Phy5645/schrodingerequationhomework2

Assume that the Hamiltonian for a system of N particles 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 \hat{H}=-\sum_{i=1}^{N}\frac{\hbar}{2m}\nabla_{i}^{2}+\sum_{i=1}^{N}\rho_{ij}[|\overrightarrow{r_{i}}-\overrightarrow{r_{j}}|]} , and $\Psi ({\overrightarrow {r_{1}}}{\overrightarrow {r_{2}}}\cdots {\overrightarrow {r_{N}}},t)$ is the wave fuction.

We define: By definition:

$\rho ({\overrightarrow {r}},t)=\sum \rho _{i}({\overrightarrow {r}},t)$

${\overrightarrow {j}}({\overrightarrow {r}},t)=\sum {\overrightarrow {j_{i}}}({\overrightarrow {r}},t)$

$\rho _{1}({\overrightarrow {r_{1}}},t)=\int \cdots \int d^{3}r_{3}d^{3}r_{3}\cdots d^{3}r_{N}\Psi ^{\star }\Psi$

${\overrightarrow {j}}({\overrightarrow {r}},t)={\frac {\hbar }{2im}}\int \cdots \int d^{3}r_{3}d^{3}r_{3}\cdots d^{3}r_{N}(\Psi ^{\star }\nabla _{1}\Psi -\Psi \nabla _{1}\Psi ^{\star })$

Prove the following relation: ${\frac {\partial \rho }{\partial t}}+\nabla \cdot {\overrightarrow {j}}=0$

Solution:

${\frac {\partial \rho }{\partial t}}={\frac {\partial }{\partial t}}\sum _{i}\rho _{i}({\overrightarrow {r}},t)$

$=\sum _{i}\int \cdots \int d^{3}r_{1}\cdots d^{3}r_{i-1}d^{3}r_{i+1}\cdots d^{3}r_{N}(\Psi ^{\star }{\frac {\partial \Psi }{\partial t}}+{\frac {\partial \Psi ^{\star }}{\partial t}}\Psi )$

$=\sum _{i}\rho _{i}({\overrightarrow {r_{i}}},t)$

$\Psi ({\overrightarrow {r_{1}}}{\overrightarrow {r_{2}}}\cdots {\overrightarrow {r_{N}}},t)$

${\begin{cases}\hbar i{\frac {\partial \Psi }{\partial t}}=\sum _{k}(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2})\Psi +\sum _{jk}v_{jk}\Psi \\-\hbar i{\frac {\partial \Psi ^{\star }}{\partial t}}=\sum _{k}(-{\frac {\hbar ^{2}}{2m}}\nabla _{k}^{2})\Psi ^{\star }+\sum _{jk}v_{jk}\Psi ^{\star }\end{cases}}$

${\frac {\partial \Psi }{\partial t}}$,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{\partial\Psi^{\star}}{\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{\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})}

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 =-\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})}

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 \nabla\cdot\overrightarrow{j}\equiv\sum_{i}\nabla_{i}\cdot\sum_{i}j_{i}(\overrightarrow{r_{i}},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 =\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}

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_{i}\cdot\overrightarrow{j_{i}}(\overrightarrow{r_{i}},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}{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})}

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}=\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})}

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}