エネルギー保存の法則
となり、 O ^ {\displaystyle {\hat {O}}} の時間発展を記述する作用素が得られる。
W ( t 1 , t 2 ) = − ∫ t 1 t 2 ∇ V ( r ( t ) ) ⋅ d r ( t ) + ∫ t 1 t 2 f ( t ) ⋅ d r ( t ) = − ∫ t 1 t 2 { ∂ V ( r ( t ) ) ∂ x d x ( t ) d t + ∂ V ( r ( t ) ) ∂ y d y ( t ) d t + ∂ V ( r ( t ) ) ∂ z d z ( t ) d t } d t + ∫ t 1 t 2 f ( t ) ⋅ d r ( t ) , {\displaystyle {\begin{aligned}W(t_{1},t_{2})&=-\int _{t_{1}}^{t_{2}}\nabla V({\boldsymbol {r}}(t))\cdot d{\boldsymbol {r}}(t)+\int _{t_{1}}^{t_{2}}{\boldsymbol {f}}(t)\cdot d{\boldsymbol {r}}(t)\\&=-\int _{t_{1}}^{t_{2}}\left\{{\frac {\partial V({\boldsymbol {r}}(t))}{\partial x}}{\frac {dx(t)}{dt}}+{\frac {\partial V({\boldsymbol {r}}(t))}{\partial y}}{\frac {dy(t)}{dt}}+{\frac {\partial V({\boldsymbol {r}}(t))}{\partial z}}{\frac {dz(t)}{dt}}\right\}dt+\int _{t_{1}}^{t_{2}}{\boldsymbol {f}}(t)\cdot d{\boldsymbol {r}}(t),\end{aligned}}}
となるので、ポテンシャルの時間についての全微分、
d V d t ( r ( t ) ) = { d x ( t ) d t ∂ ∂ x + d y ( t ) d t ∂ ∂ y + d z ( t ) d t ∂ ∂ z } V ( r ( t ) ) {\displaystyle {\frac {dV}{dt}}({\boldsymbol {r}}(t))=\left\{{\frac {dx(t)}{dt}}{\frac {\partial }{\partial x}}+{\frac {dy(t)}{dt}}{\frac {\partial }{\partial y}}+{\frac {dz(t)}{dt}}{\frac {\partial }{\partial z}}\right\}V({\boldsymbol {r}}(t))}
d d t ⟨ ψ | O ^ | ψ ⟩ = ( ∂ ∂ t | ψ ⟩ ∗ ) O ^ | ψ ⟩ + | ψ ⟩ ∗ ∂ ∂ t O ^ | ψ ⟩ = i ħ ( − ħ i ∂ ∂ t | ψ ⟩ ) ∗ O ^ | ψ ⟩ + | ψ ⟩ ∗ i ħ ħ i ∂ ∂ t O ^ | ψ ⟩ = i ħ { ( H ^ | ψ ⟩ ) ∗ O ^ | ψ ⟩ + | ψ ⟩ ∗ ħ i ( ∂ O ^ ∂ t + O ^ ∂ ∂ t ) | ψ ⟩ } = i ħ { | ψ ⟩ ∗ H ^ ∗ O ^ | ψ ⟩ + | ψ ⟩ ∗ ( ħ i ∂ O ^ ∂ t − O ^ H ^ ) | ψ ⟩ } = ⟨ ψ | ( ∂ O ^ ∂ t + i ħ { H ^ O ^ − O ^ H ^ } ) | ψ ⟩ {\displaystyle {\begin{aligned}{\frac {d}{dt}}\left\langle \psi \right\vert {\hat {O}}\left\vert \psi \right\rangle &=\left({\frac {\partial }{\partial t}}\left\vert \psi \right\rangle ^{*}\right){\hat {O}}\left\vert \psi \right\rangle +\left\vert \psi \right\rangle ^{*}{\frac {\partial }{\partial t}}{\hat {O}}\left\vert \psi \right\rangle \\&={\frac {i}{\hbar }}\left(-{\frac {\hbar }{i}}{\frac {\partial }{\partial t}}\left\vert \psi \right\rangle \right)^{*}{\hat {O}}\left\vert \psi \right\rangle +\left\vert \psi \right\rangle ^{*}{\frac {i}{\hbar }}{\frac {\hbar }{i}}{\frac {\partial }{\partial t}}{\hat {O}}\left\vert \psi \right\rangle \\&={\frac {i}{\hbar }}\left\{\left({\hat {H}}\left\vert \psi \right\rangle \right)^{*}{\hat {O}}\left\vert \psi \right\rangle +\left\vert \psi \right\rangle ^{*}{\frac {\hbar }{i}}\left({\frac {\partial {\hat {O}}}{\partial t}}+{\hat {O}}{\frac {\partial }{\partial t}}\right)\left\vert \psi \right\rangle \right\}\\&={\frac {i}{\hbar }}\left\{\left\vert \psi \right\rangle ^{*}{\hat {H}}^{*}{\hat {O}}\left\vert \psi \right\rangle +\left\vert \psi \right\rangle ^{*}\left({\frac {\hbar }{i}}{\frac {\partial {\hat {O}}}{\partial t}}-{\hat {O}}{\hat {H}}\right)\left\vert \psi \right\rangle \right\}\\&=\left\langle \psi \right\vert \left({\frac {\partial {\hat {O}}}{\partial t}}+{\frac {i}{\hbar }}\left\{{\hat {H}}{\hat {O}}-{\hat {O}}{\hat {H}}\right\}\right)\left\vert \psi \right\rangle \end{aligned}}}