Degenerate perturbation theory is a method used in quantum mechanics to analyze systems where the non-perturbed Hamiltonian operator has degenerate energy levels. In quantum mechanics, a degenerate energy level occurs when multiple distinct quantum states have the same energy. When you apply a perturbation to such a system, the perturbed energy levels and wave functions need to be determined while accounting for the degeneracy.

Here's how degenerate perturbation theory works:

Non-degenerate vs. Degenerate Systems: In non-degenerate perturbation theory, you can easily calculate the perturbed energy levels and wave functions when there is no degeneracy. The perturbation theory equations for non-degenerate systems are straightforward.
First-Order Perturbation Theory: The first step in degenerate perturbation theory is to find the first-order correction to the energy levels. This is done by calculating the matrix elements of the perturbing Hamiltonian operator within the subspace of degenerate states. The first-order energy correction is given by:
ΔE₁ = 〈ψⁿ|H' |ψⁿ〉
Here, |ψⁿ⟩ represents the unperturbed degenerate states, and H' is the perturbing Hamiltonian.
Second-Order Perturbation Theory: In the case of degeneracy, you need to go beyond first-order perturbation theory to calculate more accurate energy corrections. Second-order perturbation theory involves calculating the second-order energy correction, which accounts for interactions between the degenerate states. The second-order energy correction is given by:
ΔE₂ = ∑(n≠m) |〈ψⁿ|H'|ψᵐ〉|² / (Eⁿ - Eᵐ)
In this formula, the summation is performed over all combinations of degenerate states, and Eⁿ and Eᵐ are the unperturbed energies of the states |ψⁿ⟩ and |ψᵐ⟩, respectively.
Perturbed Wave Functions: After calculating the energy corrections, you can determine the perturbed wave functions for the degenerate states. These wave functions include contributions from other degenerate states, and they can be found by considering the matrix elements of the perturbing Hamiltonian.
Selection Rules: In degenerate perturbation theory, selection rules become important in determining which perturbed transitions are allowed and which are forbidden. Selection rules help identify the perturbed states that are coupled by the perturbing Hamiltonian.
We consider the case where the energy eigen value of Ho is gn field degenerate. We have

From 1, 2, 3 and 4

It can be shown that

Since the eigen value En is gn fold degeneracy.

There are gn orthonormal eigen kets |ψni〉; i = 1, 2, ....gn with eigen value En(0).

So we take linear combination.

From *

Multiplying both sides by

Degenerate perturbation theory is particularly useful when dealing with complex quantum systems, such as atoms and molecules, where degeneracies frequently occur. It allows for a more accurate description of energy level corrections and wave functions when multiple states share the same energy, making it a valuable tool in quantum mechanics and quantum chemistry.

This note is a part of the Physics Repository.

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