[quantum mechanics] Perturbation theory in a degenerate case

[bznm]

Homework Statement

I'm trying to understand how we can find - at the first order - the energy-shift and the eigenstates in a degenerate case.

My notes aren't clear, so I have searched in the Sakurai, but the notation is different, I have read other notes but their notation is different again... Please, tell me the formulas, at the moment I don't have to know the proof. Many thanks

 

[blue_leaf77]

In degenerate case, you have to find the set of new states in which the perturbation term is diagonal. Having found the right set of states, the first order correction to the energy formula is the same as the non-degenerate case.

I have read other notes but their notation is different again

What are you going to do if you cannot find book which uses the same notation as the ones in your notes? You just have to put up with their own notation if you want to be able to understand things.

 

[bznm]

What are you going to do if you cannot find book which uses the same notation as the ones in your notes? You just have to put up with their own notation if you want to be able to understand things.

Sure, you're right. But in this moment, I have the brain "full" and I'm not able to find the correct result.. If you could help me giving the formula and the explanation about its notation, I think that I could understood more rapidly how to "translate" my notation with the book notation... If you can help me, I'll be grateful. In this moment I'm very confused.

 

[blue_leaf77]

The formula for the first order correction is the same as that in the non-degenerate case, namely $E_m' = \langle \phi_m | H' | \phi_m \rangle$, where $\phi_m$ is one of the states in the new set in which $H'$ is diagonal. $H'$ is the perturbation term.

 

[bznm]

I have understood that I have to identify the imperturbed kets that are degenerate and find the matrix V, where V_mm'=<m|V|m'> and |m>, |m'> are eigenstates that have the same eigenvalue.

Then I have to diagonalize V and the eigenvalues are the energy-shifts at the first order. But I haven't understood what does it means "and find the related eigenstates". Do I have to find the eigenvectors related to the eigenvalues of V? But aren't they the correct zero-th order kets that the perturbed kets approaches when the perturbation approaches to zero? How I can find the first-order eingenvectors?