(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Hydrogen atom - small perturbation

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