Phy5645/Energy conservation: Difference between revisions

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hence:
hence:
<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^*\nabla\psi d^3x</math>,
=-\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 , (1) Prove the average energy equation: , where W is energy density, (2) Prove the energy conservation equation: , where is energy flux density:

Prove: the energy operator in three dimensions is: so the average energy in state is: , Using: , hence: ,

Using Gauss Theorem for the last term: , with the condition: \lim_{n \to \infty}\psi^*\nable\psi=0</math>, for infinite surface.

Hence: