Show that the two wave functions are eigenfunction

[gfxroad]

Homework Statement

Consider the dimensionless harmonic oscillator Hamiltonian
H=½ P²+½ X², P=-i d/dx.

1. Show that the two wave functions ψ₀(x)=e^-x2/2 and ψ₁(x)=xe^-x2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively.
2. Find the value of the coefficient a such that ψ₂(x)=(1+ax²)e^-x2/2 is orthogonal to ψ₀(x). Then show that ψ₂(x) is an eigenfunction of H with eigenvalue 5/2.

The Attempt at a Solution

For orthogonality the wave function product must equal to zero, and for eigenfunction we take the second derivative for both wave functions and make a comparison between the eigenvalues.
But I can't finalise the problem, so I appreciate any help in advance.

 

[Simon Bridge]

Please show working.
Your method is sort of OK.
It's a bit more than just taking the product or the second derivative.
You have to apply the definitions.