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) |

(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) |

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) |

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) |

(2.47) |

(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.