Symmetries of the Hamiltonian operator and constants of motion

TL;DR

This paper establishes a fundamental link between conserved Hermitian operators and unitary symmetries of the Hamiltonian in non-relativistic quantum mechanics, showing they are two sides of the same coin.

Contribution

It proves that every conserved Hermitian operator generates a symmetry and vice versa, clarifying the relationship between symmetries and conservation laws in quantum mechanics.

Findings

Conserved Hermitian operators generate unitary symmetries.

Unitary symmetries of the Hamiltonian are generated by conserved Hermitian operators.

The paper provides a bidirectional correspondence between conservation laws and symmetries.

Abstract

It is shown that, in the framework of non-relativistic quantum mechanics, any conserved Hermitian operator (which may depend explicitly on the time) is the generator of a one-parameter group of unitary symmetries of the Hamiltonian and that, conversely, any one-parameter family of unitary symmetries of the Hamiltonian is generated by a conserved Hermitian operator.