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 <math>\textbf{S}<math> is energy flux density: