Consider a particle moving in a potential field V ( r ) {\displaystyle V({\textbf {r}})} , (1) Prove the average energy equation: < E >= ∫ W d 3 x = ∫ [ ℏ 2 2 m ∇ ψ ∗ ⋅ ∇ ψ ] d 3 x < m a t h > , w h e r e W i s e n e r g y d e n s i t y , ( 2 ) P r o v e t h e e n e r g y c o n s e r v a t i o n e q u a t i o n :< m a t h > ∂ W ∂ t + ∇ ⋅ S = 0 {\displaystyle <E>=\int Wd^{3}x=\int \left[{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]d^{3}x<math>,whereWisenergydensity,(2)Provetheenergyconservationequation:<math>{\frac {\partial W}{\partial t}}+\nabla \cdot {\textbf {S}}=0} , where <math>\textbf{S}<math> is energy flux density:
, (1) Prove the average energy equation: ![{\displaystyle <E>=\int Wd^{3}x=\int \left[{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]d^{3}x<math>,whereWisenergydensity,(2)Provetheenergyconservationequation:<math>{\frac {\partial W}{\partial t}}+\nabla \cdot {\textbf {S}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e51751e85bee1f85dff415d25ad3caafacf7ee7a)
, where <math>\textbf{S}<math> is energy flux density: