Energy level approximation

1. Dec 5, 2007

Mjolnir

1. The problem statement, all variables and given/known data
in the first part of the problem we were told to estimate the lowest energy of a electron in a hydrogen atom for a certain orbital angular momentum l by evaluating the equation for effective potential at it's minimum. That is to say that for any given l the minimum energy state has n = l + 1. If you then evaluate V-eff = l(l+1)/2p^2 -1/p for l = n - 1 and p=p0=l(l+1) this should be a rough estimate of the actual energy. Now this is easy enough to evaluate and gives not too large of an overestimate; the next part of the problem, however, has me a little confused. For the same case I'm supposed improve the estimate by making a second order taylor expansion of V-eff about p0. The thing is I really don't see how a taylor expansion is supposed to help me, especially considering that the value at p0 will be no different than that of the original function.

2. Relevant equations
V-eff = l(l+1)/2p^2 -1/p
exact energy eigenvalues = $$\lambda$$n = 1/(2n^2), n = 1, 2, 3, ...

3. The attempt at a solution
Finding the taylor series is simple enough, giving you
(n^2-n-p)^2/(n*(n-1))^3 - 1 /(2n*(n-1))
The problem is that I don't know what to do with this now that I have it.