エーレンフェストの定理
− i ħ ∫ [ ∂ ψ ∗ ∂ t ∇ ψ + ψ ∗ ∇ ∂ ψ ∂ t ] d r = − i ħ ∫ [ − 1 i ħ ( H ^ ψ ∗ ) ∇ ψ + ψ ∗ ∇ 1 i ħ ( H ^ ψ ) ] d r = ∫ [ { − ħ 2 2 m ∇ 2 + U ( r ) } ψ ∗ ∇ ψ − ψ ∗ ∇ { − ħ 2 2 m ∇ 2 + U ( r ) } ψ ] d r = − ħ 2 2 m ∫ [ ∇ 2 ψ ∗ ∇ ψ − ψ ∗ ∇ 3 ψ ] d r + ∫ [ U ( r ) ψ ∗ ∇ ψ − ψ ∗ ∇ ( U ( r ) ψ ) ] d r {\displaystyle {\begin{aligned}-i\hbar \int \left[{\frac {\partial \psi ^{*}}{\partial t}}\nabla \psi +\psi ^{*}\nabla {\frac {\partial \psi }{\partial t}}\right]\mathrm {d} \mathbf {r} &=-i\hbar \int \left[-{\frac {1}{i\hbar }}({\hat {H}}\psi ^{*})\nabla \psi +\psi ^{*}\nabla {\frac {1}{i\hbar }}({\hat {H}}\psi )\right]\mathrm {d} \mathbf {r} \\&=\int \left[\left\{-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {r} )\right\}\psi ^{*}\nabla \psi -\psi ^{*}\nabla \left\{-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+U(\mathbf {r} )\right\}\psi \right]\mathrm {d} \mathbf {r} \\&=-{\frac {\hbar ^{2}}{2m}}\int \left[\nabla ^{2}\psi ^{*}\nabla \psi -\psi ^{*}\nabla ^{3}\psi \right]\mathrm {d} \mathbf {r} +\int \left[U(\mathbf {r} )\psi ^{*}\nabla \psi -\psi ^{*}\nabla (U(\mathbf {r} )\psi )\right]\mathrm {d} \mathbf {r} \end{aligned}}}