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Homework Help: Energy levels of hydrogen in atomic units

  1. Nov 14, 2008 #1
    [solved] Energy levels of hydrogen in atomic units

    1. The problem statement, all variables and given/known data
    The effective potential for a hydrogen atom can be thought of as the actual potential plus the centrifugal repulsion, written as:
    [tex]V_{eff} = -\frac{1}{r} + \frac{l (l+1)}{2r^{2}}[/tex]
    Remembering that you are working in atomic units, make a plot of V_eff(r) for l=0,1,2,3. Mark the energy levels for n=2,3,4. Explain why, for a given n, l cannot be larger than n-1.

    The attempt at a solution
    I have plotted the potentials for l=0,1,2,3 here: http://img385.imageshack.us/my.php?image=plotte7.gif"

    I used [tex]E=-\frac{1}{n^{2}}[/tex] for the energy levels in atomic units (is this correct?)
    As far as explaining why l <= n-1, I thought that because the energy levels would intersect the curves for l <= n-1 and the potential energies with l>=n would be unbound so that l cannot be larger than n-1. However, this does not seem to be the case from the plot I've constructed. I hope the energies I've used are incorrect, because I can't seem to extract the answer out of my plot??
    Thanks in advance!

    I realised the energy levels should be : [tex]E = -\frac{1}{2n^{2}}[/tex]
    Last edited by a moderator: Apr 23, 2017
  2. jcsd
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