Quantum Mechanics Variational Method

[cep]

Homework Statement

Consider a particle in a box in the interval [-a,a]. Use the trial wavefunction

ψ_T = x(a-x²)

to obtain an approximate energy for the first excited state of the box as a function of a.

Homework Equations

Schrödinger equation, Hamiltonian for atomic units is 1/2(d²/dx²)

Normalized energy E = ∫_-a^aψ*Hψdx/∫_-a^aψ*ψdx

The Attempt at a Solution

Right, so I just plugged in ψ_T 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?

Thanks!

 

[cep]

Okay, I found one error in my integration. The revised function is E = 1-(3/5)a / [a(a^2/7-1/15)]. Still has the same discontinuity problems, though.