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Vibrational Motion - Calculating Mean Square Displacement

  1. Feb 14, 2014 #1
    1. The problem statement, all variables and given/known data
    Calculate the mean square displacement x2 of the particle from its
    equilibrium position.


    2. Relevant equations
    ∫ from -[itex]\infty[/itex] to +[itex]\infty[/itex] of Nv2 * Hv(y) * e-y^2 dy

    Since y=x/[itex]\alpha[/itex], [itex]\alpha[/itex]dy=dx

    yHv = vHv-1 + (1/2)Hv+1


    3. The attempt at a solution
    https://www.dropbox.com/s/uiqbgzjjlqnnqwk/2014-02-14%2022.23.41.jpg

    What is boxed is where I distributed everything. That looked horrible so I applied the recursion relation once. I believe the last integral goes to zero. Integrating an odd function over a symmetrical range would be zero. But then everything would be zero and that's just wrong. There should be a relationship between v and m and kf and hbar from [itex]\alpha[/itex]. I apologize, but please explain the math as simple as possible. The math is the issue, not really the concept.

    EDIT: Should I apply the recursion relation once more in the integral as it has yHv?
     
    Last edited: Feb 14, 2014
  2. jcsd
  3. Feb 14, 2014 #2

    ShayanJ

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    Gold Member

    I don't know that much about Hermite polynomials to judge your calculations but what I know is that they are alternating in being odd and even.In fact even numbered ones are even and odd numbered ones are odd.
    Take this into account when calculating the integral!
     
  4. Feb 14, 2014 #3

    ShayanJ

    User Avatar
    Gold Member

    Equation (50) in this page may help too.
     
    Last edited: Feb 14, 2014
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