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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 Variational method (quantum mechanics) ： ウィキペディア英語版 Variational method (quantum mechanics) In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals.〔''Lorentz Trial Function for the Hydrogen Atom: A Simple, Elegant Exercise'' Thomas Sommerfeld Journal of Chemical Education 2011 88 (11), 1521–1524 〕 The basis for this method is the variational principle.〔 〕〔 〕 The method consists in choosing a "trial wavefunction" depending on one or more parameters, and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The Hartree–Fock method and the Ritz method both apply the variational method. The Harris functional method is anti-variational (it is a lower bound to the energy). == Description == Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian ''H''. Ignoring complications about continuous spectra, we look at the discrete spectrum of ''H'' and the corresponding eigenspaces of each eigenvalue λ (see spectral theorem for Hermitian operators for the mathematical background): :$\backslash sum\_\backslash lang\backslash psi\_\backslash mid\; \backslash psi\_\backslash rang=\backslash delta\_$ where $\backslash delta\_$ is the Kronecker delta :$\backslash hat\; \backslash left|\; \backslash psi\_\backslash lambda\backslash right\backslash rangle\; =\; \backslash lambda\backslash left|\backslash psi\_\backslash lambda\; \backslash right\backslash rangle.$ Physical states are normalized, meaning that their norm is equal to 1. Once again ignoring complications involved with a continuous spectrum of ''H'', suppose it is bounded from below and that its greatest lower bound is ''E''0. Suppose also that we know the corresponding state |ψ>. The expectation value of ''H'' is then :$$ \begin \left\langle\psi\mid H\mid \psi\right\rangle & = \sum_ \left\langle\psi|\psi_\right\rangle \left\langle\psi_|H|\psi_\right\rangle \left\langle\psi_|\psi\right\rangle \\ & =\sum_\lambda \left|\left\langle\psi_\lambda\mid \psi\right\rangle\right|^2\ge\sum_E_0 \left|\left\langle\psi_\lambda\mid \psi\right\rangle\right|^2=E_0 \end Obviously, if we were to vary over all possible states with norm 1 trying to minimize the expectation value of ''H'', the lowest value would be ''E''0 and the corresponding state would be an eigenstate of ''E''0. Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters ''α''''i'' (''i'' = 1, 2, ..., ''N''). The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approximations than others, therefore the choice of ansatz is important. Let's assume there is some overlap between the ansatz and the ground state (otherwise, it's a bad ansatz). We still wish to normalize the ansatz, so we have the constraints :$\backslash left\backslash langle\; \backslash psi(\backslash alpha\_i)\; \backslash mid\; \backslash psi(\backslash alpha\_i)\; \backslash right\backslash rangle\; =\; 1$ and we wish to minimize :$\backslash varepsilon(\backslash alpha\_i)\; =\; \backslash left\backslash langle\; \backslash psi(\backslash alpha\_i)|H|\backslash psi(\backslash alpha\_i)\; \backslash right\backslash rangle.$ This, in general, is not an easy task, since we are looking for a global minimum and finding the zeroes of the partial derivatives of ''ε'' over ''α''''i'' is not sufficient. If ''ψ'' (''α''''i'') is expressed as a linear combination of other functions (''α''''i'' being the coefficients), as in the Ritz method, there is only one minimum and the problem is straightforward. There are other, non-linear methods, however, such as the Hartree–Fock method, that are also not characterized by a multitude of minima and are therefore comfortable in calculations. There is an additional complication in the calculations described. As ε tends toward E0 in minimization calculations, there is no guarantee that the corresponding trial wavefunctions will tend to the actual wavefunction. This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method. A wavefunction different from the exact one is obtained by use of the method described above. Although usually limited to calculations of the ground state energy, this method can be applied in certain cases to calculations of excited states as well. If the ground state wavefunction is known, either by the method of variation or by direct calculation, a subset of the Hilbert space can be chosen which is orthogonal to the ground state wavefunction. :$\backslash left|\; \backslash psi\; \backslash right\backslash rangle\; =\; \backslash left|\backslash psi\_\}\; \backslash mid\; \backslash psi\_\}\backslash right\backslash rangle$ The resulting minimum is usually not as accurate as for the ground state, as any difference between the true ground state and $\backslash psi\_\; \backslash le\; \backslash left\backslash langle\backslash phi|H|\backslash phi\backslash right\backslash rangle.$ This holds for any trial φ since, by definition, the ground state wavefunction has the lowest energy, and any trial wavefunction will have energy greater than or equal to it. Proof: φ can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal): :$\backslash phi\; =\; \backslash sum\_n\; c\_n\; \backslash psi\_n.\; \backslash ,$ Then, to find the expectation value of the Hamiltonian: : $$ \begin & \left\langle\phi|H|\phi\right\rangle \\ & = \left\langle\sum_n c_n \psi_n |H|\sum_m c_m\psi_m\right\rangle \\ & = \sum_n\sum_m \left\langle c_n \psi_|E_m|c_m\psi_m\right\rangle \\ & = \sum_n\sum_m c_n^ *c_m E_m\left\langle\psi_n\mid\psi_m\right\rangle \\ & = \sum_ |c_n|^2 E_n. \end Now, the ground state energy is the lowest energy possible, i.e. $E\_\; \backslash ge\; E\_$. Therefore, if the guessed wave function φ is normalized: :$\backslash left\backslash langle\backslash phi|H|\backslash phi\backslash right\backslash rangle\; \backslash ge\; E\_g\; \backslash sum\_n\; |c\_n|^2\; =\; E\_g.\; \backslash ,$ 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「Variational method (quantum mechanics)」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.