ルンゲ・グロスの定理
i ∂ ∂ t ( j ( r , t ) − j ′ ( r , t ) ) | t = t 0 = ⟨ Ψ ( t 0 ) | [ j ^ ( r ) , H ^ v ( t 0 ) − H ^ v ′ ( t 0 ) ] | Ψ ( t 0 ) ⟩ = ⟨ Ψ ( t 0 ) | [ j ^ ( r ) , V ^ ( t 0 ) − V ^ ′ ( t 0 ) ] | Ψ ( t 0 ) ⟩ = i ρ ( r , t 0 ) ∇ ( v ( r , t 0 ) − v ′ ( r , t 0 ) ) {\displaystyle {\begin{aligned}i\left.{\frac {\partial }{\partial t}}{\big (}{\boldsymbol {j}}({\boldsymbol {r}},t)-{\boldsymbol {j}}'({\boldsymbol {r}},t){\big )}\right|_{t=t_{0}}&=\langle \Psi (t_{0})|[{\hat {\boldsymbol {j}}}({\boldsymbol {r}}),{\hat {H}}_{v}(t_{0})-{\hat {H}}_{v'}(t_{0})]|\Psi (t_{0})\rangle \\&=\langle \Psi (t_{0})|[{\hat {\boldsymbol {j}}}({\boldsymbol {r}}),{\hat {V}}(t_{0})-{\hat {V}}'(t_{0})]|\Psi (t_{0})\rangle \\&=i\rho ({\boldsymbol {r}},t_{0})\nabla {\big (}v({\boldsymbol {r}},t_{0})-v'({\boldsymbol {r}},t_{0}){\big )}\end{aligned}}}