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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ermakov–Lewis invariant ： ウィキペディア英語版 Ermakov–Lewis invariant Many quantum mechanical Hamiltonians are time dependent. How to solve problems where there is an explicit time dependence is an open subject nowadays. For problems of this kind it is of importance to look for constants of motion or invariants. For the (time dependent) harmonic oscillator it is possible to write several invariants, among them, the Ermakov–Lewis invariant which is developed below. The time dependent harmonic oscillator Hamiltonian reads : $\backslash hat\; =\backslash frac\backslash left().$ It is well known that an invariant for this type of interaction has the form : $$ \hat=\frac\left( \left( \frac\right) ^+(\rho\hat-\dot\hat)^\right ), where $\backslash rho$ obeys the Ermakov equation〔V.P. Ermakov, Univ. Izv. (Kiev) 20, 1 (1880)〕 :$$ \ddot+\Omega^\rho=\rho^. The above invariant is the so-called Ermakov-Lewis invariant.〔H.R. Lewis, Phys. Rev. Lett. 18, 510 (1967). http://link.aps.org/doi/10.1103/PhysRevLett.18.510〕 It is easy to show that $\backslash hat$ may be related to the time independent harmonic oscillator Hamiltonian via a unitary transformation of the form 〔H. Moya-Cessa and M. Fernández Guasti, Physics Letters A 311, 1 (2003). http://dx.doi.org/10.1016/S0375-9601(03)00461-4〕 :$\backslash hat=e^(\backslash hat\backslash hat+\backslash hat\backslash hat$ )}e^\hat^}= e^\frac} e^}\frac}, as :$\backslash frac\backslash left()=\backslash hat\backslash hat\backslash hat^.$ This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian. The first exponential in the transformation is the so-called squeeze operator. This approach may allow to simplify problems such as the Quadrupole ion trap, where an ion is trapped in a harmonic potential with time dependent frequency. The transformation presented here is then useful to take into account such effects. ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ermakov–Lewis invariant」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.