# Hamiltonian quantum mechanics

1. Jun 14, 2012

### phys2

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

A particle moves in a one dimensional potential : V(x) = 1/2(mω2x

Show that the function ψ(x) = a0exp(-βx2) is an eigenfunction for the Hamiltonian for a suitable value of β and calculate the value of energy E1

2. Relevant equations

3. The attempt at a solution

What I did was take the Hamiltonian, differentiate twice the function ψ(x) and then sub in the twice differentiated function along with the potential into the Hamiltonian. But there I get confused. Apparently taking a peek at the solutions, it argues that you should set the Hamiltonian to zero and then since it is supposed to be a constant, the x^2 terms cancel out and your final answer is β=mω/2hbar. I have no idea what they have done though! Any one care to explain? Thanks!

2. Jun 14, 2012

### vela

Staff Emeritus
Show us what you got when you did that. If $\psi$ is an eigenfunction of $\hat{H}$, what does $\hat{H}\psi$ have to equal?

3. Jun 14, 2012

### phys2

So I got -h(bar)2/2m (4β2x2) ψ + 1/2mω2x2ψ = Hamiltonian

Hψ=Eψ?

4. Jun 14, 2012

### vela

Staff Emeritus
You're missing a term. You didn't differentiate correctly, perhaps.

Right, so after applying the Hamiltonian, you should be able to write the result as a constant times the original wave function. To do that, β will have to take on a specific value.