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Symmetries and conservation laws

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