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Show that function is not an energy eigenfunction

  1. Mar 13, 2013 #1
    1. The problem statement, all variables and given/known data

    We are considering the Superposition state:

    ψ(x,t) = 1/2 u1(x)e(-i/hbar)E1t + √3/2 u4(x) e(-i/hbar)E4t

    We had to verify that ψ is a solution of the time dependent Schrodinger Equation, which I have done. Next we are asked to show that ψ is NOT an energy eigenfunction and therefore not a solution to the time independent Schrodinger Equation.

    2. Relevant equations

    Time Independent Schrodinger Equation: H(ψ(x,t)) = Enψ(x,t)
    Hamiltonian operator: H= (-hbar/2m)∇2+V(x)

    3. The attempt at a solution

    At first I tried to see if I could somehow show that the first equation did not hold and tried to apply the Hamiltonian on the left hand side, but I was just left with ∇2u(x) terms and I couldn't think of any way of simplifying them. Then I considered the time dependent Schrodinger which states that the Hamiltonian of ψ is equal to (i*hbar)dψ/dt. So I then set the right hand sides of each Schr. equation equal to each other (as the left hand side of both is the Hamiltonian) and tried to go about showing that this equality did not hold. I can't seem to get anywhere with this method. I'm wondering if there's a simpler way of doing this?
     
  2. jcsd
  3. Mar 13, 2013 #2

    tiny-tim

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    hi beth92! :smile:

    hint: H is additive :wink:
     
  4. Mar 13, 2013 #3
    Hmm okay. So can I just say that ψ is a superposition of two states ψ1 and ψ4 and that the time independent Schrodinger Equation then becomes:

    H(ψ14) = En*(ψ14)

    which, due to the fact that H is additive means that:

    H(ψ1)+H(ψ4) = E1ψ1+E4ψ4

    And therefore En is not an energy eigenvalue but is rather a sum/superposition of two separate energies E1 and E4...or does En just not exist at all? I feel like I almost understand it but I am struggling to come up with words to explain the conclusion.
     
  5. Mar 13, 2013 #4

    tiny-tim

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    hi beth92! :smile:
    that's right …

    now you have to prove that that is not a scalar multiple of ψ1 + ψ4 :wink:
     
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