variational method (quantum mechanics)

In quantum mechanics, the variational method is one way of finding approximations for the lowest energy eigenstate. 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 for each eigenvalue. So,

\mathbf{1}=\sum_{\lambda\in Spec(H)}|\psi_\lambda><\psi_\lambda|

H|\psi_\lambda>=\lambda|\psi_\lambda>

<\psi_{\lambda_1}|\psi_{\lambda_2}>=\delta_{\lambda_1,\lambda_2}

The completeness relation. "Physical" states are normalized, meaning their norm is 1. (once again, we ignore complications involving a continuous spectrum) Suppose the spectrum of H is bounded from below and its greatest lower bound is E₀. Suppose we have such a state |ψ>. Let's look at the expectation value of H.

<\psi|H|\psi>=\sum_{\lambda_1,\lambda_2 \in Spec{H}}<\psi|\psi_{\lambda_1}><\psi_{\lambda_1}|H|\psi_{\lambda_2}><\psi_{\lambda_2}|\psi>=\sum_{\lambda\in Spec(H)}\lambda |<\psi_\lambda|\psi>|^2\ge\sum_{\lambda \in Spec(H)}E_0 |<\psi_\lambda|\psi>|^2=E_0

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. But varying over the entire Hilbert space is often too complicated for physical calculations. So, let's look at a subspace of the entire Hilbert space, parametrized by some (real) differentiable parameters α. The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approximations than others, so it's important to choose the ansatz carefully. Let's assume there is some overlap between the ansatz and the ground state. Otherwise, it's simply a bad ansatz. We still wish to normalize the ansatz, so we have the constraints

<\psi(\alpha)|\psi(\alpha)>=1

We wish to minimize

<\psi(\alpha)|H|\psi(\alpha)>

subject to the constraint above. Using the method of Lagrange multipliers,

{\partial \over \partial \alpha}<\psi(\alpha)|H|\psi(\alpha)>=\lambda {\partial \over \partial \alpha}<\psi(\alpha)|\psi(\alpha)>

where λ is the Lagrange multiplier. Of course, this only gives the stationary points and we still have to check if we have the global minimum. The lowest expectation value we get is an upper bound to E₀. Better ansatzes give better bounds.

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