Phy5645/Energy conservation: Difference between revisions
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Using: <math>\psi^*\nabla^2\psi=\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi </math>, | Using: <math>\psi^*\nabla^2\psi=\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi </math>, | ||
hence: | hence: | ||
<math><E>=\iiint\left(-\frac{\hbar^2}{2m}\right)\left{\nabla\left(\psi^*\psi\right)-\nabla\psi^*\nabla\psi\right} d^3x+\iiint\psi^*\nabla\psi d^3x </math>, | <math><E>=\iiint\left(-\frac{\hbar^2}{2m}\right) | ||
\left{\nabla\left(\psi^*\psi\right)-\nabla\psi^*\nabla\psi\right} d^3x+\iiint\psi^*\nabla\psi d^3x </math>, | |||
Revision as of 16:32, 9 December 2009
Example 1
Consider a particle moving in a potential field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\textbf{r})} , (1) Prove the average energy equation: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <E>=\int W d^3x=\int\left[\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi\right]d^3x} , where W is energy density, (2) Prove the energy conservation equation: , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{S}} is energy flux density:
Prove: the energy operator in three dimensions is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <H>=-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi} so the average energy in state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi } is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <E>=\iiint \psi^*H\psi d^3x=\iiint \psi^*\left(-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi\right) d^3x } , Using: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi^*\nabla^2\psi=\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi } , hence: Failed to parse (syntax error): {\displaystyle <E>=\iiint\left(-\frac{\hbar^2}{2m}\right) \left{\nabla\left(\psi^*\psi\right)-\nabla\psi^*\nabla\psi\right} d^3x+\iiint\psi^*\nabla\psi d^3x } ,