Consider a particle in a box in the interval [-a,a]. Use the trial wavefunction
ψT = x(a-x2)
to obtain an approximate energy for the first excited state of the box as a function of a.
Schrodinger equation, Hamiltonian for atomic units is 1/2(d2/dx2)
Normalized energy E = ∫-aaψ*Hψdx/∫-aaψ*ψdx
The Attempt at a Solution
Right, so I just plugged in ψT to the energy equation, and after evaluating the integrals, got E = 1-(3/5)a2 / [a(a2/7-1/15)]. The problem is, this function is discontinuous. It is discontinuous at 0, which makes sense to me, and at a = √(7/15), which I don't understand. I'm pretty sure my integrations were done properly... can anyone either explain this discontinuity to me, or tell me why my set-up was flawed?