Emmy Noether Lagrangian differentiable symmetry conservation law Hamiltonian Hermitian operators unitary
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.
