# Show that the two wave functions are eigenfunction…

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

Consider the dimensionless harmonic oscillator Hamiltonian

*=½*

**H**

**P**^{2}+½

**X**^{2},

**=-**

*P**i*d/dx.

- Show that the two wave functions ψ
_{0}(x)=e^{-x}^{2}/2 and ψ_{1}(x)=xe^{-x}^{2}/2 are eigenfunction ofwith eigenvalues ½ and 3/2, respectively.*H* - Find the value of the coefficient
*a*such that ψ_{2}(x)=(1+*a*x^{2})e^{-x}^{2}/2 is orthogonal to ψ_{0}(x). Then show that ψ_{2}(x) is an eigenfunction ofwith eigenvalue 5/2.**H**

**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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