Transition amplitudes and relation between wavefunctions

[stunner5000pt]

bb] The dipole transition amplitude for the transition (nlm) -> (n'l'm') is given by [/b]

$$\int \psi_{n'l'm'}^* \vec{r} \psi_{nlm} d\tau$$
Is the dipole transition amplitude simply a measure of how likely a certain transiton is??

Heres another question
In converting $$\psi_{nlm_{l}m_{s}} = C_{n}\psi_{nljm_{j}}$$
my prof said that finding the Cn would involve a rather messy calculation involving group theory... how does that come about?? How does a simple relation like j = l + s bring about something like that??

thanks for your input!

 

[dextercioby]

bb] The dipole transition amplitude for the transition (nlm) -> (n'l'm') is given by [/b]

$$\int \psi_{n'l'm'}^* \vec{r} \psi_{nlm} d\tau$$
Is the dipole transition amplitude simply a measure of how likely a certain transiton is??

It's a probability amplitude. Its square modulus gives the transition probability.

Heres another question
In converting $$\psi_{nlm_{l}m_{s}} = C_{n}\psi_{nljm_{j}}$$
my prof said that finding the Cn would involve a rather messy calculation involving group theory... how does that come about?? How does a simple relation like j = l + s bring about something like that??

The Clebsch-Gordan coefficients that you need are a result of group theory. Angular momentum theory (including the addition of angular momenta) is a result of group theory.