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Variational method (quantum mechanics)

From Biocrawler, the free encyclopedia.

The variational method is, in quantum mechanics, one way of finding approximations to the lowest energy eigenstate or ground state. The basis for this method is the variational principle.

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Introduction

Suppose we're given a Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the spectrum of H and the corresponding eigenspaces of each eigenvalue λ:

$\mathbf{1}=\sum_{\lambda\in \mathrm{Spec}(H)}|\psi_\lambda\rang\lang\psi_\lambda|$

with

$\lang\psi_{\alpha}|\psi_{\beta}\rang=\delta_{\alpha\beta}$

and

$\left. \hat{H}|\psi_\lambda\right \rangle = \left. \lambda|\psi_\lambda \right\rangle$ .

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 E₀. Suppose also that we know the corresponding state |ψ>. The expectation value of H is then

$\left\langle\psi|H|\psi\right\rangle = \sum_{\lambda_1,\lambda_2 \in \mathrm{Spec}(H)} \left\langle\psi|\psi_{\lambda_1}\right\rangle \left\langle\psi_{\lambda_1}|H|\psi_{\lambda_2}\right\rangle \left\langle\psi_{\lambda_2}|\psi\right\rangle$

$=\sum_{\lambda\in \mathrm{Spec}(H)}\lambda |\left\langle\psi_\lambda|\psi\right\rangle|^2\ge\sum_{\lambda \in \mathrm{Spec}(H)}E_0 |\left\langle\psi_\lambda|\psi\right\rangle|^2=E_0$

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Ansatz

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 E₀ and the corresponding state would be an eigenstate of E₀. 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 teh 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

$\left\langle \psi(\alpha_i) | \psi(\alpha_i) \right\rangle = 1$

and we wish to minimize

$\varepsilon(\alpha_i) \left\langle \psi(\alpha_i)|H|\psi(\alpha_i) \right\rangle$ .

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 function (α_i being the ceofficients), 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 calcualtions.

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Variational method (quantum mechanics)

From Biocrawler, the free encyclopedia.

Introduction

Ansatz

See also

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