翻訳と辞書 Words near each other ・ Rivaldo Costa Amaral Filho ・ Rivaldo González ・ Rivalen ・ Rivalen am Steuer ・ Ritz Ballroom, Bridgeport ・ Ritz Ballroom, Kings Heath ・ Ritz Brothers ・ Ritz Camera Centers ・ Ritz Cinema, Barrow-in-Furness ・ Ritz Crackers ・ Ritz Dakota Digital ・ Ritz Fizz ・ Ritz FM ・ Ritz Guitars ・ Ritz Hotel Project, Washington, D.C. ・ Ritz method ・ Ritz Metro ・ Ritz Model A ・ Ritz Newspaper ・ Ritz Plaza Hotel ・ Ritz Sidecar ・ Ritz Theater (Newburgh, New York) ・ Ritz Theatre ・ Ritz Theatre (Elizabeth, New Jersey) ・ Ritz Theatre (Haddon Township, New Jersey) ・ Ritz Theatre (Jacksonville) ・ Ritz's Equation ・ Ritz-Carlton Astana ・ Ritz-Carlton Atlantic City ・ Ritz-Carlton Club and Residences Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ritz method ： ウィキペディア英語版 Ritz method The Ritz method is a direct method to find an approximate solution for boundary value problems. The method is named after Walter Ritz. In quantum mechanics, a system of particles can be described in terms of an "energy functional" or Hamiltonian, which will measure the energy of any proposed configuration of said particles. It turns out that certain privileged configurations are more likely than other configurations, and this has to do with the eigenanalysis ("analysis of characteristics") of this Hamiltonian system. Because it is often impossible to analyze all of the infinite configurations of particles to find the one with the least amount of energy, it becomes essential to be able to approximate this Hamiltonian in some way for the purpose of numerical computations. The Ritz method can be used to achieve this goal. In the language of mathematics, it is exactly the finite element method used to compute the eigenvectors and eigenvalues of a Hamiltonian system. == Discussion == As with other variational methods, a trial wave function, $\backslash Psi$, is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration. It can be shown that the ground state energy, $E\_0$, satisfies an inequality: :$E\_0\; \backslash le\; \backslash frac.$ That is, the ground-state energy is less than this value. The trial wave-function will always give an expectation value larger than or equal to the ground-energy. If the trial wave function is known to be orthogonal to the ground state, then it will provide a boundary for the energy of some excited state. The Ritz ansatz function is a linear combination of ''N'' known basis functions $\backslash left\backslash lbrace\backslash Psi\_i\backslash right\backslash rbrace$, parametrized by unknown coefficients: :$\backslash Psi\; =\; \backslash sum\_^N\; c\_i\; \backslash Psi\_i.$ With a known Hamiltonian, we can write its expected value as :$\backslash varepsilon\; =\; \backslash frac\; \backslash left|\; \backslash displaystyle\backslash sum\_^Nc\_i\backslash Psi\_i\; \backslash right\backslash rangle\}^Nc\_i\backslash Psi\_i\; \backslash right\backslash rangle\}\; =\; \backslash frac^Nc\_i^$ *c_jH_}^Nc_i^ *c_jS_} \equiv \frac. The basis functions are usually not orthogonal, so that the overlap matrix ''S'' has nonzero nondiagonal elements. Either $\backslash left\backslash lbrace\; c\_i\; \backslash right\backslash rbrace$ or $\backslash left\backslash lbrace\; c\_i^$ * \right\rbrace (the conjugation of the first) can be used to minimize the expectation value. For instance, by making the partial derivatives of $\backslash varepsilon$ over $\backslash left\backslash lbrace\; c\_i^$ * \right\rbrace zero, the following equality is obtained for every ''k'' = 1, 2, ..., ''N'': :$\backslash frac\; =\; \backslash frac-\backslash varepsilon\; S\_)\}\; =\; 0,$ which leads to a set of ''N'' secular equations: :$\backslash sum\_^N\; c\_j\; \backslash left(\; H\_\; -\; \backslash varepsilon\; S\_\; \backslash right)\; =\; 0\; \backslash quad\; \backslash text\; \backslash quad\; k\; =\; 1,2,\backslash dots,N.$ In the above equations, energy $\backslash varepsilon$ and the coefficients $\backslash left\backslash lbrace\; c\_j\; \backslash right\backslash rbrace$ are unknown. With respect to ''c'', this is a homogeneous set of linear equations, which has a solution when the determinant of the coefficients to these unknowns is zero: :$\backslash det\; \backslash left(\; H\; -\; \backslash varepsilon\; S\; \backslash right)\; =\; 0,$ which in turn is true only for ''N'' values of $\backslash varepsilon$. Furthermore, since the Hamiltonian is a hermitian operator, the ''H'' matrix is also hermitian and the values of $\backslash varepsilon\_i$ will be real. The lowest value among $\backslash varepsilon\_i$ (i=1,2,..,N), $\backslash varepsilon\_0$, will be the best approximation to the ground state for the basis functions used. The remaining ''N-1'' energies are estimates of excited state energies. An approximation for the wave function of state ''i'' can be obtained by finding the coefficients $\backslash left\backslash lbrace\; c\_j\; \backslash right\backslash rbrace$ from the corresponding secular 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「Ritz method」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.