# Variational Principle Problem

1. Nov 11, 2013

### bmb2009

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

A particle of mass m is in a potential of V(x) = Kx4 and the wave function is given as ψ(x)= e^-(ax2) use the variational principle to estimate the ground state energy.

Part B:
The true ground state energy wave function for this potential is a symmetric function of x i.e. ψ0(x)=ψ0(-x). Use the result that <ψ0lψ(β)> = 0 along with an approximately chosen wave function, to estimate the energy of the first excited state.

2. Relevant equations

3. The attempt at a solution

Ok so I know how to compute variational method approximations and I have proven the identity <ψ0lψ(β)> = 0 earlier on my assignment and understand the identity as well. What I don't understand is the part that says "Use the result that <ψ0lψ(β)> = 0 along with an approximately chosen wave function, to estimate the energy of the first excited state."

Again I know that when <ψ0lψ(β)> = 0 the variational principle becomes E1≤ <ψlHlψ>/<ψlψ> but does the problem want me to chose a different wavefunction? And if so how to I go about choosing this new wave-function?

2. Nov 11, 2013

### qbert

The point is to pick a wavefunction which is orthogonal to your first
and use that as a trial to read the first excited energy [again variationally].
The clue is telling you to look at the parity, an even wf. is orthogonal to an odd wf,
so your trial function should be chosen odd.