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Modified Harmonic Oscillator probabilities

  1. Jul 25, 2014 #1
    1. The problem statement, all variables and given/known data

    The e-functions for n=0,1,2 e-energies are given as

    psi_0 = 1/(pi^1/4 * x0^1/2)*e^(x^2/(2*x0^2)

    psi_1 =...

    psi_2 =...

    The factor x0 is instantaneously changed to y= x0/2. This means the initial wavefunction does not change.

    Find the expansions coefficients of the initial state to the three lowest states of the modified potential. What energy measurements would you find and with what probability.

    2. Relevant equations

    Prob En = c_n ^2

    En = hbar*omega*(n+0.5)

    3. The attempt at a solution

    I can find the new energies:

    En = hbar*omega*(n+0.5) = m*omega^2*x_0^2*(n+1/2)

    Putting in y = x_0 instead we get:

    En = m*omega^2*y^2*(n+1/2)
    = 1/4 m*omega^2*x_0^2*(n+1/2)
    = 1/4 hbar*omega*(n+0.5)


    For the expansion coefficients I am not sure what to put into the formula:

    cn = integral from -inf to inf: phi* * psi dx

    I know that the modified eqn to psi_0 is:

    psi_m0 = sqrt(2)*e^2 * psi_0

    But if I put this in I just get sqrt(2)*e^4 and that doesn't make sense for the probs (>1)
     
    Last edited: Jul 25, 2014
  2. jcsd
  3. Jul 25, 2014 #2

    BiGyElLoWhAt

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    Gold Member

    Yes, assuming that ##\dot{x}_0## = 0. Otherwise the phase shift will change.
    Is this a typo? ##\Psi^* \Psi##
    Honestly I'm not sure what you're trying to do, if that was supposed to be a complex conjucate (that's what it looks like to me) Your psi functions are real, so... Again, though, I really don't know what your trying to do. I'm not going to guarantee to be able to help you, but could you give a clearer explaination of the problem?

    It almost looks as thought you're trying to normalize a function to solve for the coefficients, but I'm not sure.
     
  4. Jul 25, 2014 #3

    TSny

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    Homework Helper
    Gold Member

    This isn't correct. You can't factor out e2 like that.
     
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