The variational method

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 E0 is the lowest possible energy eigenvalue.  For a normalized |ψ> we have

Therefore

E0 ≤ < ψ|H|ψ>.

If | ψ> is not normalized we write

The variational method consists of evaluating the matrix element <ψ|H|ψ> 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.).

Generalization: Ritz theorem

The mean value <ψ|H|ψ>, considered as a function of all the vectors of the state space, is stationary if |ψ> is an eigenvector with a discrete (non-continuous) eigenvalue.

is an eigenvector of H with a discrete eigenvalue.

Proof:

Let Then

or




If | ψ> is an eigenvector of H then <H> is its eigenvalue and δ<H> = 0 
since <ψ|(H-<H>) and  (H-<H>)|ψ> are zero.
If δ<H> = 0 for any |δψ>, then <ψ|(H-<H>)|δψ> + <δψ|(H-<H>)|ψ> must also be zero for any |δψ>. 
Let us define |φ>=(H-<H>)|ψ>. Then  <φ|δψ> + <δψ|φ> = 0 for any |δψ> → 0.
Let |δψ>→ δλ|φ>.  Then 2δλ<φ|φ> = 0, or |φ> = 0.
Therefore 
i.e. |ψ> is an eigenvector of H with eigenvalue <H>.

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 E0 by picking a trial function that is orthogonal to the one found for E0.
 

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