Obtaining the Probability Vector Current Density in Canonical Quantum Mechanics by Linear Superposition

TL;DR

This paper develops a systematic method to construct the probability vector current density in quantum mechanics using linear superposition, clarifying its physical interpretation and relation to classical physics.

Contribution

It introduces a unique, closed-form linear-superposition approach to derive the probability vector current density from its divergence, resolving ambiguities and linking it to classical counterparts.

Findings

Constructs probability vector current density solely from its divergence.

Shows the method applies to all classical Hamiltonians via linear superposition.

Identifies the Ehrenfest subclass related to classical physics.

Abstract

The quantum mechanics status of the probability vector current density has long seemed to be marginal. On one hand no systematic prescription for its construction is provided, and the special examples of it that are obtained for particular types of Hamiltonian operator could conceivably be attributed to happenstance. On the other hand this concept's key physical interpretation as local average particle flux, which flows from the equation of continuity that it is supposed to satisfy in conjunction with the probability scalar density, has been claimed to breach the uncertainty principle. Given the dispiriting impact of that claim, we straightaway point out that the subtle directional nature of the uncertainty principle makes it consistent with the measurement of local average particle flux. We next focus on the fact that the unique closed-form linear-superposition quantization of any classical Hamiltonian function yields in tandem the corresponding unique linear-superposition closed-form divergence of the probability vector current density. Because the probability vector current density is linked to the quantum physics only through the occurrence of its divergence in the equation of continuity, it is theoretically most appropriate to construct this vector field exclusively from its divergence -- analysis of the best-known "textbook" special example of a probability vector current density shows that it is thus constructed. That special example in fact leads to the physically interesting "Ehrenfest subclass" of probability vector current densities, which are closely related to their classical peers.