An arbitrary state vector can be expanded in terms of eigenvectors of the Hamiltonian of the system.
The expectation value of H in the state ψ> is
Here E_{0} is the lowest possible energy eigenvalue. For a normalized ψ> we have
Therefore
E_{0} ≤ < ψHψ>.
If  ψ> is not normalized we write
The variational
method consists of evaluating the matrix element <ψHψ>
with a trial vector  ψ>
that fulfills all boundary conditions, and that depends on a number
of parameters and varying these parameters until the expectation value
of H is a minimum. (δ<H>=0).
The result is an upper limit for the ground state energy of the system,
which is likely to be close to actual ground state energy if the trial
vector resembles the eigenvector (i.e. it has the right number of nodes
etc.).
is
an eigenvector of H with a discrete eigenvalue.
Proof:Let Then or

The variational method consists
of making sure that δ<H> = 0
for some δψ>.
It can be generalized to find an upper limit for energy eigenvalues other
than E_{0} by picking a trial function that is orthogonal
to the one found for E_{0}.
copyright: http://electron6.phys.utk.edu/qm2/modules/m8/variational.htm
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