久保公式
よって線形次数で摂動による期待値 A ^ ( t ) {\displaystyle {\hat {A}}(t)} を得る。
⟨ A ^ ( t ) ⟩ = ⟨ A ^ ⟩ 0 − i ∫ t 0 t d t ′ 1 Z 0 ∑ n e − β E n ⟨ n ( t 0 ) | A ^ ( t ) V ^ ( t ′ ) − V ^ ( t ′ ) A ^ ( t ) | n ( t 0 ) ⟩ = ⟨ A ^ ⟩ 0 − i ∫ t 0 t d t ′ ⟨ [ A ^ ( t ) , V ^ ( t ′ ) ] ⟩ 0 {\displaystyle {\begin{aligned}\langle {\hat {A}}(t)\rangle &=\langle {\hat {A}}\rangle _{0}-i\int _{t_{0}}^{t}dt'{\frac {1}{Z_{0}}}\sum _{n}e^{-\beta E_{n}}\langle n(t_{0})|{\hat {A}}(t){\hat {V}}(t')-{\hat {V}}(t'){\hat {A}}(t)|n(t_{0})\rangle \\&=\langle {\hat {A}}\rangle _{0}-i\int _{t_{0}}^{t}dt'\langle [{\hat {A}}(t),{\hat {V}}(t')]\rangle _{0}\end{aligned}}}
ここで括弧 ⟨ ⟩ 0 {\displaystyle \langle \rangle _{0}} は熱平衡状態 H ^ 0 {\displaystyle {\hat {H}}_{0}} での平均値を表す。