Perturbation theory is a mathematical approach for approximating solutions to complex problems by starting from a known exact solution.Real physical systems are rarely solvable exactly. Perturbation theory is a recipe for approximate calculations, where the problem is split into an ideal, exactly solvable 'skeleton' and weak complicating factors (perturbations). The answer is sought as a chain of corrections, each successive one smaller than the previous. This approach dominates quantum mechanics, astrophysics, and field theory—anywhere a small parameter exists. For example, in quantum electrodynamics, the interaction of electrons with light is calculated as a series in the fine-structure constant (a number around 1/137).
Henri Poincaré in the 1890s developed methods of series in a small parameter for the three-body problem in celestial mechanics. In the 1920s, Paul Dirac formulated time-dependent perturbation theory for calculating quantum transitions. John Archibald Wheeler applied perturbation theory in general relativity and for analyzing the scattering of gravitational waves.
In stationary non-relativistic perturbation theory, the first-order energy correction is E_n^(1) = ⟨n^(0)|V|n^(0)⟩, where |n^(0)⟩ are the eigenvectors of H0. In second order, E_n^(2) = Σ_{m≠n} |⟨m^(0)|V|n^(0)⟩|^2/(E_n^(0) - E_m^(0)). The series are often asymptotic, so they converge only for sufficiently small perturbation. In quantum electrodynamics, the expansion in α≈1/137 gives excellent convergence; in quantum chromodynamics at low energies, the coupling constant is large, and non-perturbative methods (lattice calculations) are needed. Thus, like an astronomer refining Uranus's orbit correction by correction, the physicist step by step approaches the truth until small additions no longer change the result.
Today, satellite orbits, the behavior of atoms in laser light, and the evolution of galactic clusters are calculated this way.
Using perturbation theory, the anomalous magnetic moment of the electron was calculated, and the result matched experiment to the 12th decimal place — as if you measured the distance from the Earth to the Moon with an error equal to the thickness of a hair.
| E_n | variable | Perturbed energy of state n — energy joule |
| E_n^{(0)} | variable | Unperturbed energy of state n — energy joule |
| E_m^{(0)} | variable | Unperturbed energy of state m — energy joule |
| H' | variable | Perturbation Hamiltonian — hamiltonian joule |
| |n^{(0)}\rangle | variable | Unperturbed state n — quantum state |
| |m^{(0)}\rangle | variable | Unperturbed state m — quantum state |
| \sum_{m \neq n} | operator | Sum over all states m except n — summation |
| \langle m^{(0)} | H' | operator | Matrix element of perturbation between unperturbed states — matrix element |
