Phy5645/Energy conservation

Example 1

Consider a particle moving in a potential field $V({\textbf {r}})$, (1) Prove the average energy equation: $<E>=\int Wd^{3}x=\int \left[{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]d^{3}x$, where W is energy density, (2) Prove the energy conservation equation: ${\frac {\partial W}{\partial t}}+\nabla \cdot {\textbf {S}}=0$, where ${\textbf {S}}$ is energy flux density: ${\textbf {S}}=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial \psi ^{*}}{\partial t}}\nabla \psi +{\frac {\partial \psi }{\partial t}}\nabla \psi ^{*}\right)$

Prove: the energy operator in three dimensions is: $<H>=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi$ so the average energy in state $psi$ is: $<E>=iiint\psi ^{*}Hpsid^{3}x=iiintpsi^{*}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi \right)d^{3}x$