Variational method (quantum mechanics)

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

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[edit] Introduction

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):

\sum_{\lambda_1,\lambda_2\in\mathrm{Spec}(H)}\lang\psi_{\lambda_1}|\psi_{\lambda_2}\rang=\delta_{\lambda_1\lambda_2}

where δi,j is the Kronecker delta

\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[1]

\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

[edit] 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 αi (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

\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 αi is not sufficient. If ψ (αi) is expressed as a linear combination of other functions (αi 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.

| ψ > = | ψtest > − < ψgr | ψtest > | ψgr >

The resulting minimum is usually not as accurate as for the ground state, as any difference between the true ground state and ψgr results in a lower excited energy. This defect is worsened with each higher excited state.

[edit] See also

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[edit] Notes

  1. ^ These arguments can be found in "Quantum Mechanics -- an Introduction," fourth edition by W. Greiner.