Time-Independent Perturbation Theory

Exactly-solvable systems like the harmonic oscillator and hydrogen atom are rare. The workhorse method for everything else is perturbation theory: treat the Hamiltonian as a solvable part plus a small correction,

$$\hat{H}={\hat{H}}_0+\lambda \hat{V},$$

where the spectrum of ${\hat{H}}_0$ is known and $\lambda$ is a bookkeeping parameter tracking the order of smallness. This page develops the corrections to stationary energies and states; time-dependent perturbations (transitions) are treated separately.

1. The Non-Degenerate Series

Suppose ${\hat{H}}_0|n^{(0)}⟩=E_n^{(0)}|n^{(0)}⟩$ with the level $E_n^{(0)}$ non-degenerate. Expand the perturbed eigenpair in powers of $\lambda$:

$$E_n=E_n^{(0)}+\lambda E_n^{(1)}+{\lambda }^2{E}_n^{(2)}+\cdots ,|n⟩=|n^{(0)}⟩+\lambda |n^{(1)}⟩+\cdots .$$

Substituting into $\hat{H}|n⟩=E_n|n⟩$ and matching order by order gives the standard results.

First-order energy — the expectation of the perturbation in the unperturbed state:

$$\boxed{E_n^{(1)}=⟨n^{(0)}|\hat{V}|n^{(0)}⟩.}$$

First-order state — admixture of the other levels, weighted by energy denominators:

$$|n^{(1)}⟩=\sum _{k\ne n}\frac{⟨k^{(0)}|\hat{V}|n^{(0)}⟩}{E_n^{(0)}-E_k^{(0)}}|k^{(0)}⟩.$$

Second-order energy — always negative for the ground state, since every denominator $E_0^{(0)}-E_k^{(0)}<0$:

$$E_n^{(2)}=\sum _{k\ne n}\frac{|⟨k^{(0)}|\hat{V}|n^{(0)}⟩{|}^2}{E_n^{(0)}-E_k^{(0)}}.$$

The method requires $|⟨k^{(0)}|\hat{V}|n^{(0)}⟩|\ll |E_n^{(0)}-E_k^{(0)}|$: the perturbation must be small compared to level spacings. It fails outright when a denominator vanishes — the case of degeneracy.

2. Degenerate Perturbation Theory

If $E_n^{(0)}$ is $g$-fold degenerate, the naive series diverges: the "correct" zeroth-order states are unknown linear combinations within the degenerate subspace. The fix: diagonalize the perturbation inside the degenerate subspace first. Form the $g\times g$ matrix

$$V_{ab}=⟨n_a^{(0)}|\hat{V}|n_b^{(0)}⟩,$$

and its eigenvalues are the first-order energy shifts $E^{(1)}$; its eigenvectors are the good zeroth-order states. A perturbation that is diagonalized this way generically lifts the degeneracy — the mechanism by which symmetry-breaking terms split spectral lines.

3. Worked Applications (Hydrogen Fine Structure)

The hydrogen degeneracy is split by several small terms, each an exercise in the above:

• Relativistic kinetic correction $-{\hat{p}}^4/8m^3{c}^2$: a non-degenerate-style shift $\sim {\alpha }^2{E}_n$ depending on $\ell$.
• Spin–orbit coupling $\propto \hat{\mathbf{L}}\cdot \hat{\mathbf{S}}$: diagonal in the coupled $|j,m_j⟩$ basis obtained by adding angular momenta, splitting levels by total $j$.
• Zeeman effect (external $\mathbf{B}$): $\hat{V}=-\mathbf{\mu }\cdot \mathbf{B}\propto ({\hat{L}}_z+2{\hat{S}}_z)B$ lifts the $m$-degeneracy; weak-field vs. strong-field (Paschen–Back) limits correspond to which term dominates.
• Stark effect (external $\mathbf{E}$): $\hat{V}=e\mathbf{E}\cdot \hat{\mathbf{r}}$; the linear shift requires degenerate perturbation theory because states of opposite parity are degenerate in hydrogen, so $⟨\hat{\mathbf{r}}⟩$ need not vanish.

4. Practical Remarks

• Selection rules kill most matrix elements: $⟨k^{(0)}|\hat{V}|n^{(0)}⟩$ vanishes unless $\hat{V}$ connects states of compatible symmetry (parity, $\Delta m$, $\Delta \ell$). Computing which elements survive is usually more than half the work.
• The perturbation series is generally asymptotic, not convergent — useful term by term but ultimately divergent. When no small parameter exists, use the variational method (bounds) or WKB (semiclassical) instead.

See also

• Time-dependent perturbation theory — transitions and rates.
• Variational method — non-perturbative ground-state bounds.
• Hydrogen atom — the unperturbed spectrum being corrected.
• Addition of angular momenta — the spin–orbit basis.