Validity of Perturbation Theory

When can you trust the series?

Perturbation theory produces a power series in $\lambda$ , but writing a series is not the same as that series being accurate — or even convergent. This lesson collects the practical tests for when the expansion is trustworthy.

The basic smallness condition

Look back at the first-order state correction,

|n^{(1)}\rangle = \sum_{m \neq n} \frac{\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}}\,|m^{(0)}\rangle.

Each admixture coefficient must be small for the corrected state to remain close to the unperturbed one. The governing condition is

\left|\frac{\langle m^{(0)}|\hat{H}'|n^{(0)}\rangle}{E_n^{(0)} - E_m^{(0)}}\right| \ll 1 \quad \text{for all } m \neq n.

In words: the matrix element of the perturbation must be much smaller than the energy gap to every state it couples to. Two failure modes follow directly.

Failure mode 1: a large coupling

If $\hat{H}'$ is simply not small — a strong field, a deep extra well — the numerators are large and the series terms do not shrink. No amount of cleverness rescues a non-perturbative problem; you need exact diagonalization, variational methods, or numerics.

Failure mode 2: a small (or zero) gap

Even a tiny $\hat{H}'$ breaks the expansion if two unperturbed levels are nearly degenerate, because the denominator $E_n^{(0)} - E_m^{(0)}$ approaches zero. This is the situation that forces degenerate perturbation theory: diagonalize within the near-degenerate subspace first, then perturb. Near-degeneracy (small but nonzero gap) is the most delicate regime and often needs the gap and coupling treated on equal footing.

Convergence is subtle even when terms are small

A surprising fact: many perturbation series in physics are asymptotic, not convergent. The terms shrink at first, reach a smallest size, then grow without bound. The classic example is the ground-state energy of the quartic anharmonic oscillator, whose perturbation series in the coupling has zero radius of convergence — yet the first few terms give superb numerical accuracy. The lesson is pragmatic: a truncated perturbation series can be highly accurate even if the full series diverges, as long as you stop near the term of smallest magnitude.

A practical checklist

Before trusting a perturbative result, confirm:

Dimensionless smallness. Form the ratio coupling / gap and check it is $\ll 1$ .
No accidental degeneracy. Identify states with nearly equal unperturbed energy that $\hat{H}'$ connects; if any exist, switch to the degenerate treatment.
Decreasing corrections. Compare $E^{(1)}$ , $E^{(2)}$ , $\dots$ ; they should fall off rapidly.
A cross-check when possible. Compare against an exactly solvable limit, a variational bound, or a numerical diagonalization on a truncated basis.

The takeaway

Perturbation theory is reliable precisely when the perturbation is weak compared to the energy gaps it bridges. It fails for strong couplings and for (near-)degeneracies, and even when it "works" the underlying series may be asymptotic. Knowing these limits is what separates blindly applying a formula from using it responsibly.