Table of Contents
- Introduction
- Why Degeneracy Requires Special Treatment
- The Failure of Non-Degenerate Formulas
- Basic Setup of Degenerate Perturbation Theory
- Perturbation Within the Degenerate Subspace
- Constructing the Effective Hamiltonian
- Diagonalizing the Perturbation Matrix
- First-Order Corrections to Energy and States
- Physical Interpretation
- Example: Zeeman Effect in Hydrogen
- Example: Stark Effect in Degenerate Systems
- Higher-Order Corrections
- Applications and Importance
- Limitations and Alternatives
- Conclusion
1. Introduction
In quantum mechanics, degenerate perturbation theory is used when the unperturbed Hamiltonian of a system has multiple eigenstates corresponding to the same energy. These degenerate states require a special treatment since standard perturbation theory produces undefined expressions due to division by zero. Degenerate perturbation theory helps correctly describe how these energy levels and states evolve under a small perturbation.
2. Why Degeneracy Requires Special Treatment
Degenerate energy levels occur in systems with symmetries. Because the energy is the same for multiple eigenstates, any linear combination of these states is also an eigenstate. Perturbations typically lift this degeneracy and introduce energy splittings. However, standard perturbation theory fails in this scenario, necessitating a more careful diagonalization of the perturbation within the degenerate subspace.
3. The Failure of Non-Degenerate Formulas
The standard first-order correction to the wavefunction is given by:
\[
|\psi_n^{(1)}\rangle = \sum_{k \ne n} \frac{\langle \psi_k^{(0)} | \hat{H}’ | \psi_n^{(0)} \rangle}{E_n^{(0)} – E_k^{(0)}} |\psi_k^{(0)}\rangle
\]
This expression diverges when \( E_k^{(0)} = E_n^{(0)} \), which occurs in degenerate systems. Moreover, choosing a specific \( |\psi_n^{(0)}\rangle \) from the degenerate set is ambiguous without further constraints.
4. Basic Setup of Degenerate Perturbation Theory
Suppose \( \hat{H}0 \) has a degenerate eigenvalue \( E^{(0)} \) with \( d \) linearly independent eigenstates \( \{ |\phi_a^{(0)}\rangle \}{a=1}^d \). The goal is to find linear combinations of these states that remain eigenstates when the perturbation \( \hat{H}’ \) is added.
5. Perturbation Within the Degenerate Subspace
We construct the matrix of the perturbation \( \hat{H}’ \) within the degenerate subspace:
\[
H’_{ab} = \langle \phi_a^{(0)} | \hat{H}’ | \phi_b^{(0)} \rangle
\]
This Hermitian matrix represents the perturbation’s action on the degenerate space.
6. Constructing the Effective Hamiltonian
We define the effective Hamiltonian:
\[
\hat{H}_{\text{eff}} = P \hat{H}’ P
\]
Where \( P \) is the projector onto the degenerate subspace. The eigenvectors of \( \hat{H}_{\text{eff}} \) give the proper combinations of the original degenerate states that diagonalize \( \hat{H}’ \).
7. Diagonalizing the Perturbation Matrix
Solving the eigenvalue problem:
\[
\sum_{b=1}^d H’_{ab} \chi_b^{(k)} = E_k^{(1)} \chi_a^{(k)}
\]
yields:
- \( E_k^{(1)} \): first-order energy corrections.
- \( \chi^{(k)} \): coefficients for forming new orthonormal states:
\[
|\psi_k^{(0)}\rangle = \sum_{a=1}^d \chi_a^{(k)} |\phi_a^{(0)}\rangle
\]
8. First-Order Corrections to Energy and States
These corrections depend on the eigenvalues and eigenvectors of the matrix \( H’ \) within the degenerate subspace. These become the new “unperturbed” states to which higher-order corrections can be applied if needed.
9. Physical Interpretation
Degenerate perturbation theory explains:
- How symmetry breaking lifts degeneracy.
- Why degeneracies split into multiple nearby energy levels.
- How the “direction” of perturbation in Hilbert space determines the new eigenstates.
10. Example: Zeeman Effect in Hydrogen
When an external magnetic field \( B \) is applied, the degenerate \( m_l \) levels of hydrogen split due to the interaction term:
\[
\hat{H}’ = -\mu_B B \hat{L}_z
\]
Constructing and diagonalizing \( H’ \) within the \( n = 2 \) degenerate manifold reveals the Zeeman energy shifts and the new magnetic quantum number eigenstates.
11. Example: Stark Effect in Degenerate Systems
The hydrogen atom’s \( n=2 \) subspace includes 2s and 2p states. A uniform electric field in the \( z \)-direction leads to:
\[
\hat{H}’ = -e E z
\]
Only states with \( \Delta \ell = \pm 1 \) and \( \Delta m = 0 \) mix. Solving the eigenvalue problem for this restricted \( 2 \times 2 \) matrix gives new states and energies that predict the linear Stark effect.
12. Higher-Order Corrections
Once degeneracy is resolved, standard (non-degenerate) second-order perturbation theory may be used. Corrections beyond first order require projecting out contributions from both inside and outside the degenerate space.
13. Applications and Importance
- Atomic fine structure.
- Spectroscopy of atoms in magnetic/electric fields.
- Level splitting in solid-state systems.
- Symmetry breaking in molecular configurations.
14. Limitations and Alternatives
- Complex for high-dimensional degenerate subspaces.
- Breakdown for strong perturbations.
- Requires precise knowledge of matrix elements.
- Alternatives include variational and numerical methods.
15. Conclusion
Degenerate perturbation theory provides a systematic method for understanding how degenerate quantum systems behave under small perturbations. It resolves mathematical ambiguities and gives physical insight into symmetry breaking, energy level splitting, and new state formation. It is a cornerstone of quantum mechanics, with wide applications in atomic, molecular, and condensed matter physics.
