Straightforward Derivation of the Schr{\"o}dinger Equation from Classical Mechanics and the Planck Postulate

TL;DR

This paper presents a simple derivation of the Schrödinger equation from classical mechanics principles and the Planck postulate, challenging the belief that it cannot be derived from first principles.

Contribution

It offers a novel derivation of the Schrödinger equation using only the Hamilton-Jacobi equation and the Planck postulate, avoiding assumptions of probabilistic or statistical nature.

Findings

Derivation of Schrödinger equation from classical mechanics and Planck postulate

Avoids ad hoc assumptions about micro-scale phenomena

Provides a straightforward, principle-based derivation

Abstract

According to the widely accepted notion, the Schr{\"o}dinger equation (SE) is not derivable in principle. Contrary to this belief, we present here a straightforward derivation of SE. It is based on only two fundamentals of mechanics: the classical Hamilton-Jacobi equation(HJE) and the Planck postulate about the discrete transfer of energy at micro-scales. Our approach is drastically different from the other published derivations of SE which either employ an ad hoc underlying assumption about the probabilistic or the statistical nature of the micro-scale phenomena, or rely on the prior knowledge of SE and arrive at it by introducing a new postulate - neither present in classical mechanics nor following from experiments - with a suitable but physically unjustifiable choice of a key arbitrary constant.