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: $<E>=\int Wd^{3}x=\int \left[{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]d^{3}x$, 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 ${\textbf {S}}$ 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)}

Proof:

(1):the energy operator in three dimensions is: $H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V$ 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: $<E>=\iiint \psi ^{*}H\psi d^{3}x=\iiint \psi ^{*}\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi \right)d^{3}x$, 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: $<E>=\iiint \left(-{\frac {\hbar ^{2}}{2m}}\right)\left(\nabla \left(\psi ^{*}\nabla \psi \right)-\nabla \psi ^{*}\nabla \psi \right)d^{3}x+\iiint \psi ^{*}\nabla \psi d^{3}x$ $=-{\frac {\hbar ^{2}}{2m}}\iiint \nabla \left(\psi ^{*}\nabla \psi \right)d^{3}x+{\frac {\hbar ^{2}}{2m}}\iiint \nabla \psi ^{*}\nabla \psi d^{3}x+\iiint \psi ^{*}V\psi d^{3}x$,

Using Gauss Theorem for the last term: $-{\frac {\hbar ^{2}}{2m}}\iiint \nabla \left(\psi ^{*}\nabla \psi \right)d^{3}x=\iint \psi ^{*}\nabla \psi \cdot d{\textbf {S}}$, with the condition: $\lim _{r\to \infty }\psi ^{*}\nabla \psi =0$, for infinite surface.

Hence:$<E>=\int Wd^{3}x=\int \left[{\frac {\hbar ^{2}}{2m}}\nabla \psi ^{*}\cdot \nabla \psi \right]d^{3}x$

(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}} , $={\frac {\hbar ^{2}}{2m}}\left(\nabla \cdot \left(\nabla \psi ^{*}\cdot {\frac {\partial \psi }{\partial t}}+{\frac {\partial \psi ^{*}}{\partial t}}\cdot \nabla \psi \right)-\left({\frac {\partial \psi }{\partial t}}\nabla ^{2}\psi ^{*}+{\frac {\partial \psi ^{*}}{\partial t}}\nabla ^{2}\psi \right)\right)+{\frac {\partial \psi ^{*}}{\partial t}}\nabla \psi +\psi ^{*}\nabla {\frac {\partial \psi }{\partial t}}$ 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: $i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +\nabla \psi$, 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: ${\textbf {S}}=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial \psi ^{*}}{\partial t}}\nabla \psi +{\frac {\partial \psi }{\partial t}}\nabla \psi ^{*}\right)$,

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: ${\frac {\partial W}{\partial t}}+\nabla \cdot {\textbf {S}}=0$