Derivation of the Schr\"odinger equation from the Hamilton-Jacobi equation in Feynman's path integral formulation of quantum mechanics

TL;DR

This paper derives the Schrödinger equation from Feynman's path integral formulation and the Hamilton-Jacobi equation, connecting classical and quantum mechanics through a simple, foundational approach.

Contribution

It provides a straightforward derivation of the Schrödinger equation from classical mechanics principles within Feynman's framework, highlighting the role of the Hamilton-Jacobi equation.

Findings

Derivation of Schrödinger equation from path integrals and Hamilton-Jacobi equation

Critical discussion of de Broglie's matter waves

Review of Schrödinger's original derivations

Abstract

It is shown how the time-dependent Schr\"{o}dinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schr\"{o}dinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi equation, are also reviewed. The derivation of the time-dependent equation is based on an {\it a priori} assumption equivalent to Feynman's dynamical postulate. De Broglie's concepts of 'matter waves' and their phase and group velocities are also critically discussed.