Phy5645/Energy conservation: Difference between revisions
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== Example 1 == | == Example 1 == | ||
Consider a particle moving in a potential field <math>V(\textbf{r})</math>, (1) Prove the average energy equation: <math><E>=\int W d^3x=\int\left[\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi\right]d^3x</math>, | Consider a particle moving in a potential field <math>V(\textbf{r})</math>, (1) Prove the average energy equation: <math><E>=\int W d^3x=\int\left[\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi\right]d^3x</math>, | ||
where W is energy density, (2) Prove the energy conservation equation: <math>\frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0</math>, where <math>\textbf{S}<math> is energy flux density: | where W is energy density, (2) Prove the energy conservation equation: <math>\frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0</math>, where <math>\textbf{S}</math> is energy flux density: | ||
Revision as of 15:57, 9 December 2009
Example 1
Consider a particle moving in a potential field , (1) Prove the average energy equation: , where W is energy density, (2) Prove the energy conservation equation: , where is energy flux density:
![{\displaystyle <E>=\int Wd^{3}x=\int \left[{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]d^{3}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7a2a78ffd153d91faa93bd06e59ed7112944bb)
