where, due to time evolution of operators in the Heisenberg picture, O ( t ) = U † † ( t , 0 ) O ( 0 ) U ( t , 0 ) . The time-evolution unitary operator U ( t 2 , t 1 ) is the time-ordered exponential of an integral, U ( t 2 , t 1 ) = T ( e − − i ∫ ∫ t 1 t 2 H ( t ′ ) d t ′ ) . (Note that if the Hamiltonian at one time commutes with the Hamiltonian at different times, then this can be simplified to U ( t 2 , t 1 ) = e − − i ∫ ∫ t 1 t 2 H ( t ′ ) d t ′ .)