3.1: Time-Independent Degenerate Perturbation Theory

We have seen how we can find approximate solutions for a system whose Hamiltonian is of the form

$$\hat{H} = \hat{H}_0 + \hat{V} \nonumber$$

When we assumed that $\hat{H}$ and $\hat{H}_0$ possess discrete, non-degenerate eigenvalues only. This led to a mixing of states where

$$|\phi_0 \rangle = |n_0 \rangle + \sum_{k \neq 0} \frac{V_{k0}}{(E_0 − E_k)} |n_k \rangle \nonumber$$

Clearly, if $E_0 = E_k$ this diverges. As do the higher order energy shifts (see 2.4). Thus for the degenerate case we cannot associate a particular perturbed state $|\phi_0 \rangle$ with a particular unperturbed state $|n_0 \rangle$: we need to take a different approach. In fact, the approximation we make is completely different: we assume that the small perturbation only mixes those states which are degenerate. We then solve the problem exactly for that subset of states.

Assume that $\hat{H}_0$ possesses $N$ degenerate eigenstates $|m \rangle$ with eigenvalue $E_{deg}$. It may also possesses non-degenerate eigenstates, which can be treated separately by non-degenerate perturbation theory. We write a perturbed eigenstate $|\phi_j \rangle$ as an linear expansion in the unperturbed degenerate eigenstates only:

$$|\phi_j \rangle = \sum_i |m_i \rangle \langle m_i | \phi_j \rangle = \sum_i c_{ji} | m_i \rangle \nonumber$$

Where $i$ here runs over degenerate states only. The TISE now becomes:

$$[\hat{H}_0 + \hat{V} ] |\phi_j \rangle = [\hat{H}_0 + \hat{V} ] \sum_i c_{ji} |m_i \rangle = E_j \sum_i c_{ji} |m_i \rangle \nonumber$$

but we know that for all degenerate eigenstates $\hat{H}_0|m_i \rangle = E_{deg} |m_i \rangle$. So we obtain:

$$\sum_i c_{ji} \hat{V} | m_i \rangle = (E_j - E_{deg}) \sum_i c_{ji} | m_i \rangle \nonumber$$

premultiplying by some unperturbed state $\langle m_k|$ gives

$$\sum_i c_{ji} \left[ \langle m_k | \hat{V} | m_i \rangle - \delta_{ik} (E_j - E_{deg}) \right] = 0 \nonumber$$

We can get a similar equation from each unperturbed state $|m_k \rangle$. We thus have an eigenvalue problem: the eigenvector has elements $c_{ji}$ and the eigenvalues are $\Delta E_j = E_j − E_{deg}$. Writing the matrix elements between the $i^{th}$ and $k^{th}$ unperturbed degenerate states as $V_{ik} \equiv \langle m_i |\hat{V} |m_k \rangle$ we recover the determinantal equation:

$$\begin{vmatrix} V_{11} − \Delta E_j & V_{12} & ... & V_{1N} \\ V_{21} & V_{22} − \Delta E_j & ... & V_{2N} \\ ... & ... & ... & ... \\ V_{N1} & V_{N2} & ... & V_{NN} − \Delta E_j \end{vmatrix} = 0 \nonumber$$

The $N$ eigenvalues obtained by solving this equation give the shifts in energy due to the perturbation, and the eigenvectors give the perturbed states $|\phi \rangle$ in the unperturbed, degenerate basis set $|m \rangle$.