# Perturbation Theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. Perturbation theory is applicable if the problem at hand can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.

Perturbation theory leads to an expression for the desired solution in terms of a formal power series in some "small" parameter – known as a perturbation series – that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we have for the approximation to the full solution A, a series in the small parameter (here called ), like the following:

In this example, would be the known solution to the exactly solvable initial problem and, ... represent the higher-order terms which may be found iteratively by some systematic procedure. For small these higher-order terms in the series become successively smaller. An approximate "perturbation solution" is obtained by truncating the series, usually by keeping only the first two terms, the initial solution and the "first-order" perturbation correction:

### Other articles related to "perturbation theory, theory, perturbations, perturbation":

Electron Liquid
... Perturbation theory can be used in this regime to discuss the properties of weakly interacting electron systems ... Perturbation theory fails for electron liquids where ... Most common metals are in the regime and hence the metallic regime is poorly described by perturbation theory ...
Cosmological Perturbation Theory - 1+3 Covariant Gauge-invariant Perturbation Theory
... usual to use the gauge-invariant covariant perturbation theory developed by Hawking (1966) and Ellis and Bruni (1989) ... and systematically reduces the theory down to one that is linear around a particular background ... This approach to perturbation theory produces differential equations that are of just the right order needed to describe the true physical degrees of freedom and as such no non-phys ...
Apsidal Precession - Perturbation Theory
... The expected rate of apsidal precession can be calculated more accurately using the methods of perturbation theory. ...
Cosmological Perturbation Theory - Gauge-invariant Perturbation Theory
... The gauge-invariant perturbation theory is based on developments by Bardeen (1980), Kodama and Sasaki (1984) building on the work of Lifshitz (1946) ... This is the standard approach to perturbation theory of general relativity for cosmology ... has as it starting point the FRW background around which perturbations are developed ...
Perturbation Theory in Chemistry
... Many of the ab initio quantum chemistry methods use perturbation theory directly or are closely related methods ... Møller–Plesset perturbation theory uses the difference between the Hartree–Fock Hamiltonian and the exact non-relativistic Hamiltonian as the perturbation ...

### Famous quotes containing the word theory:

There never comes a point where a theory can be said to be true. The most that one can claim for any theory is that it has shared the successes of all its rivals and that it has passed at least one test which they have failed.
—A.J. (Alfred Jules)