Next: Molecular Quantum Mechanics
Up: Approximate Methods
Previous: Perturbation Theory
Contents
The variational method is the other main approximate method used in
quantum mechanics. Compared to perturbation theory, the variational
method can be more robust in situations where it's hard to determine a
good unperturbed Hamiltonian (i.e., one which makes the perturbation small
but is still solvable). On the other hand, in cases where there is a
good unperturbed Hamiltonian, perturbation theory can be more
efficient than the variational method.
The basic idea of the variational method is to guess a ``trial''
wavefunction for the problem, which consists of some adjustable
parameters called ``variational parameters.'' These parameters are
adjusted until the energy of the trial wavefunction is minimized. The
resulting trial wavefunction and its corresponding energy are
variational method approximations to the exact wavefunction and
energy.
Why would it make sense that the best approximate trial wavefunction
is the one with the lowest energy? This results from the Variational
Theorem, which states that the energy of any trial wavefunction is
always an upper bound to the exact ground state energy .
This can be proven easily. Let the trial wavefunction be denoted
. Any trial function can formally be expanded as a linear
combination of the exact eigenfunctions . Of course,
in practice, we don't know the , since we're assuming that
we're applying the variational method to a problem we can't solve
analytically. Nevertheless, that doesn't prevent us from using the
exact eigenfunctions in our proof, since they certainly exist and form
a complete set, even if we don't happen to know them. So, the trial
wavefunction can be written

(146) 
and the approximate energy corresponding to this wavefunction is

(147) 
Substituting the expansion over the exact wavefuntions,

(148) 
Since the functions are the exact eigenfunctions of , we can use
to obtain

(149) 
Now using the fact that eigenfunctions of a Hermitian operator form an
orthonormal set (or can be made to do so),

(150) 
We now subtract the exact ground state energy from both
sides to obtain

(151) 
Since every term on the righthand side is greater than or equal to
zero, the lefthand side must also be greater than or equal to zero,
or

(152) 
In other words, the energy of any approximate wavefunction is always
greater than or equal to the exact ground state energy .
This explains the strategy of the variational method: since the energy
of any approximate trial function is always above the true energy,
then any variations in the trial function which lower its energy are
necessarily making the approximate energy closer to the exact answer.
(The trial wavefunction is also a better approximation to the true
ground state wavefunction as the energy is lowered, although not
necessarily in every possible sense unless the limit
is reached).
One example of the variational method would be using the Gaussian
function
as a trial function for the
hydrogen atom ground state. This problem could be solved by the
variational method by obtaining the energy of as a function
of the variational parameter , and then minimizing
to find the optimum value . The variational theorem's
approximate wavefunction and energy for the hydrogen atom would then
be
and
.
Frequently, the trial function is written as a linear combination
of basis functions, such as

(153) 
This leads to the linear variation method, and the variational
parameters are the expansion coefficients . The energy for this
approximate wavefunction is just

(154) 
which can be simplified using the notation
to yield

(157) 
Differentiating this energy with respect to the expansion coefficients
yields a nontrivial solution only if the following ``secular
determinant'' equals 0.

(158) 
If an orthonormal basis is used, the secular equation is greatly
simplified because is 1 for and
0 for . In this case, the secular determinant is

(159) 
In either case, the secular determinant for basis functions gives
an th order polynomial in which is solved for different
roots, each of which approximates a different eigenvalue.
The variational method lies behind HartreeFock theory and the
configuration interaction method for the electronic structure of
atoms and molecules.
Next: Molecular Quantum Mechanics
Up: Approximate Methods
Previous: Perturbation Theory
Contents
David Sherrill
20060815