# Quantum Mechanics Variational Method

1. Oct 29, 2011

### cep

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

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

Normalized energy E = ∫-aaψ*Hψdx/∫-aaψ*ψdx

3. 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?

Thanks!

2. Oct 29, 2011

### 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.