3.1: Time-Independent Degenerate Perturbation Theory

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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 \(|m \rangle\).