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Assume that the Hamiltonian for a system of N particles is <math>\hat{H}=-\sum_{i=1}^{N}\frac{\hbar}{2m}\nabla_{i}^{2}+\sum_{i=1}^{N}\rho_{ij}[|\overrightarrow{r_{i}}-\overrightarrow{r_{j}}|]</math>, and <math>\Psi(\overrightarrow{r_{1}}\overrightarrow{r_{2}}\cdots\overrightarrow{r_{N}},t)</math> is the wave fuction.
We define:
<math>\rho(\overrightarrow{r},t)=\sum\rho_{i}(\overrightarrow{r},t)</math>
<math>\overrightarrow{j}(\overrightarrow{r},t)=\sum\overrightarrow{j_{i}}(\overrightarrow{r},t)</math>
<math>\rho_{1}(\overrightarrow{r_{1}},t)=\int\cdots\int d^{3}r_{3}d^{3}r_{3}\cdots d^{3}r_{N}\Psi^{\star}\Psi</math>
<math>\overrightarrow{j}(\overrightarrow{r},t)=\frac{\hbar}{2im}\int\cdots\int d^{3}r_{3}d^{3}r_{3}\cdots d^{3}r_{N}(\Psi^{\star}\nabla_{1}\Psi-\Psi\nabla_{1}\Psi^{\star})</math>
Prove the following relation: <math>\frac{\partial\rho}{\partial t}+\nabla\cdot\overrightarrow{j}=0</math>
Solution:
By definition:
By definition:


<math>\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial t}\sum_{i}\rho_{i}(\overrightarrow{r},t)</math>
<math>\frac{\partial\rho}{\partial t}=\frac{\partial}{\partial t}\sum_{i}\rho_{i}(\mathbf{r},t)</math>


<math>=\sum_{i}\int\cdots\int d^{3}r_{1}\cdots d^{3}r_{i-1}d^{3}r_{i+1}\cdots d^{3}r_{N}(\Psi^{\star}\frac{\partial\Psi}{\partial t}+\frac{\partial\Psi^{\star}}{\partial t}\Psi)</math>
<math>=\left.\sum_{i}\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\left (\Psi^{\star}\frac{\partial\Psi}{\partial t}+\frac{\partial\Psi^{\star}}{\partial t}\Psi\right )\right |_{\mathbf{r}_i=\mathbf{r}}</math>


<math>=\sum_{i}\rho_{i}(\overrightarrow{r_{i}},t)      \quad (1)</math>
<math>=\sum_{i}\left. \rho_{i}(\mathbf{r}_{i},t)\right |_{\mathbf{r}_i=\mathbf{r}}     \quad (1)</math>


The wave function of many particles system <math>\Psi(\overrightarrow{r_{1}}\overrightarrow{r_{2}}\cdots\overrightarrow{r_{N}},t)</math> satisfies the Schrodinger equation for many particles system:
The wave function of the many-particle system <math>\Psi(\textbf{r}_{1},\textbf{r}_{2},\ldots,\textbf{r}_{N};t)</math> satisfies the following Schrödinger equation:


<math>\begin{cases}
<math>\begin{cases}
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-i\hbar\frac{\partial\Psi^{\star}}{\partial t}=\sum_{k}(-\frac{\hbar^{2}}{2m}\nabla_{k}^{2})\Psi^{\star}+\sum_{jk}v_{jk}\Psi^{\star}\end{cases}</math>
-i\hbar\frac{\partial\Psi^{\star}}{\partial t}=\sum_{k}(-\frac{\hbar^{2}}{2m}\nabla_{k}^{2})\Psi^{\star}+\sum_{jk}v_{jk}\Psi^{\star}\end{cases}</math>


Substitute <math>\frac{\partial\Psi}{\partial t}</math> and <math>\frac{\partial\Psi^{\star}}{\partial t}</math> in to formula <math>(1)</math>
If we substitute <math>\frac{\partial\Psi}{\partial t}</math> and <math>\frac{\partial\Psi^{\star}}{\partial t}</math> into formula <math>(1)</math>, we obtain
 
<math>\frac{\partial\rho_{i}}{\partial t}=\frac{i\hbar}{2m}\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\sum_{k}(\Psi^{\star}\nabla_{k}^{2}\Psi-\Psi\nabla_{k}^{2}\Psi^{\star})</math>
 
<math>=\frac{i\hbar}{2m}\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\sum_{k}\nabla_{k}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star}),</math>


<math>\frac{\partial\rho_{i}}{\partial t}=-\int\cdots\int d^{3}r_{1}\cdots d^{3}r_{i-1}d^{3}r_{i+1}\cdots d^{3}r_{N}\cdot\sum_{k}\frac{\hbar}{2im}(\Psi^{\star}\nabla_{k}^{2}\Psi-\Psi\nabla_{k}^{2}\Psi^{\star})</math>
or, taking the sum over <math>i</math>,


<math>=-\int\cdots\int d^{3}r_{1}\cdots d^{3}r_{i-1}d^{3}r_{i+1}\cdots d^{3}r_{N}\cdot\sum_{k}\frac{\hbar}{2im}\nabla_{k}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star})</math>
<math>\frac{\partial\rho}{\partial t}=\frac{i\hbar}{2m}\sum_{i}\left.\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\sum_{k}\nabla_{k}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star})\right |_{\mathbf{r}_i=\mathbf{r}}.</math>


<math>\nabla\cdot\overrightarrow{j}\equiv\sum_{i}\nabla_{i}\cdot\sum_{i}j_{i}(\overrightarrow{r_{i}},t)</math>
Let us now consider terms for which <math>i\neq k.</math>  In these cases, we may use Gauss' Theorem, along with the requirement that <math>\lim_{r_k\rightarrow\infty}\Psi^{\ast}\nabla_{k}\Psi=0</math> for all <math>k,</math> to show that these terms must vanish.  Therefore,


<math>=\nabla_{1}\cdot\overrightarrow{j_{1}}(\overrightarrow{r_{1}},t)+\nabla_{2}\cdot\overrightarrow{j_{2}}(\overrightarrow{r_{2}},t)+\cdots\nabla_{i}\cdot\overrightarrow{j_{i}}(\overrightarrow{r_{i}},t)\cdots</math>
<math>\frac{\partial\rho}{\partial t}=\frac{i\hbar}{2m}\sum_{i}\left.\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\nabla_{i}\cdot(\Psi^{\star}\nabla_{i}\Psi-\Psi\nabla_{i}\Psi^{\star})\right |_{\mathbf{r}_i=\mathbf{r}}</math>


<math>=\sum_{i}\nabla_{i}\cdot\overrightarrow{j_{i}}(\overrightarrow{r_{i}},t)</math>
<math>=-\sum_{i}\nabla\cdot\mathbf{j}_{i}(\mathbf{r},t)=-\nabla\cdot\mathbf{j}(\mathbf{r},t),</math>


<math>=\frac{\hbar}{2im}\sum_{i}\int\cdots\int d^{3}r_{1}\cdots d^{3}r_{i-1}d^{3}r_{i+1}\cdots d^{3}r_{N}\times\nabla_{j}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star})</math>
or


<math>\frac{\partial\rho}{\partial t}=\sum_{i}\frac{\partial\rho}{\partial t}=\sum_{i}\int\cdots\int d^{3}r_{1}\cdots d^{3}r_{i-1}d^{3}r_{i+1}\cdots d^{3}r_{N}\times\sum_{k}\frac{\hbar}{2im}\nabla_{k}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star})</math>
<math>\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}=0.</math>


<math>i\neq k</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

By definition:

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\rho}{\partial t}=\frac{\partial}{\partial t}\sum_{i}\rho_{i}(\mathbf{r},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 =\left.\sum_{i}\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\left (\Psi^{\star}\frac{\partial\Psi}{\partial t}+\frac{\partial\Psi^{\star}}{\partial t}\Psi\right )\right |_{\mathbf{r}_i=\mathbf{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 =\sum_{i}\left. \rho_{i}(\mathbf{r}_{i},t)\right |_{\mathbf{r}_i=\mathbf{r}} \quad (1)}

The wave function of the many-particle system 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(\textbf{r}_{1},\textbf{r}_{2},\ldots,\textbf{r}_{N};t)} satisfies the following 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 \begin{cases} i\hbar\frac{\partial\Psi}{\partial t}=\sum_{k}(-\frac{\hbar^{2}}{2m}\nabla^{2})\Psi+\sum_{jk}v_{jk}\Psi\\ -i\hbar\frac{\partial\Psi^{\star}}{\partial t}=\sum_{k}(-\frac{\hbar^{2}}{2m}\nabla_{k}^{2})\Psi^{\star}+\sum_{jk}v_{jk}\Psi^{\star}\end{cases}}

If we substitute 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\Psi}{\partial t}} 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 \frac{\partial\Psi^{\star}}{\partial t}} into formula 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 (1)} , 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\rho_{i}}{\partial t}=\frac{i\hbar}{2m}\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\sum_{k}(\Psi^{\star}\nabla_{k}^{2}\Psi-\Psi\nabla_{k}^{2}\Psi^{\star})}

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{i\hbar}{2m}\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\sum_{k}\nabla_{k}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star}),}

or, taking the sum over 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} ,

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\rho}{\partial t}=\frac{i\hbar}{2m}\sum_{i}\left.\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\sum_{k}\nabla_{k}\cdot(\Psi^{\star}\nabla_{k}\Psi-\Psi\nabla_{k}\Psi^{\star})\right |_{\mathbf{r}_i=\mathbf{r}}.}

Let us now consider terms for which 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\neq k.} In these cases, we may use Gauss' Theorem, along with the requirement that 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_k\rightarrow\infty}\Psi^{\ast}\nabla_{k}\Psi=0} for all 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 k,} to show that these terms must vanish. Therefore,

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\rho}{\partial t}=\frac{i\hbar}{2m}\sum_{i}\left.\int\cdots\int d^{3}\mathbf{r}_{1}\,\cdots\,d^{3}\mathbf{r}_{i-1}\,d^{3}\mathbf{r}_{i+1}\,\cdots\,d^{3}\mathbf{r}_{N}\nabla_{i}\cdot(\Psi^{\star}\nabla_{i}\Psi-\Psi\nabla_{i}\Psi^{\star})\right |_{\mathbf{r}_i=\mathbf{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 =-\sum_{i}\nabla\cdot\mathbf{j}_{i}(\mathbf{r},t)=-\nabla\cdot\mathbf{j}(\mathbf{r},t),}

or

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\rho}{\partial t}+\nabla\cdot\mathbf{j}=0.}

Back to Relation Between the Wave Function and Probability Density

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