Phy5645/Energy conservation

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(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 \left\langle E\right\rangle=\iiint \Psi^{\ast}H\Psi\,d^3\textbf{r}=\iiint \Psi^{\ast}\left (-\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi\right )\,d^3\textbf{r}}

Using the identity, 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\cdot\left(\Psi^*\nabla\Psi\right)-\nabla\Psi^{\ast}\cdot\nabla\Psi,} we obtain

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\langle E\right\rangle=-\frac{\hbar^2}{2m}\iiint\left (\nabla\cdot\left (\Psi^{\ast}\nabla\Psi\right)-\nabla\Psi^{\ast}\cdot\nabla\Psi\right )\,d^3\textbf{r}+\iiint\Psi^{\ast}V\Psi\,d^3\textbf{r} } 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\cdot\left (\Psi^{\ast}\nabla\Psi\right)\,d^3\textbf{r}+\frac{\hbar^2}{2m}\iiint\nabla\Psi^{\ast}\cdot\nabla\Psi\,d^3\textbf{r}+\iiint\Psi^{\ast}V\Psi\,d^3\textbf{r}}

If we apply Gauss' Theorem to the first 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^{\ast}\nabla\Psi\right )\,d^3\textbf{r}=\iint\Psi^{\ast}\nabla\Psi\cdot d\textbf{S},}

as well as 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^{\ast}\nabla\Psi=0,} we obtain

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\langle E\right\rangle=\int W\,d^3\textbf{r}=\int\left [\frac{\hbar^2}{2m}\nabla\Psi^{\ast}\cdot\nabla\Psi+\Psi^{\ast}V\Psi\right ]d^3\textbf{r}}

(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}

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