Revisiting Dirac and Schr\"odinger: A Proof Offered for the Non-Relativistic Time-Dependent Schr\"odinger Equation

TL;DR

This paper constructs an axiomatic framework using Dirac formalism to prove that the non-relativistic time-dependent Schr"odinger equation is a theorem rather than a postulate, strengthening the theoretical foundation.

Contribution

It provides a rigorous proof that the Schr"odinger equation can be derived from axioms, challenging its traditional status as a postulate in quantum mechanics.

Findings

Schr"odinger equation is derivable from axioms

Strengthens the theoretical foundation of quantum mechanics

Reveals the equation's dependence on other axioms

Abstract

In this paper, three plausible axioms together with two definitions are employed to build an axiomatic framework, and then with the help of the Dirac formalism, it is demonstrated that the time-dependent Schr\"odinger wave equation is no longer a postulate for the whole theory, but a theorem. Subsequently, a proof for the theorem is presented. The result implies that the other remaining axioms of the picture essentially involve the Schr\"odinger equation, and this consequence lets the whole theory become stronger.