Phy5645/Energy conservation: Difference between revisions
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<math><E>=\iiint\left(-\frac{\hbar^2}{2m}\right)d^3x </math>, | <math><E>=\iiint\left(-\frac{\hbar^2}{2m}\right)d^3x </math>, | ||
<math>\left{\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi\right} </math> | <math>\left{\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi\right} </math>, | ||
So: | So: | ||
<math>\iiint\psi^*\nabla\psi d^3x </math>, | <math>\iiint\psi^*\nabla\psi d^3x </math>, | ||
Revision as of 16:41, 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: 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 \frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0} , 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: 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}=-\frac{\hbar^2}{2m}\left(\frac{\partial\psi^*}{\partial t}\nabla\psi + \frac{\partial\psi}{\partial t}\nabla\psi^*\right)}
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 (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\left(-\frac{\hbar^2}{2m}\right)d^3x } ,
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 \left{\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi\right} } ,
So: 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 \iiint\psi^*\nabla\psi d^3x } ,