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: <math>\textbf{S}=-\frac{hbar^2}{2m}\left(frac{\partial\psi^*}{\partial t}\nabla\psi + frac{\partial\psi}{\partial t}\nabla\psi^*\right)