Summary: Symmetry considerations may allow us to extend the variational method to certain excited states.
Looking again at the expression , and recalling that the are the squares of the overlap between the trial function and the actual eigenstates of the system, we see that we can only find bounds on excited states if we can arrange for the overlap of the trial wave function with all lower states to be zero. Usually this is not possible.
However an exception occurs where the states of the system can be separated into sets with different symmetry properties or other quantum numbers. Examples include parity and (in 3 dimensions) angular momentum. For example the lowest state with odd parity will automatically have zero overlap with the (even-parity) ground state, and so an upper bound can be found for it as well.
For the square well, the relevant symmetry is reflection about the midpoint of the well. If we choose a trial function which is antisymmetric about the midpoint, it must have zero overlap with the true ground state. So we can get a good bound on the first excited state, since . Using we get .
If we wanted a bound on , we’d need a wave function which was orthogonal to both the ground state and the first excited state. The latter is easy by symmetry, but as we don’t know the exact ground state (or so we are pretending!) we can’t ensure the first. We can instead form a trial wave function which is orthogonal to the best trial ground state, but we will no longer have a strict upper bound on the energy , just a guess as to its value.
In this case we can choose with a new value of which gives orthogonality to the previous state, and then we get (as opposed to 9 for the actual value).