Near-Degeneracy and Fine Structure

When two levels are not exactly degenerate but have a small energy gap compared to the perturbation, non-degenerate perturbation theory gives poor results. Consider two levels with:

The solution is quasi-degenerate perturbation theory: treat the near-degenerate subspace exactly (like degenerate PT) and the rest perturbatively. The effective $2 \times 2$ matrix becomes:

where $W_{ij} = \langle i|\hat{H}'|j\rangle$. The eigenvalues are:

This reduces to the degenerate result when $E_1^{(0)} = E_2^{(0)}$ and to the non-degenerate result when $|E_1^{(0)} - E_2^{(0)}| \gg |W_{12}|$. It smoothly interpolates between the two limits.

A beautiful consequence of near-degenerate perturbation theory is the avoided crossing (also called anti-crossing). If we tune a parameter that would make two levels cross in the absence of coupling ($W_{12} = 0$), the off-diagonal coupling $W_{12} \neq 0$ prevents the levels from actually crossing:

This is a manifestation of the von Neumann-Wigner no-crossing theorem: for a generic Hermitian matrix depending on a single real parameter, eigenvalues do not cross. Crossings require special symmetry (e.g., different symmetry quantum numbers).

The hydrogen fine structure consists of two perturbative corrections to the non-relativistic Hamiltonian, both arising from relativistic effects at order $(v/c)^2$:

The relativistic energy-momentum relation $E = \sqrt{p^2c^2 + m^2c^4}$, expanded to next order beyond the non-relativistic kinetic energy, gives:

The first-order energy correction is (using $\hat{p}^2 = 2m_e(\hat{H}_0 - \hat{V})$ to avoid computing $\langle p^4\rangle$ directly):

Using the expectation values $\langle 1/r\rangle = 1/(n^2 a_0)$ and $\langle 1/r^2\rangle = 1/(n^3(l+1/2)a_0^2)$:

The electron's magnetic moment interacts with the magnetic field produced by the nucleus (in the electron's rest frame). After the Thomas precession correction:

To evaluate $\langle\hat{\vec{S}}\cdot\hat{\vec{L}}\rangle$, use the total angular momentum $\hat{\vec{J}} = \hat{\vec{L}} + \hat{\vec{S}}$:

The first-order correction (for $l \neq 0$):

Remarkably, when the relativistic and spin-orbit corrections are combined, the result depends only on $n$ and $j$ (not on $l$ separately):

where $\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137$ is the fine structure constant. The fine structure corrections are smaller than the Bohr energies by a factor of $\alpha^2 \approx 5.3 \times 10^{-5}$.

Key consequence: States with the same $n$ and $j$ but different $l$ remain degenerate. For example, the $2S_{1/2}$ ($l=0, j=1/2$) and $2P_{1/2}$ ($l=1, j=1/2$) states have the same fine structure energy.

The residual degeneracy between $2S_{1/2}$ and $2P_{1/2}$ predicted by the Dirac equation is actually broken by quantum electrodynamic (QED) effects. The Lamb shift, measured by Willis Lamb in 1947, showed:

This tiny splitting arises from:

• Vacuum fluctuations: The electron interacts with virtual photons of the quantum electromagnetic field
• Self-energy: The electron's interaction with its own field
• Vertex correction: QED corrections to the electron-photon coupling

The Lamb shift was one of the key experimental results that motivated the development of renormalized quantum electrodynamics by Feynman, Schwinger, and Tomonaga.