Why Perturbation Theory

Most Hamiltonians cannot be solved exactly

Only a short list of quantum systems can be solved in closed form: the infinite square well, the harmonic oscillator, the hydrogen atom, a few others. Real systems almost never match these idealized models. An atom sits in a stray electric field; an electron feels relativistic corrections; a molecule is squeezed by its neighbors. Each of these adds a term to the Hamiltonian that destroys the exact solvability.

The central idea of perturbation theory is to write the true Hamiltonian as a piece we can solve plus a small correction we cannot:

\hat{H} = \hat{H}^{(0)} + \lambda \hat{H}',

where $\hat{H}^{(0)}$ is the unperturbed Hamiltonian with known eigenvalues $E_n^{(0)}$ and eigenstates $|n^{(0)}\rangle$ , $\hat{H}'$ is the perturbation, and $\lambda$ is a bookkeeping parameter that tracks "how many powers of the small term" appear in each expression. At the end we set $\lambda = 1$ ; its only job is to organize the expansion.

Expanding in a small parameter

If $\hat{H}'$ is small compared with the spacing of the unperturbed levels, then the true energies and states should be close to the unperturbed ones. Perturbation theory makes this precise by writing each quantity as a power series in $\lambda$ :

E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots,

|n\rangle = |n^{(0)}\rangle + \lambda |n^{(1)}\rangle + \lambda^2 |n^{(2)}\rangle + \cdots.

The terms $E_n^{(1)}, E_n^{(2)}, \dots$ are the first-order, second-order, ... energy corrections, and similarly for the state. We will derive compact formulas for each correction in the lessons that follow. The first-order energy shift, for instance, turns out to be just the expectation value of the perturbation in the unperturbed state, $E_n^{(1)} = \langle n^{(0)} | \hat{H}' | n^{(0)} \rangle$ — remarkably simple.

When the recipe works

Perturbation theory is an approximation, not an identity. It is useful when the corrections shrink rapidly: $|E_n^{(1)}| \ll |E_n^{(0)}|$ , and higher orders smaller still. Roughly, the controlling ratio is the matrix element of $\hat{H}'$ divided by the energy gap to nearby states,

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

When two unperturbed levels are degenerate this denominator vanishes and the naive series blows up — a situation that demands the special treatment we develop in the lesson on degenerate perturbation theory.

Why it matters

Perturbation theory is the workhorse of practical quantum mechanics. The fine structure of hydrogen, the Stark and Zeeman effects, the Lamb shift, and the energy levels of superconducting qubits in a weak drive are all computed this way. Even when more powerful numerical methods exist, the perturbative result gives the physical intuition — which term causes a splitting, and how it scales with field strength. The rest of this module builds the machinery step by step.