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In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.[1][2] A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts.[3] In perturbation theory, the solution is expressed as a power series in a small parameter ε {\displaystyle \varepsilon } . [1][2] The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of ε {\displaystyle \varepsilon } usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.

Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. The field in general remains actively and heavily researched across multiple disciplines.

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