A.7 Accuracy of the variational method
This note has a closer look at the accuracy of the variational method.
Any approximate ground state solution may always be written as
a sum of the eigenfunctions :
where, if the approximation is any good at all, the coefficient
of the ground state is close to one, while
, , ...are small.
The condition that is normalized, 1,
works out to be
since the eigenfunctions are orthonormal.
Similarly, the expectation energy
works out to be
Eliminating using the normalization condition above gives
One of the things this expression shows is that any approximate wave
function (not just eigenfunctions) has more expectation energy than
the ground state . All other terms in the sum above are
positive since is the lowest energy value.
The expression above also shows that while the deviations of the wave
function from the exact ground state are proportional to the
coefficients , the errors in
energy are proportional to the squares of those coefficients.
And the square of any reasonably small quantity is much smaller than
the quantity itself. So the approximate ground state energy is much
more accurate than would be expected from the wave function errors.
Still, if an approximate system is close to the ground state energy,
then the wave function must be close to the ground state wave
function. More precisely, if the error in energy is a small number,
call it , then the amount of
polluting approximate ground
state must be no more than
. And that is in the worst case
scenario that all the error in the expectation value of energy is due
to the second eigenfunction.
As a measure of the average combined error in wave function, you can
use the magnitude or norm of the combined pollution:
That error is no more than . To
verify it, note that
(Of course, if the ground state wave function would be degenerate,
would be . But in that case you do not care about
the error in , since then and are
equally good ground states, and becomes .)
The bottom line is that the lower you can get your expectation energy,
the closer you will get to the true ground state energy, and the small
error in energy will reflect in a small error in wave function.