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Hydrogen atom - small perturbation

  1. Jul 28, 2008 #1
    1. The problem statement, all variables and given/known data

    A hydrogen atom is perturbed with the potential [tex]V(r) = \frac{\alpha}{r^{2}}[/tex] ([tex]\alpha[/tex] is small). Find first-order perturbation corrections to the energy levels and then exact levels of the perturbed system.


    2. Relevant equations

    The unperturbed hydrogen atom radial equation is:

    [tex]-\frac{\hbar^{2}}{2m} \frac{d^{2}u}{dr^{2}} + [-\frac{e^{2}}{4\pi \epsilon_{0}} \frac{1}{r} + \frac{\hbar^{2}}{2m} \frac{l(l+1)}{r^{2}}]u = Eu[/tex]

    where [tex]l[/tex] is an integer.


    3. The attempt at a solution

    I don't know how to find the exact energy levels of the perturbed system. Because the perturbation is proportional to [tex]\frac{1}{r^{2}}[/tex], in the radial equation for the perturbed atom I can introduce a new parameter [tex]k[/tex] such that:

    [tex]\frac{\hbar^{2}}{2m} \frac{k(k+1)}{r^{2}} = \frac{\hbar^{2}}{2m} \frac{l(l+1)}{r^{2}} + \frac{\alpha}{r^{2}}[/tex].

    Then new energy levels will be just energy levels of an unperturbed hydrogen atoms with [tex]k[/tex] in place of [tex]l[/tex]. But then, [tex]k[/tex] has to be an integer for the hydrogen solutions to make sense, and it is at the same time a function of the parameter [tex]\alpha[/tex], so it need not be an integer. What's wrong here? How to find the exact energy levels?
     
  2. jcsd
  3. Jul 29, 2008 #2
    It looks like your attempt at a solution is trying to find the exact levels. Take a look at Chapter 6 in Griffiths' QM book for finding the first-order perturbation corrections. You could probably also look up time-independent perturbation theory on the web and find a wikipedia page or something.

    I'm not completely sure about finding the exact levels, but I think you have a good idea there. This perturbation won't affect the spherical harmonics at all since it is r-dependent. It seems like it might work to just replace [tex]n=j_{max}+l+1[/tex] with [tex]n=j_{max}+k+1[/tex] and plug the n into the energy formula, but I'm not so sure there because of the issue of k not having to be an integer.
     
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