Quantum Mechanics Course

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← Part VI/Degenerate Perturbation

2. Degenerate Perturbation Theory

Reading time: ~32 minutes | Pages: 8

Special treatment needed when unperturbed levels have the same energy.

The Degeneracy Problem

Non-degenerate perturbation theory breaks down when:

$$E_n^{(0)} = E_m^{(0)} \quad \text{for } n \neq m$$

Denominators $E_n^{(0)} - E_m^{(0)}$ vanish, leading to divergences

Degenerate Subspace

Suppose $g$-fold degeneracy at energy $E_n^{(0)}$:

$$\hat{H}_0|\psi_n^{(0),\alpha}\rangle = E_n^{(0)}|\psi_n^{(0),\alpha}\rangle, \quad \alpha = 1, 2, \ldots, g$$

The $g$ states span a degenerate subspace

The Key Insight

Any linear combination is also an eigenstate:

$$|\phi\rangle = \sum_{\alpha=1}^g c_\alpha|\psi_n^{(0),\alpha}\rangle$$

Goal: Find the "right" linear combinations that work well with perturbation theory

Good Basis Selection

Choose basis states that diagonalize the perturbation within the degenerate subspace:

$$W_{\alpha\beta} = \langle\psi_n^{(0),\alpha}|\hat{H}'|\psi_n^{(0),\beta}\rangle$$

$W$ is a $g \times g$ matrix called the perturbation matrix

Diagonalization Procedure

Step 1: Construct the $W$ matrix

Step 2: Find eigenvalues and eigenvectors of $W$:

$$\sum_\beta W_{\alpha\beta} c_\beta^{(i)} = E^{(1),i} c_\alpha^{(i)}$$

Step 3: Eigenvalues are first-order energy corrections

Step 4: Eigenvectors give "good" basis states:

$$|\psi_n^{(0),i}\rangle = \sum_\alpha c_\alpha^{(i)}|\psi_n^{(0),\alpha}\rangle$$

Splitting of Degeneracy

Perturbation typically lifts degeneracy:

$$E_n^i = E_n^{(0)} + E^{(1),i} + E^{(2),i} + \cdots$$

Originally degenerate levels split into $g$ distinct (or partially split) levels

Example: Two-Fold Degeneracy

For $g = 2$, the $W$ matrix is:

$$W = \left(\begin{array}{cc}W_{11} & W_{12}\\W_{21} & W_{22}\end{array}\right)$$

Eigenvalues:

$$E^{(1)}_\pm = \frac{W_{11} + W_{22}}{2} \pm \sqrt{\left(\frac{W_{11} - W_{22}}{2}\right)^2 + |W_{12}|^2}$$

Example: Stark Effect in Hydrogen (n=2)

At $n=2$, hydrogen has 4-fold degeneracy: $|2,0,0\rangle$, $|2,1,0\rangle$, $|2,1,1\rangle$, $|2,1,-1\rangle$

Electric field perturbation:

$$\hat{H}' = eE_0 z$$

Selection rules: only $|2,0,0\rangle$ and $|2,1,0\rangle$ mix

Energy shifts (linear in field):

$$E^{(1)} = \pm 3eE_0 a_0$$

Symmetry Breaking

Degeneracy often protected by symmetry. Perturbation breaks symmetry:

  • Spherical symmetry → Cylindrical: Magnetic field along z-axis
  • Rotational → Translational: Crystal field
  • Time-reversal: Magnetic perturbations

Accidental vs Essential Degeneracy

Essential degeneracy:

  • Required by symmetry (e.g., angular momentum $m$ values)
  • Cannot be lifted without breaking symmetry

Accidental degeneracy:

  • Not required by symmetry (e.g., hydrogen $n,\ell$ degeneracy)
  • Can be lifted by perturbations respecting system symmetry

When to Use Degenerate Theory

  • Always start with degenerate theory when degeneracy present
  • After diagonalization, may proceed with non-degenerate methods for higher orders
  • Watch for near degeneracies: $|E_n^{(0)} - E_m^{(0)}| \ll |H'|$
  • Zeeman effect, Stark effect, crystal field splitting all require this approach
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