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Stationary state of two body wave function

  • Thread starter Piano man
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  • #1
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The ultimate goal of the problem is to show that the stationary state two body wave function can be written as
[tex] \Psi(\vec{r_1},\vec{r_2},t)=e^{i\vec{P}\cdot\vec{R}/\hbar}\psi_E(\vec{r})e^{-i\left[\frac{P^2}{2M}+E\right]t/\hbar}[/tex]

So far, I have separated the variables in the time independent equation:

[tex] \Psi(\vec{r_1},\vec{r_2})=\Psi(\vec{R})\psi_E(\vec{r})[/tex]

where [tex]\vec{R}=[/tex]centre of mass
and [tex]\vec{r}=\vec{r_1}-\vec{r_2}[/tex]

and have the two equations
[tex] \frac{-\hbar^2}{2M}\nabla^2_R\Psi(\vec{R})=E_{com}\Psi(\vec{R})[/tex] and

[tex] \left[\frac{-\hbar^2}{2\mu}\nabla^2_r+V(\vec{r})\right]\psi_E(\vec{r})=E\psi_E(\vec{r})[/tex]
where [tex]M=m_1+m_2[/tex]
and [tex]\mu=[/tex]reduced mass

...So...what I think I have to do is write
[tex] \Psi(\vec{r_1},\vec{r_2},t)=\Psi(\vec{R})\psi_E(\vec{r})\phi(t)[/tex]

and solve each equation separately, but I'm not sure how to get the answer in the first equation I wrote. For example, how do I introduce time?

Any suggestions would be great!
 

Answers and Replies

  • #2
G01
Homework Helper
Gold Member
2,665
16
Start from the time dependent Schrodinger Equation, not the time independent equation.
 

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