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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 Gross–Pitaevskii equation ： ウィキペディア英語版 Gross–Pitaevskii equation The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model. In the Hartree–Fock approximation the total wave-function $\backslash Psi$ of the system of $N$ bosons is taken as a product of single-particle functions $\backslash psi$, :$$ \Psi(\mathbf_1,\mathbf_2,\dots,\mathbf_N)=\psi(\mathbf_1)\psi(\mathbf_2)\dots\psi(\mathbf_N) where $\backslash mathbf\_i$ is the coordinate of the $i$-th boson. The pseudopotential model Hamiltonian of the system is given as :$$ H=\sum_^N \left(-+V(\mathbf_i)\right) +\sum_\delta(\mathbf_i-\mathbf_j), where $m$ is the mass of the boson, $V$ is the external potential, $a\_s$ is the boson-boson scattering length, and $\backslash delta(\backslash mathbf)$ is the Dirac delta-function. If the single-particle wave-function satisfies the Gross–Pitaevski equation, :$$ \left(-\frac + V(\mathbf) + \vert\psi(\mathbf)\vert^2\right)\psi(\mathbf)=\mu\psi(\mathbf), the total wave-function minimizes the expectation value of the model Hamiltonian under normalization condition $\backslash int\; dV\; |\backslash Psi|^2=N.$ It is a model equation for the single-particle wavefunction in a Bose–Einstein condensate. It is similar in form to the Ginzburg–Landau equation and is sometimes referred to as a nonlinear Schrödinger equation. A Bose–Einstein condensate (BEC) is a gas of bosons that are in the same quantum state, and thus can be described by the same wavefunction. A free quantum particle is described by a single-particle Schrödinger equation. Interaction between particles in a real gas is taken into account by a pertinent many-body Schrödinger equation. If the average spacing between the particles in a gas is greater than the scattering length (that is, in the so-called dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles. This is made evident by setting the coupling constant of interaction in the Gross–Pitaevskii equation to zero (see the following section): thereby, the single-particle Schrödinger equation describing a particle inside a trapping potential is recovered. ==Form of equation== The equation has the form of the Schrödinger equation with the addition of an interaction term. The coupling constant, g, is proportional to the scattering length $a\_s$ of two interacting bosons: :$g=\backslash frac$, where $\backslash hbar$ is the reduced Planck's constant and m is the mass of the boson. The energy density is :$\backslash mathcal=\backslash frac\backslash vert\backslash nabla\backslash Psi(\backslash mathbf)\backslash vert^2\; +\; V(\backslash mathbf)\backslash vert\backslash Psi(\backslash mathbf)\backslash vert^2\; +\; \backslash fracg\backslash vert\backslash Psi(\backslash mathbf)\backslash vert^4,$ where $\backslash Psi$ is the wavefunction, or order parameter, and V is an external potential. The time-independent Gross–Pitaevskii equation, for a conserved number of particles, is :$\backslash mu\backslash Psi(\backslash mathbf)\; =\; \backslash left(-\backslash frac\backslash nabla^2\; +\; V(\backslash mathbf)\; +\; g\backslash vert\backslash Psi(\backslash mathbf)\backslash vert^2\backslash right)\backslash Psi(\backslash mathbf)$ where $\backslash mu$ is the chemical potential. The chemical potential is found from the condition that the number of particles is related to the wavefunction by :$N\; =\; \backslash int\backslash vert\backslash Psi(\backslash mathbf)\backslash vert^2\; \backslash ,\; d^3r.$ From the time-independent Gross–Pitaevskii equation, we can find the structure of a Bose–Einstein condensate in various external potentials (e.g. a harmonic trap). The time-dependent Gross–Pitaevskii equation is :$i\backslash hbar\backslash frac\; =\; \backslash left(-\backslash frac\backslash nabla^2\; +\; V(\backslash mathbf)\; +\; g\backslash vert\backslash Psi(\backslash mathbf,t)\backslash vert^2\backslash right)\backslash Psi(\backslash mathbf,t).$ From the time-dependent Gross–Pitaevskii equation we can look at the dynamics of the Bose–Einstein condensate. It is used to find the collective modes of a trapped gas. 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「Gross–Pitaevskii equation」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.