L2.3 Degenerate Perturbation theory: Example and setup

TL;DR

Degenerate perturbation theory fails when trying to apply non-degenerate formulas to degenerate states, leading to incorrect results.

Transcript

PROFESSOR: OK, degenerate perturbation theory. OK, we've done nicely our non-degenerate case. So we'll get degenerate done in a very clear way, I think. One more blackboard. So the first thing I want to say about degenerate perturbation theories, when it fails and why it fails and what goes wrong. So degenerate perturbation theory. Trivial example ... Read More

Key Insights

• 🚱 Degenerate perturbation theory fails when non-degenerate formulas are applied to degenerate states.
• 🍳 Degenerate states have multiple possible eigenvectors, and the choice of eigenvectors for the perturbed Hamiltonian breaks the degeneracy.
• ❓ The formulas for state corrections in degenerate perturbation theory can only be fully obtained by considering multiple equations.

Questions & Answers

Q: Why does degenerate perturbation theory fail when using non-degenerate formulas?

Degenerate perturbation theory fails when using non-degenerate formulas because degenerate states have multiple possible eigenvectors, and the formulas do not take this into account.

Q: What happens to the eigenvalues of a degenerate Hamiltonian under a perturbation?

The eigenvalues of a degenerate Hamiltonian can split under a perturbation, breaking the degeneracy.

Q: How are the eigenvectors of a perturbed Hamiltonian related to the eigenvectors of the original matrix?

The eigenvectors of a perturbed Hamiltonian are one possible choice of eigenvectors for the original matrix, and the perturbation breaks the degeneracy by making some of the equivalent eigenvectors preferred.

Q: How does degenerate perturbation theory differ from non-degenerate perturbation theory in terms of calculating state corrections?

In degenerate perturbation theory, the calculation of state corrections is more complicated because the part in the degenerate subspace cannot be calculated solely from the first equation. The second equation needs to be used to find the missing part of the state correction.

Summary & Key Takeaways

• Degenerate perturbation theory fails when using non-degenerate formulas for degenerate states.

• The formulas give incorrect results because they do not account for the fact that degenerate states have multiple possible eigenvectors.

• The eigenvectors of a perturbed Hamiltonian are one possible choice of eigenvectors for the original matrix, and the perturbation breaks the degeneracy.