# Eigenvalue for 1D Quantum Harmonic Oscillator

1. Feb 3, 2013

### theojohn4

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

Show that the following is an eigenfunction of $$\hat{H}_{QHO}$$ and hence find the corresponding eigenvalue:

$$u(q)=A (1-2q^2) e^\frac{-q^2} {2}$$

2. Relevant equations

Hamiltonian for 1D QHO of mass m
$$\hat{H}_{QHO} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 x^2$$

3. The attempt at a solution

I don't know where to start

2. Feb 3, 2013

### cosmic dust

What means that a function is an eigen-function of some operator? Doesn't this mean that when you act with that operator on that function, the result will be the original function multiplied by a constant (where this constant is the eigenvalue)?

So, try to operate with HQHO on u(q) and see what happens!

(the momentum operator is repsresented by -i$\hbar$d/dx)