A system can be represented by $\displaystyle H = -\frac{d^2}{dx^2} + x^2$ (H is the Hamiltonian operator).

Show $\displaystyle Axe^{\frac{-x^2}{2}}$ is an eigenfunction of $\displaystyle H$ and calculate the eigenvalue. Then, by normalizing, find $\displaystyle A$.

I have the answer, but not sure how they get it. It is:

Eigenvalue = 3

$\displaystyle A=2^{\frac{1}{2}}\pi^{-\frac{1}{4}}$