Phy5645/Energy conservation

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:(1): 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 } 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 + \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: 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 \lim_{r \to \infty}\psi^*\nabla\psi=0} , 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}

(2):first we find the time derivative of energy 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 \frac{\partial W}{\partial t}=\frac{\partial}{\partial t}\left(\nabla\psi^*\nabla\psi+\psi^*\nabla\psi\right) =\frac{\hbar^2}{2m}\left(\nabla\psi^*\nabla\frac{\partial\psi}{\partial t} + \nabla\frac{\partial\psi^*}{\partial t}\nabla\psi\right) + \frac{\partial\psi^*}{\partial t}\nabla\psi+\psi^*\nabla\frac{\partial\psi}{\partial t}} , 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}\left(\nabla\cdot\left(\nabla\psi^*\cdot\frac{\partial\psi}{\partial t} + \frac{\partial\psi^*}{\partial t}\cdot\nabla\psi\right) - \left(\frac{\partial\psi}{\partial t}\nabla^2\psi^*+\frac{\partial\psi^*}{\partial t}\nabla^2\psi\right)\right)+\frac{\partial\psi^*}{\partial t}\nabla\psi+\psi^*\nabla\frac{\partial\psi}{\partial t}} 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}\nabla\cdot\left(\nabla\psi^*\cdot\frac{\partial\psi}{\partial t} + \frac{\partial\psi^*}{\partial t}\cdot\nabla\psi\right)+\frac{\partial\psi^*}{\partial t}\left(-\frac{\hbar^2}{2m}\nabla^2\psi+\nabla\psi\right)+\frac{\partial\psi}{\partial t}\left(-\frac{\hbar^2}{2m}\nabla^2\psi^*+\nabla\psi^*\right)} ,

Using Schrodinger Equations: 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 i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi+\nabla\psi} , and, 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 -i\hbar\frac{\partial\psi^*}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi^*+\nabla\psi^*} ,

Also the energy flux density 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 \textbf{S}=-\frac{\hbar^2}{2m}\left(\frac{\partial\psi^*}{\partial t}\nabla\psi + \frac{\partial\psi}{\partial t}\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 \frac{\partial W}{\partial t}=-\nabla\cdot\textbf{S}+\frac{\partial\psi^*}{\partial t}\frac{\partial\psi}{\partial t}-\frac{\partial\psi}{\partial t}\frac{\partial\psi^*}{\partial t}=-\nabla\cdot\textbf{S}} , 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 \frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0}