Editing Quantum mechanics (section)

=== Symmetries and conservation laws ===
{{Main|Noether's theorem}}

The Hamiltonian <math>H</math> is known as the ''generator'' of time evolution, since it defines a unitary time-evolution operator <math>U(t) = e^{-iHt/\hbar}</math> for each value of <math>t</math>. From this relation between <math>U(t)</math> and <math>H</math>, it follows that any observable <math>A</math> that commutes with <math>H</math> will be <em>conserved</em>: its expectation value will not change over time.<ref name="Zwiebach2022" />{{rp|471}} This statement generalizes, as mathematically, any Hermitian operator <math>A</math> can generate a family of unitary operators parameterized by a variable <math>t</math>. Under the evolution generated by <math>A</math>, any observable <math>B</math> that commutes with <math>A</math> will be conserved. Moreover, if <math>B</math> is conserved by evolution under <math>A</math>, then <math>A</math> is conserved under the evolution generated by <math>B</math>. This implies a quantum version of the result proven by [[Emmy Noether]] in classical ([[Lagrangian mechanics|Lagrangian]]) mechanics: for every [[differentiable]] [[Symmetry (physics)|symmetry]] of a Hamiltonian, there exists a corresponding [[conservation law]].