How to Show the Eigenvalue for v=1 in a Harmonic Oscillator?

[ahhppull]

Homework Statement

Write down the v=1 eigenfunction for the harmonic oscillator. Substitute this eigenfunction into the Schrödinger equation and show that the eigenvalue is (3/2)hν.

Homework Equations

The Attempt at a Solution

I'm not really sure on how to to this, but here's what I did...

The wavefunction of an oscillator is ψv(x) = N_vH_v(y)e^−y2/2. For V = 1, H_v(y) = 2y.

ψv(x) = N_v(2y)e^−y2/2.

Then I normalized it by squaring it and finding out what N is. I tried plugging in the Schrödinger's equation but it doesn't work. What am I doing wrong?

 

[vela]

So far so good, but we can't help you if you don't show the rest of your work.