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Simple Harmonic Motion

  1. Sep 17, 2009 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m moves in the one dimensional Poschel-Teller potential [tex]V(x)[/tex]. Find an expression for the natural frequency of small oscillations.



    2. Relevant equations

    [tex]V(x) = -V_{0}sech^{2}(x/\lambda)[/tex]



    3. The attempt at a solution

    I am making the uncertain assumption that this would be the angular frequency. But, I do not know how to derive it based on just the potential alone. I have tried to determine the period as well, graphically, but this function doesn't seem to be the type associated with normal periodic motion.
     
  2. jcsd
  3. Sep 17, 2009 #2

    kuruman

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    The potential that you have is not a harmonic potential, however any potential that is an even function can be expanded in Taylor series about its minimum and will have a leading term that is quadratic in x. The words "small oscillations" is a hint that you need to make this expansion. First plot your potential to make sure that it has a minimum, then expand about x = 0. Ignore the constant, that's the "zero of energy". The leading term will be

    [tex]\frac{1}{2}\;\frac{d^{2}V}{dx^{2}}\big|_{x=0}\;x^{2}[/tex]

    That's a term that can be related to a harmonic potential of the form

    [tex]\frac{1}{2}\;kx^{2}[/tex]

    whose frequency you can easily extract from the effective spring constant k.
     
  4. Sep 18, 2009 #3
    I figured it out. Thanks!

    But just one tiny question, if the leading term is going to be a quadratic, then the original expansion has to be out to the fourth power?
     
    Last edited: Sep 18, 2009
  5. Sep 18, 2009 #4
    I must not be doing something right, the whole thing went to zero.
     
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