The purpose of this note is to verify directly that the variation of the expectation energy is zero at any energy eigenstate, not just the ground state.
Suppose that you are trying to find some energy eigenstate
with eigenvalue , and that you are close to it, but no
cigar. Then the wave function can be written as
The normalization condition 1 is, using
The expectation energy is
Now, if you make small changes in the wave function, the values of will slightly change, by small amounts that will be indicated by , and you get
The bottom line is that if you locate the nearest wave function for which 0 for all acceptable small changes in that wave function, well, if you are in the vicinity of an energy eigenfunction, you are going to find that eigenfunction.
One final note. If you look at the expression above, it seems like none of the other eigenfunctions are eigenfunctions. For example, the ground state would be the case that is one, and all the other coefficients zero. So a small change in would seem to produce a change in expectation energy, and the expectation energy is supposed to be constant at eigenstates.
The problem is the normalization condition, whose differential form