# Stationary state of two body wave function

• Piano man

#### Piano man

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

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

$$\Psi(\vec{r_1},\vec{r_2})=\Psi(\vec{R})\psi_E(\vec{r})$$

where $$\vec{R}=$$centre of mass
and $$\vec{r}=\vec{r_1}-\vec{r_2}$$

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

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

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

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!

Start from the time dependent Schrodinger Equation, not the time independent equation.