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Hamiltonian quantum mechanics

  1. Jun 14, 2012 #1
    1. The problem statement, all variables and given/known data

    A particle moves in a one dimensional potential : V(x) = 1/2(mω2x

    Show that the function ψ(x) = a0exp(-βx2) is an eigenfunction for the Hamiltonian for a suitable value of β and calculate the value of energy E1

    2. Relevant equations



    3. The attempt at a solution

    What I did was take the Hamiltonian, differentiate twice the function ψ(x) and then sub in the twice differentiated function along with the potential into the Hamiltonian. But there I get confused. Apparently taking a peek at the solutions, it argues that you should set the Hamiltonian to zero and then since it is supposed to be a constant, the x^2 terms cancel out and your final answer is β=mω/2hbar. I have no idea what they have done though! Any one care to explain? Thanks!
     
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  3. Jun 14, 2012 #2

    vela

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    Show us what you got when you did that. If ##\psi## is an eigenfunction of ##\hat{H}##, what does ##\hat{H}\psi## have to equal?

     
  4. Jun 14, 2012 #3
    So I got -h(bar)2/2m (4β2x2) ψ + 1/2mω2x2ψ = Hamiltonian

    Hψ=Eψ?
     
  5. Jun 14, 2012 #4

    vela

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    You're missing a term. You didn't differentiate correctly, perhaps.

    Right, so after applying the Hamiltonian, you should be able to write the result as a constant times the original wave function. To do that, β will have to take on a specific value.
     
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