A Heuristic Derivation Of The Schr\"odinger Equation and the Momentum Operator

TL;DR

This paper provides a heuristic derivation of the Schrödinger equation and the momentum operator using de Broglie's hypothesis, experimental results, and theoretical insights to clarify their origins for students.

Contribution

It offers a new heuristic approach that explains the derivation of the Schrödinger equation and momentum operator, addressing gaps in traditional pedagogical methods.

Findings

Clarifies the origin of the Schrödinger equation

Derives the momentum operator heuristically

Enhances understanding of quantum foundations

Abstract

Students in a quantum mechanics course are often introduced to the Schr\"odinger equation as the standard mathematical tool. However, rarely do students develop an understanding as to why the equation is the choice for modeling quantum phenomena or where it came from. While many books do have a heuristic derivation, they sometimes differ in their approach and often fail to give a satisfying explanation to the so-called canonical substitution of momentum. In this paper, we use de Broglie's hypothesis along with experimental results and theories to provide a heuristic derivation of the Schr\"odinger equation and of the the momentum operator.