The Degeneracy

Lifting degeneracy with perturbation theory.

The Problem

Perturbation theory works when the unperturbed energy levels are distinct. It does not work when two or more states share the same energy. In that case, the denominators in the second-order correction — the energy differences $E_n^{(0)} - E_m^{(0)}$ — vanish, the formula diverges, and the method breaks down. The states are degenerate. The perturbation cannot be treated as a small correction to a single state, because the perturbation mixes the degenerate states completely.

This is not a minor issue. Degeneracy is common. Symmetry produces it. The hydrogen atom's $n=2$ level is fourfold degenerate (ignoring spin). The three $p$ orbitals are degenerate in the absence of a magnetic field. Crystal field splitting, spin-orbit coupling, external fields — almost every physical system has degenerate or nearly degenerate states at some point. Perturbation theory must handle them.

The Resolution

The resolution is simple in principle and requires a change in perspective. Instead of treating the perturbation as a correction to a single state $|n^{(0)}\rangle$, you treat it as a correction to the entire degenerate subspace. You diagonalize the perturbation within that subspace. The eigenvectors of the perturbation matrix become the "correct" zeroth-order states, and the eigenvalues give you the first-order energy corrections.

Concretely: if states $|a^{(0)}\rangle, |b^{(0)}\rangle, |c^{(0)}\rangle$ all have the same unperturbed energy $E^{(0)}$, you build the matrix

$$W_{ij} = \langle i^{(0)} | V | j^{(0)} \rangle$$

for all pairs within the degenerate subspace. You diagonalize this matrix. The eigenvalues are the first-order corrections to the energy. The eigenvectors are the linear combinations of degenerate states that the perturbation selects as its preferred basis.

This is called lifting the degeneracy. The perturbation splits the degenerate level into distinct levels, each shifted by a different amount. The splitting is the first-order correction. It is often the dominant effect.

The Selection

Not all bases are equal. The unperturbed Hamiltonian $H_0$ treats all degenerate states equally — they all have the same energy, so $H_0$ cannot distinguish them. The perturbation $V$ breaks this equivalence. It acts as a discriminator. Some linear combinations of the degenerate states are eigenstates of $V$; others are not. The eigenstates of $V$ are the states that the perturbation respects. These are the correct zeroth-order states, because in these states, the perturbation is already diagonal at first order.

This is the key insight of degenerate perturbation theory: you must choose the basis that diagonalizes the perturbation before you can apply perturbation theory. If you choose the wrong basis — if you pick states that are eigenstates of $H_0$ but not of $V$ — the perturbative corrections will blow up, because the matrix elements between the states you picked will be large and the energy denominators will be zero.

The perturbation selects the basis. The basis is not unique — if the perturbation matrix has degenerate eigenvalues, there is still residual degeneracy — but it is determined by the physics. The perturbation is the selector.

The Partial Lifting

Often, the perturbation does not lift the degeneracy completely. A fourfold degenerate level might split into two doubly degenerate levels. A threefold degeneracy might split into one singlet and one doublet. The pattern of splitting depends on the symmetry of both $H_0$ and $V$. If $V$ preserves some of the symmetry of $H_0$, the degeneracy is only partially lifted. The remaining degeneracy corresponds to the symmetry that both Hamiltonians share.

This is useful. The pattern of splitting encodes information about the symmetries of the problem. By observing how a degenerate level splits in a perturbation, you can infer what symmetries the perturbation breaks and which it preserves. Degeneracy lifting is a diagnostic tool.

The Near-Degeneracy

Sometimes the energy levels are not exactly degenerate, but nearly so. $E_a^{(0)} - E_b^{(0)} = \epsilon$, where $\epsilon$ is small but non-zero. In this case, the second-order correction is large:

$$E_a^{(2)} = \frac{|\langle a^{(0)} | V | b^{(0)} \rangle|^2}{\epsilon}$$

If $\epsilon$ is comparable to or smaller than the matrix element, the perturbative expansion breaks down even though the levels are technically non-degenerate. The remedy is the same: treat the near-degenerate pair as an effective two-level system, diagonalize the $2 \times 2$ matrix, and use the result as the starting point for higher-order corrections.

This is called the near-degenerate approximation, and it is one of the most practically useful applications of degenerate perturbation theory. Molecular vibrations, avoided crossings, and resonances all involve near-degenerate states, and in all of them, the two-level diagonalization gives the correct leading behavior.

The Lesson

Degeneracy is not a failure of perturbation theory. It is a signal that the problem requires a different starting basis. The perturbation theory itself does not change — you still expand in powers of $\lambda$ — but the zeroth-order states must be chosen to diagonalize the perturbation within the degenerate subspace. Once you make that change, the theory works as before. The degeneracy is lifted. The corrections are well-defined. The answer emerges.

The lesson is one of flexibility: when the obvious starting point fails, look for the basis that the problem itself suggests. The perturbation is telling you which states are physically relevant. Listen to it.