Why Does the H_{\alpha} Line of the Balmer Series Split?

[Reshma]

I need to show the splitting of the H_{\alpha} line of the Balmer series of a Hydrogen atom using the spin hypothesis.

Here is my work on this.
H_{\alpha} is obtained from the transition of the electron from the third orbit to the second i.e. from n=3 to n=2.
Seletion rule for Hydrogen atom is: \Delta l = \pm 1 & \Delta m_l = \pm 1 , 0.
This implies transition from 3s to 2s is forbidden but transition from 3p to 2s is allowed. How do I explain the splitting of the spectral line?

 

[dextercioby]

You need to do perturbation theory to the H atom using the spin-orbit coupling hamiltonian which should be assumed to be a time independent perturbation. One more thing: is the energy level involved in a H_alpha degenerate or not ?

Daniel.

 

[Reshma]

You need to do perturbation theory to the H atom using the spin-orbit coupling hamiltonian which should be assumed to be a time independent perturbation. One more thing: is the energy level involved in a H_alpha degenerate or not ?

Daniel.

I haven't studied perturbation theory. I cannot say whether the energy level of H_alpha is degenerate. They can involve transition from 3 ²P_1/2 or 3 ²P_3/2.

However, since the Hydrogen atom consists of only 1 electron, so s = +1/2 or -1/2. Is it possible to show the splitting using spin hypothesis?