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. A critical feature of the technique is a middle step that breaks the problem into “solvable” and “perturbative” parts. In perturbation theory, the solution is expressed as a power series in a small parameter ε. The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of ε 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.

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  1. shinichi Post author

    Perturbation theory

    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.

  2. shinichi Post author



    上記のような複数天体間、複数粒子間に相互作用が働くときの運動は数学的に厳密に解くことができないことが知られている(多体問題)。これらの数学的に厳密に解くことのできない問題の近似解を求める手法の1つに、摂動論(せつどうろん、 英語: perturbation theory)がある。具体的には、次のような手順で近似解を求める。

    • 考えている問題Aを、厳密に解ける問題Bに小さな変更(摂動)が加えられた問題であるとみなす。
    • 問題Aの近似解は、問題Bの厳密解に、摂動が加わったことによって生じる小さな補正(摂動項)を加えたものであると考える。
    • ここで求めるべき摂動項は、問題Bの厳密解の組み合わせ、すなわち一次結合の形で表現出来ると考え、その係数を与えられた条件から順次求める。

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