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.
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
Energy of 1D Harmonic Oscillator using a polynomial trail wave function.*
* Example: 1D H.O. using Gaussian.*
Jim Branson 2013-04-22