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# 2.4 Variational principles

In section 2.1 we outlined the basic principles of quantum mechanics, and in particular noted the rôle of the quantity as the expectation value of the observable corresponding to the operator . In that section, mention was briefly made of the relationship:

 (2.41)

which is simply derived from equations 2.5, 2.6 and 2.8. If we relax the restriction on orthonormalisation, the expression for the expectation value becomes
 (2.42)

We now consider the expectation value of the Hamiltonian operator for the electrons, defined in equation 2.19 and reproduced here:

 (2.43)

in which the electronic energy is now labelled , and the dependence on the nuclear coordinates is suppressed since the nuclei are assumed to be static following the conclusions of section 2.2. This equation is an eigenvalue equation for a linear Hermitian operator, and as such can always be recast in the form of finding the stationary points of a functional subject to a constraint.

Consider the expectation value of the Hamiltonian which is a functional of the wave-function, and make a small variation to the state-vector: . The change in is given by

 (2.44)

neglecting changes which are second-order or higher in in the last line. Thus the quantity is stationary ( ) when is an eigenstate of and the eigenvalue is ,
 (2.45)

and this equation is the time-independent Schrödinger equation. The eigenvalues of can therefore be found by finding the stationary values of i.e. finding the stationary values of subject to the constraint that is constant. In this procedure, the eigenvalue plays the rôle of a Lagrange multiplier used to impose the constraint.

In this dissertation we will only be interested in finding the electronic ground-state which is the eigenstate of the Hamiltonian with the lowest eigenvalue . Suppose that we have a state close to the ground-state, but with some small error. Since the eigenstates of the Hamiltonian form a complete set, the error can be expanded as a linear combination of the excited eigenstates. The whole state can thus be written as

 (2.46)

where
 (2.47)

We now calculate the value of :
 (2.48)

By definition, for , so that we note two points:
• , with equality only when (i.e. for ),
• the error in the estimate of is second-order in the error in the wave-function (i.e. ).

The importance of such a variational principle is now clear. To calculate the ground-state energy , we can minimise the functional with respect to all states which are antisymmetric under exchange of particles. The value of this functional gives an upper bound to the value of , and even a relatively poor estimate of the ground-state wave-function gives a relatively good estimate of . Eigenstates corresponding to excited states of the Hamiltonian can be found by minimising the functional with respect to states which are constructed to be orthogonal to all lower-lying states (which is usually achieved by considering the symmetries of the states) but in this work we will only ever be interested in the ground-state, and so there are no restrictions on the states other than antisymmetry.

Next: 3. Quantum Mechanics of Up: 2. Many-body Quantum Mechanics Previous: 2.3 Identical particles   Contents
Peter Haynes