In the passive transformation picture, translational invariance requires, [ T ^ ^ ( x ) , H ^ ^ ] = 0 It follows that [ T ^ ^ ( x ) , U ^ ^ ( t ) ] = 0 where U ^ ^ ( t ) is the unitary time evolution operator. When the Hamiltonian is time independent, U ^ ^ ( t ) = exp ⁡ ⁡ ( − − i H ^ ^ t ℏ ℏ ) If the Hamiltonian is time dependent, the above commutation relation is satisfied if p ^ ^ or T ^ ^ ( x ) commutes with H ^ ^ ( t ) for all t.