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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 Quantum stochastic calculus ： ウィキペディア英語版 Quantum stochastic calculus Quantum stochastic calculus is a generalization of stochastic calculus to noncommuting variables. The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing measurement, as in quantum trajectories. Just as the Lindblad master equation provides a quantum generalization to the Fokker-Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical Langevin equations. For the remainder of this article ''stochastic calculus'' will be referred to as ''classical stochastic calculus'', in order to clearly distinguish it from quantum stochastic calculus. == Heat baths == An important physical scenario in which a quantum stochastic calculus is needed is the case of a system interacting with a heat bath. It is appropriate in many circumstances to model the heat bath as an assembly of harmonic oscillators. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following Hamiltonian: :$H=H\_\backslash mathrm(\backslash mathbf)+\backslash frac\backslash sum\_n\backslash left((p\_n-\backslash kappa\_nX)^2+\backslash omega\_n^2q\_n^2\backslash right)\backslash ,,$ where $H\_\backslash mathrm$ is the system Hamiltonian, $\backslash mathbf$ is a vector containing the system variables corresponding to a finite number of degrees of freedom, $n$ is an index for the different bath modes, $\backslash omega\_n$ is the frequency of a particular mode, $p\_n$ and $q\_n$ are bath operators for a particular mode, $X$ is a system operator, and $\backslash kappa\_n$ quantifies the coupling between the system and a particular bath mode. In this scenario the equation of motion for an arbitrary system operator $Y$ is called the ''quantum Langevin equation'' and may be written as:〔 where $()$ and $\backslash$ denote the commutator and anticommutator (respectively), the memory function $f$ is defined as: :$f(t)\backslash equiv\backslash sum\_n\backslash kappa\_n^2\backslash cos(\backslash omega\_nt)\backslash ,,$ and the time dependent noise operator $\backslash xi$ is defined as: :$\backslash xi(t)\backslash equiv\; i\backslash sum\_n\backslash kappa\_n\backslash sqrt\}\backslash left(-a\_n(t\_0)e^+a^\backslash dagger\_n(t\_0)e^\backslash right)\backslash ,,$ where the bath annihilation operator $a\_n$ is defined as: :$a\_n\backslash equiv\backslash frac\{\backslash sqrt\{2\backslash hbar\backslash omega\_n\}\}\backslash ,.$ Oftentimes this equation is more general than is needed, and further approximations are made to simplify the equation. 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「Quantum stochastic calculus」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.