Derivation of the Schrodinger equation from fundamental principles

TL;DR

This paper rigorously derives the Schrödinger equation from fundamental principles, clarifying its connection to particle energy, momentum, and probability amplitude without relying on heuristic methods.

Contribution

It provides a formal derivation of the Schrödinger equation based on basic assumptions about probability amplitudes and wave relations, moving beyond heuristic approaches.

Findings

Derivation of Schrödinger equation from fundamental assumptions

Clarification of the relation between energy, momentum, and wave properties

Formal basis for the wave function as a probability amplitude

Abstract

Schrodinger path to the quantum mechanical wave equation was heuristic and guided more by physical intuition than formal deduction. Here we derive the Schrodinger equation for the particle wave function, assuming that it has a meaning of the probability amplitude to find the particle at time t at point r and the relations E=hw, p=hk expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave.