Need help regarding transition dipole moment

[semc]

Hi, I have been trying to get the expression for the transition dipole moment of hydrogen but I am not able to get the expression. Hope someone can help me with that.

I want to evaluate $\vec{d}(v)=<v|\hat{r}|0>$ where v is the free state and 0 is the 1s wave function for hydrogen.

$\vec{d}(v)=\int dτ\frac{\alpha^{3/4}}{\pi^{1/2}}exp(-\sqrt{\alpha}r)(-ih\frac{d}{dp})exp(-ipr)$

After integration by parts i got $\frac{\alpha^{3/4}}{\pi^{1/2}}\left(\frac{1}{\sqrt{\alpha}+ip}\right)^{2}$

However, it should be $\vec{d}(v)=i\left(\frac{2^{7/2}}{\pi}\alpha^{5/4}\right)\frac{p}{\left( p^2+\alpha\right)^3}$.

Can someone point out where I went wrong?

 

[Meir Achuz]

exp(-ipr)should be $exp(-i{\bf p\cdot r})$,
and $-i\hbar\partial_p$ can just be ${\bf r}$.
Then you need angular integration.

 

[semc]

Actually I am having trouble doing the integration due to the plane wave part since there is a dot product over there. I have been looking up the meaning of dipole moment and transition dipole moment but can't find much information. Is there any recommended books to read up on this? Thanks!

 

[Meir Achuz]

${\bf p\cdot r}=pr\cos(\theta)$. Then the angular integration is easy.