# 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, . The parameters 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 . That
is, we can write the trial wave function as
In that case, the minimisation problem of the energy reduces to solving
the *generalised eigenvalue problem*: where
stands for the *vector* with elements and the matrices
and have elements For the special case of an orthonormal basis ,
, the Dirac delta-function, and the generalised
eigenvalue problem reduces to an ordinary eigenvalue problem: