Phy5645/Energy conservation: Difference between revisions
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<math><E>=\iiint\left(-\frac{\hbar^2}{2m}\right)\left(\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi\right)d^3x +\iiint\psi^*\nabla\psi d^3x | <math><E>=\iiint\left(-\frac{\hbar^2}{2m}\right)\left(\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi\right)d^3x +\iiint\psi^*\nabla\psi d^3x | ||
=-\frac{\hbar^2}{2m}\iiint\nabla\left(\psi^*\nabla\psi\right)d^3x + \frac{\hbar^2}{2m}\iiint\nabla\psi^*\nabla\psi d^3x + \iiint\psi^* | =-\frac{\hbar^2}{2m}\iiint\nabla\left(\psi^*\nabla\psi\right)d^3x + \frac{\hbar^2}{2m}\iiint\nabla\psi^*\nabla\psi d^3x + \iiint\psi^*V\psi d^3x</math>, | ||
Using Gauss Theorem for the last term: | Using Gauss Theorem for the last term: | ||
<math>-\frac{\hbar^2}{2m}\iiint\nabla\left(\psi^*\nabla\psi\right) d^3x=\iint\psi^*\nabla\psi\cdot d\textbf{S}</math>, | <math>-\frac{\hbar^2}{2m}\iiint\nabla\left(\psi^*\nabla\psi\right) d^3x=\iint\psi^*\nabla\psi\cdot d\textbf{S}</math>, | ||
with the condition: \lim_{n \to \infty}\psi^*\nable\psi=0</math>, for infinite surface. | |||
Hence:<math><E>=\int W d^3x=\int\left[\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi\right]d^3x</math> | |||
Revision as of 20:50, 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+V} 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)\left(\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi\right)d^3x +\iiint\psi^*\nabla\psi d^3x =-\frac{\hbar^2}{2m}\iiint\nabla\left(\psi^*\nabla\psi\right)d^3x + \frac{\hbar^2}{2m}\iiint\nabla\psi^*\nabla\psi d^3x + \iiint\psi^*V\psi d^3x} ,
Using Gauss Theorem for the last term: 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{\hbar^2}{2m}\iiint\nabla\left(\psi^*\nabla\psi\right) d^3x=\iint\psi^*\nabla\psi\cdot d\textbf{S}} , with the condition: \lim_{n \to \infty}\psi^*\nable\psi=0</math>, for infinite surface.
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>=\int W d^3x=\int\left[\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi\right]d^3x}