Classical probability density distributions with uncertainty relations for ground states of simple non-relativistic quantum-mechanical systems

TL;DR

This paper demonstrates that incorporating the Heisenberg uncertainty principle into classical probability distributions significantly improves their agreement with quantum ground state distributions, surpassing traditional methods like WKB.

Contribution

It introduces a simple method to enhance classical distributions with uncertainty relations, achieving closer alignment with quantum distributions for ground states.

Findings

Classical distributions are greatly improved by including uncertainty relations.

The method yields better results than the WKB approximation.

The approach is applicable to excited states as well.

Abstract

The probability density distributions for the ground states of certain model systems in quantum mechanics and for their classical counterparts are considered. It is shown, that classical distributions are remarkably improved by incorporating into them the Heisenberg uncertainty relation between position and momentum. Even the crude form of this incorporation makes the agreement between classical and quantum distributions unexpectedly good, except for the small area, where classical momenta are large. It is demonstrated that the slight improvement of this form, makes the classical distribution very similar to the quantum one in the whole space. The obtained results are much better than those from the WKB method. The paper is devoted to ground states, but the method applies to excited states too.