Show that the two wave functions are eigenfunction…

1. The problem statement, all variables and given/known data
Consider the dimensionless harmonic oscillator Hamiltonian
HP2X2, 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.

3. 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.

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