Because the ground state has the lowest possible energy,
we can vary a test wavefunction, minimizing the energy,
to get a good estimate of the ground state energy.
To do this, we will add a variable amount of an arbitrary function
to the energy eigenstate.
That is,
is stationary (2nd order changes only) with respect to variation in
.
Conversely, it can be shown that
is only stationary for eigenfunctions
.
We can use the variational principle to approximately find
and to find an upper bound on
.
* Example:
Energy of 1D Harmonic Oscillator using a polynomial trail wave function.*
* Example:
1D H.O. using Gaussian.*
Jim Branson 2013-04-22