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Eigenvalues containing the variable x

  1. Aug 25, 2014 #1
    1. The problem statement, all variables and given/known data
    The wave function ψ(x)=Ae-b2x2/2 where A and b are real constants, is a normalized eigenfunction of the schrodinger eqn for a free particle of mass m and energy E. Then find the value of E


    2. Relevant equations



    3. The attempt at a solution
    Substituting the wave function in the time independent schrodinger equation in 1D for free particle (V(x)=0), I got
    [tex]E=\frac{\hbar^2 b^2 (1-b^2 x^2)}{2m}[/tex]
    But i thought eigenvalues should not contain the variable x and i don't know how to get rid of it. Apparently, the correct answer is
    [tex]E=\frac{\hbar^2 b^2 }{2m}[/tex]
    which is obtained by taking x=0 in my answer. But how can i simply substitute x=0?
     
  2. jcsd
  3. Aug 25, 2014 #2

    nrqed

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    That wave function is not an eigenstate for the free Hamiltonian, it is an eigenstate for the 1D harmonic oscillator. If they said that it describes a free particle, they made a mistake. Use the 1D harmonic oscillator and you will see that you get an x independent energy, as it should be (you are correct about that point)
     
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