- #1

gfxroad

- 20

- 0

## Homework Statement

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^{-x2/2}and ψ_{1}(x)=xe^{-x2/2}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^{-x2/2}is orthogonal to ψ_{0}(x). Then show that ψ_{2}(x) is an eigenfunction ofwith eigenvalue 5/2.**H**

## 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.