Perturbation theory

Perturbation theory approximates solutions to difficult problems by starting from a solvable model and accounting systematically for small changes to it.

Asymptotic expansion: An approximation expressed as a sequence of terms whose errors shrink in a specified limiting regime. Perturbation series often describe limiting behavior even when they do not converge.
Small parameter: A dimensionless quantity whose magnitude is much less than one in a specified regime. It measures the influence being added and organizes the successive correction terms.
Rayleigh–Schrödinger perturbation theory: A quantum-mechanical method for approximating energy levels and states when a Hamiltonian differs slightly from a solvable one. It applies systematic corrections to atomic and molecular energy predictions.
Variational method: A method that estimates a solution by optimizing a quantity over a chosen family of trial states or functions. It can give bounds or optimized estimates without constructing a perturbation series.
Regular perturbation: A perturbation method in which the solution varies smoothly with a small parameter and ordinary power-series expansion applies. This is the simplest case: expanding in the small influence yields consistent approximations.
Taylor series: A representation of a function by powers of its displacement from a point, using its derivatives there. Many regular perturbation calculations derive corrections by expanding equations in the small parameter.
Time-independent perturbation theory: A method for calculating stationary energy and state corrections when a quantum Hamiltonian changes by a small fixed amount. It handles fixed changes to quantum systems without modeling their time evolution.
Numerical analysis: The study of algorithms for computing approximate solutions to mathematical problems. Numerical methods approximate solutions directly rather than correcting an analytic reference solution.
Singular perturbation: A perturbation problem in which setting the small parameter to zero changes the character or effective order of the equations. It shows why straightforward expansions can fail when a small change causes a structural shift.
Linearization: An approximation of a nonlinear system near a reference state by its linear terms. The first correction often reduces to a linear equation around the known solution.

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