Eigenvalue for 1D Quantum Harmonic Oscillator

[theojohn4]

Homework Statement

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

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

Homework Equations

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

Not sure about any others

The Attempt at a Solution

I don't know where to start

 

[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 H_QHO on u(q) and see what happens!(the momentum operator is repsresented by -i$\hbar$d/dx)