エーレンフェストの定理
∫ ∇ 2 ψ ∗ r ψ d r = [ ∇ ψ ∗ r ψ ] − ∞ + ∞ − ∫ ∇ ψ ∗ ∇ ( r ψ ) d r = − [ ψ ∗ ∇ ( r ψ ) ] − ∞ + ∞ + ∫ ψ ∗ ∇ 2 ( r ψ ) d r = ∫ ψ ∗ ∇ ( ∇ r ψ + r ∇ ψ ) d r = ∫ [ ψ ∗ ∇ ψ + ψ ∗ ∇ ( r ∇ ψ ) ] d r = ∫ [ 2 ψ ∗ ∇ ψ + ψ ∗ r ∇ 2 ψ ] d r {\displaystyle {\begin{aligned}\int \nabla ^{2}\psi ^{*}\mathbf {r} \psi \ \mathrm {d} \mathbf {r} &=[\nabla \psi ^{*}\mathbf {r} \psi ]_{-\infty }^{+\infty }-\int \nabla \psi ^{*}\nabla (\mathbf {r} \psi )\mathrm {d} \mathbf {r} \\&=-[\psi ^{*}\nabla (\mathbf {r} \psi )]_{-\infty }^{+\infty }+\int \psi ^{*}\nabla ^{2}(\mathbf {r} \psi )\mathrm {d} \mathbf {r} \\&=\int \psi ^{*}\nabla (\nabla \mathbf {r} \psi +\mathbf {r} \nabla \psi )\mathrm {d} \mathbf {r} \\&=\int \left[\psi ^{*}\nabla \psi +\psi ^{*}\nabla (\mathbf {r} \nabla \psi )\right]\mathrm {d} \mathbf {r} \\&=\int \left[2\psi ^{*}\nabla \psi +\psi ^{*}\mathbf {r} \nabla ^{2}\psi \right]\mathrm {d} \mathbf {r} \end{aligned}}}