Quantum Physics - Fine Structure

[rabbit44]

Homework Statement

Estimate the size of the fi ne structure of the 2p con guration in hydrogen-like
sodium (i.e. sodium ions with all but the last electron removed). Why can't we use
perturbation theory to calculate the effect of the spin-orbit interaction in hydrogen-like uranium?

Homework Equations

The Attempt at a Solution

I don't understand what the question means! Is it looking for a literal size of some sort, or does it want the size of the energy change from the energy of a Gross Structure state to a petrubed state?

How would you read this question?

Thanks

 

[nickjer]

For sodium you have Z=11. So you can use the same math for the hydrogen atom except replace Z with 11.

Also, perturbation theory only works for very small terms. Once the energies in perturbation theory start getting to the same scale as the initial energies, then it doesn't apply. And look at uranium, it has Z=92. Since the fine structure constant is on the order of Z^2, you get a factor of Z^2 = 8464 which isn't small anymore.