3.4. Summary

In this chapter, we have studied a successful method for approximating the ground state solution of a complicated quantum mechanical problem: variational calculus. This method simply consists of finding the minimum of the expectation value of the Hamiltonian in a restricted space, where the term 'restricted' indicates that the space is a subset of the full Hilbert space of the quantum problem. It is trivial to see that this variational solution yields an energy equal to or larger than the exact value.

The space in which we search for the solution usually is a set which we can represent by parametrised trial solutions, $\psi_{\text{T}} = \psi_{\text{T}}(\alpha_1, \alpha_2, \ldots, \alpha_N)$. The parameters $\alpha_j$ enter in a possibly non-linear way into the wave function. Solving non-linear minimisation problems is non-trivial, but numerical routines are available for this. Analytical non-linear variational problems are usually restricted to one or two variational parameters with respect to which the expectation value of the Hamiltonian is to be minimised.

A special case is the one in which the trial wave functions depend linearly on the variational parameters, which we now call $C_p$. That is, we can write the trial wave function as $$\left| \psi_{\text{T}} \right\rangle= \sum_{p=1}^N C_p \left| \chi_p\right\rangle.$$ In that case, the minimisation problem of the energy reduces to solving the generalised eigenvalue problem: $${\bf HC} = E {\bf S C},$$ where ${\bf C}$ stands for the vector with elements $C_p$ and the matrices ${\bf H}$ and ${\bf S}$ have elements $$\begin{aligned} H_{pq} &= \left\langle\chi_p \left| H \right| \chi_q \right\rangle; \\ S_{pq} &= \left\langle\chi_p | \chi_q \right\rangle. \end{aligned}$$ For the special case of an orthonormal basis $\chi_p$, $S_{pq} = \delta_{pq}$, the Dirac delta-function, and the generalised eigenvalue problem reduces to an ordinary eigenvalue problem: $${\bf H C} = E {\bf C}.$$