Quantum mechanics of a free particle from properties of the Dirac delta function

TL;DR

This paper derives fundamental quantum properties of a free particle, including eigenfunctions, commutation relations, and Ehrenfest theorem, solely from the properties of the Dirac delta function, highlighting its foundational role.

Contribution

It introduces a novel derivation of key quantum mechanics principles for a free particle based entirely on delta function properties, without additional assumptions.

Findings

Eigenfunction form derived from delta function properties

Canonical commutation relation obtained through differentiation of delta function

Ehrenfest theorem demonstrated using delta function properties

Abstract

Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, $\bra{x} p > = \exp(ipx/\hbar)/\sqrt{2\pi\hbar}$. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of $\bra{x} p >$.