momentum expectation values in various bound states and show that the expectation value really turns out to be zero, a consequence of the fact that the momentum expectation value is real. We comment briefly on the status of the angular variables in quantum mechanics and the problems related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg's equation of motion for the radial component of the momentum for the Hydrogen atom.