グリーン–久保公式
i ħ ∂ ρ ′ ∂ t = [ H 0 , ρ ′ ] + [ H ′ , ρ 0 ] {\displaystyle i\hbar {\partial \rho ' \over {\partial t}}=[H_{0},\rho ']+[H',\rho _{0}]}
次にρ′(t) を次のように表現し直す。
ρ ̄ ′ ≡ exp  ( i H 0 t ħ ) ρ ′ ( t ) exp  ( − i H 0 t ħ ) {\displaystyle {\bar {\rho }}'\equiv \exp \left({iH_{0}t \over \hbar }\right)\rho '(t)\exp \left({-{iH_{0}t \over \hbar }}\right)}
この時間発展は次のようになる。
i ħ ∂ ρ ̄ ′ ( t ) ∂ t = [ H ̄ ′ ( t ) , ρ 0 ] {\displaystyle i\hbar {\partial {\bar {\rho }}'(t) \over {\partial t}}=[{\bar {H}}'(t),\rho _{0}]}
H ̄ ′ ( t ) = exp  ( i H 0 t ħ ) H ′ ( t ) exp  ( − i H 0 t ħ ) {\displaystyle {\bar {H}}'(t)=\exp \left({iH_{0}t \over \hbar }\right)H'(t)\exp \left({-{iH_{0}t \over \hbar }}\right)}