Degenerate pertubation theory when the first order fails

[znbhckcs]

The basic algorithm of degenerate perturbation theory is quite simple:
1.Write the perturbed Hamiltonian as a matrix in the degenerate subspace.
2.Diagonalize it.
3.The eigenstates are the 'correct' states to which the system will go as the perturbation ->0.

But what to do if the first order does not break the degeneracy?
For instance, if the perturbation matrix elements are all 0 in the degenerate subspace.

It seems unlikely to me that there is nothing else to be done in that case...

 

[chafelix]

Actually when you diagonalize you solve the problem EXACTLY to all orders(in the framework of the set of states you use). It's when you have both degenerate and nondegenerate that order makes sense. To your question, if in the degerenate subspace the perturbation is 0, then to get a nonzero result you need more states.
Example: Take a static electric field (z-axis) and the n=2 level of atomic hydrogen.
210 and 200 get mixed, but 211 and 21-1 are not affected. So to affect them, you need to
include more states, e.g. n=3, when they an mix with 321 and 32-1 for example.
This is not degenerate anymore, of course

 

[Entropia]

I would like to clarify that degenerate perturbation theory is a powerful tool for understanding and predicting the behavior of quantum systems when a perturbation is introduced. However, as with any theory or algorithm, there are limitations and cases where it may not provide a satisfactory solution.

In the case where the first order perturbation does not break the degeneracy, it is important to carefully examine the system and the perturbation matrix elements. One possible solution could be to consider higher order perturbation terms, as they may contribute to breaking the degeneracy. Additionally, it may be necessary to reconsider the choice of basis states used in the perturbation theory calculations.

Furthermore, it is important to keep in mind that perturbation theory is an approximation and may not always provide an exact solution. In such cases, it may be necessary to use other techniques such as variational methods or numerical simulations to obtain a more accurate understanding of the system.

In conclusion, while degenerate perturbation theory is a valuable tool in the study of quantum systems, it is important to carefully consider its limitations and explore alternative approaches in cases where the first order perturbation does not break the degeneracy. As scientists, it is our responsibility to continue to push the boundaries of understanding and to find innovative solutions to challenging problems.