Phy5645/Energy conservation: Difference between revisions

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==  Example 1  ==
(1) The energy operator in three dimensions is: <math>H=-\frac{\hbar^2}{2m}\nabla^2+V</math>
Consider a particle moving in a potential field <math>V(\textbf{r})</math>, (1) Prove the average energy equation: <math><E>=\int W d^3x=\int\left[\frac{\hbar^2}{2m}\nabla\psi^*\cdot\nabla\psi\right]d^3x</math>,
so the average energy in state <math> \Psi </math> is:
where W is energy density, (2) Prove the energy conservation equation: <math>\frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0</math>, where <math>\textbf{S}</math> is energy flux density: <math>\textbf{S}=-\frac{\hbar^2}{2m}\left(\frac{\partial\psi^*}{\partial t}\nabla\psi + \frac{\partial\psi}{\partial t}\nabla\psi^*\right)</math>
<math>\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}</math>


Prove:(1):
Using the identity, <math>\Psi^*\nabla^2\Psi=\nabla\cdot\left(\Psi^*\nabla\Psi\right)-\nabla\Psi^{\ast}\cdot\nabla\Psi,</math> we obtain
the energy operator in three dimensions is: <math>H=-\frac{\hbar^2}{2m}\nabla^2+V</math>
so the average energy in state <math> \psi </math> is:
<math><E>=\iiint \psi^*H\psi d^3x=\iiint \psi^*\left(-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi\right) d^3x </math>,
Using: <math>\psi^*\nabla^2\psi=\nabla\left(\psi^*\nabla\psi\right)-\nabla\psi^*\nabla\psi </math>,
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>
<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:
<math>\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} </math>
<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\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}</math>
with the condition: <math>\lim_{r \to \infty}\psi^*\nabla\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>
If we apply Gauss' Theorem to the first term,


(2):first we find the time derivative of energy density:
<math>-\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},</math>
<math>\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}</math>,
as well as the condition, <math>\lim_{r \to \infty}\Psi^{\ast}\nabla\Psi=0,</math> we obtain
<math>=\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}</math>
 
<math>=\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)</math>,
<math>\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}</math>
 
(2) We first find the time derivative of energy density:
 
<math>\frac{\partial W}{\partial t}=\frac{\partial}{\partial t}\left (\nabla\Psi^{\ast}\cdot\nabla\Psi+\Psi^{\ast}V\Psi\right )
=\frac{\hbar^2}{2m}\left (\nabla\Psi^{\ast}\cdot\nabla\frac{\partial\Psi}{\partial t} + \nabla\frac{\partial\Psi^{\ast}}{\partial t}\cdot\nabla\Psi\right ) + \frac{\partial\Psi^{\ast}}{\partial t}V\Psi+\Psi^{\ast}V\frac{\partial\Psi}{\partial t}</math>
<math>=\frac{\hbar^2}{2m}\left [\nabla\cdot\left (\nabla\Psi^{\ast}\frac{\partial\psi}{\partial t} + \frac{\partial\Psi^{\ast}}{\partial t}\nabla\Psi\right) - \left (\frac{\partial\Psi}{\partial t}\nabla^2\Psi^{\ast}+\frac{\partial\Psi^{\ast}}{\partial t}\nabla^2\Psi\right )\right ]+\frac{\partial\Psi^{\ast}}{\partial t}V\Psi+\Psi^{\ast}V\frac{\partial\Psi}{\partial t}</math>
<math>=\frac{\hbar^2}{2m}\nabla\cdot\left (\nabla\Psi^{\ast}\frac{\partial\Psi}{\partial t}+\frac{\partial\Psi^{\ast}}{\partial t}\nabla\Psi\right)+\frac{\partial\Psi^{\ast}}{\partial t}\left (-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi\right )+\frac{\partial\Psi}{\partial t}\left (-\frac{\hbar^2}{2m}\nabla^2\Psi^{\ast}+\nabla\Psi^{\ast}\right )</math>,
 
Using the Schrödinger equation,
<math>i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi,</math>
 
and its complex conjugate,
<math>-i\hbar\frac{\partial\Psi^{\ast}}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi^{\ast}+V\Psi^{\ast},</math>
 
and defining the energy flux density as <math>\textbf{S}=-\frac{\hbar^2}{2m}\left(\frac{\partial\Psi^{\ast}}{\partial t}\nabla\Psi + \frac{\partial\Psi}{\partial t}\nabla\Psi^{\ast}\right ),</math>
 
We obtain
<math>\frac{\partial W}{\partial t}=-\nabla\cdot\textbf{S}+\frac{\partial\Psi^{\ast}}{\partial t}\frac{\partial\Psi}{\partial t}-\frac{\partial\Psi}{\partial t}\frac{\partial\Psi^{\ast}}{\partial t}=-\nabla\cdot\textbf{S},</math>
 
or, rearranging,
<math>\frac{\partial W}{\partial t}+\nabla \cdot \textbf{S}=0.</math>
 
Back to [[Relation Between the Wave Function and Probability Density#Problems|Relation Between the Wave Function and Probability Density]]

Latest revision as of 13:21, 18 January 2014

(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) We first 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^{\ast}\cdot\nabla\Psi+\Psi^{\ast}V\Psi\right ) =\frac{\hbar^2}{2m}\left (\nabla\Psi^{\ast}\cdot\nabla\frac{\partial\Psi}{\partial t} + \nabla\frac{\partial\Psi^{\ast}}{\partial t}\cdot\nabla\Psi\right ) + \frac{\partial\Psi^{\ast}}{\partial t}V\Psi+\Psi^{\ast}V\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^{\ast}\frac{\partial\psi}{\partial t} + \frac{\partial\Psi^{\ast}}{\partial t}\nabla\Psi\right) - \left (\frac{\partial\Psi}{\partial t}\nabla^2\Psi^{\ast}+\frac{\partial\Psi^{\ast}}{\partial t}\nabla^2\Psi\right )\right ]+\frac{\partial\Psi^{\ast}}{\partial t}V\Psi+\Psi^{\ast}V\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^{\ast}\frac{\partial\Psi}{\partial t}+\frac{\partial\Psi^{\ast}}{\partial t}\nabla\Psi\right)+\frac{\partial\Psi^{\ast}}{\partial t}\left (-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi\right )+\frac{\partial\Psi}{\partial t}\left (-\frac{\hbar^2}{2m}\nabla^2\Psi^{\ast}+\nabla\Psi^{\ast}\right )} ,

Using the Schrödinger 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 i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi,}

and its complex conjugate, 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^{\ast}}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi^{\ast}+V\Psi^{\ast},}

and defining the energy flux density as 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^{\ast}}{\partial t}\nabla\Psi + \frac{\partial\Psi}{\partial t}\nabla\Psi^{\ast}\right ),}

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 \frac{\partial W}{\partial t}=-\nabla\cdot\textbf{S}+\frac{\partial\Psi^{\ast}}{\partial t}\frac{\partial\Psi}{\partial t}-\frac{\partial\Psi}{\partial t}\frac{\partial\Psi^{\ast}}{\partial t}=-\nabla\cdot\textbf{S},}

or, rearranging, 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.}

Back to Relation Between the Wave Function and Probability Density

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