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Show that the two wave functions are eigenfunction

  1. Mar 29, 2014 #1
    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.
  2. jcsd
  3. Mar 30, 2014 #2

    Simon Bridge

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    Science Advisor
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    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.
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