(a) Prove the following corollary to the variational principle: If 〈ψ | ψgs〉 = 0, then...

(a) Prove the following corollary to the variational principle: If 〈ψ | ψgs〉 = 0, then 〈H〉 ≥ Efe, where Efe is the energy of the first excited state.

Thus, if we can find a trial function that is orthogonal to the exact ground state, we can get an upper bound on the first excited state. In general, it's difficult to be sure that  is orthogonal to ψgs, since (presumably) we don't know the latter. However, if the potential V(x) is an even function of x, then the ground state is likewise even, and hence any odd trial function will automatically meet the condition for the corollary.

(b) Find the best bound on the first excited state of the one-dimensional harmonic oscillator using the trial function

Ψ (x)= Axe–bx.