How to Find Energy Corrections for a Perturbed Hydrogen Atom?

[neworder1]

Homework Statement

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

Homework Equations

The unperturbed hydrogen atom radial equation is:

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

where $$l$$ is an integer.

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 $$\frac{1}{r^{2}}$$, in the radial equation for the perturbed atom I can introduce a new parameter $$k$$ such that:

$$\frac{\hbar^{2}}{2m} \frac{k(k+1)}{r^{2}} = \frac{\hbar^{2}}{2m} \frac{l(l+1)}{r^{2}} + \frac{\alpha}{r^{2}}$$.

Then new energy levels will be just energy levels of an unperturbed hydrogen atoms with $$k$$ in place of $$l$$. But then, $$k$$ has to be an integer for the hydrogen solutions to make sense, and it is at the same time a function of the parameter $$\alpha$$, so it need not be an integer. What's wrong here? How to find the exact energy levels?

 

[badphysicist]

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 $$n=j_{max}+l+1$$ with $$n=j_{max}+k+1$$ 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.