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Derivation of orbital period - Hydrogen

  1. May 21, 2015 #1
    Hey everyone!
    1. The problem statement, all variables and given/known data

    I've been giving the equation for a gaussian wave packet and from that I have to derive this formula:
    [itex]T_{Kepler}=2\pi \bar n ^3[/itex] by doing a first order taylor series approximation at [itex]\bar n[/itex] of the phase:
    [itex]f(x)=f(\bar n)+\frac{df}{dx}|_{\bar n}(x-\bar n)+\sigma(x^2)[/itex]

    The given wave packet:
    [itex]\psi(t)=\frac{1}{(2\pi\sigma_n^2)^{1/4}}\sum_{n=1}^\infty exp\left(-\frac{(n-\bar n)^2}{4\sigma_n^2}\right)r\psi_{n,l,m}exp\left(i\frac{t}{2n^2}\right)[/itex]

    2. Relevant equations
    I assume I've posted all the relevant ones in 1.

    3. The attempt at a solution
    Why is the dimension of [itex]T_{Kepler}[/itex] not seconds?
    Anyways I tried doing the taylor approximation with t as the differentiation variable and got this (assuming [itex]i\frac{t}{2n^2}[/itex] is the phase):
    [itex]i\frac{t}{2n^2}=i\frac{\bar n}{2n^2}+i\frac{t-\bar n}{2n^2}+\sigma(t^2)=i\frac{it}{2n^2}[/itex] (Strange)

    Am I supposed to do it for the variable n? Or how do I proceed from here?
    Any hints are very appreciated.
    Kind regards
    Alex
     
    Last edited: May 21, 2015
  2. jcsd
  3. May 26, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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