The Physical Origin of Schr\"odinger Equation

TL;DR

This paper derives the Schr"odinger equation and fundamental operators in quantum mechanics from classical symmetry principles, providing a physical origin for quantum formalism.

Contribution

It offers a derivation of the Schr"odinger equation and operators based solely on translation invariance, linking quantum mechanics to classical Hamiltonian symmetry.

Findings

Derivation of Schr"odinger equation from symmetry principles

Direct obtaining of commutation relations from operator forms

Insight into the physical origin of quantum mechanics

Abstract

Quantum mechanics is one of the basic theories of modern physics. Here, the famous Schr\"odinger equation and the differential operators representing mechanical quantities in quantum mechanics are derived, just based on the principle that the translation invariance (symmetry) of a system in Hamiltonian mechanics should be preserved in quantum mechanics. Moreover, according to the form of the differential operators, the commutation relation in quantum mechanics between the generalized coordinate and the generalized momentum can be directly obtained. We believe that the results in this paper are very useful for understanding the physical origin of quantum mechanics.