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## Homework Statement

We are to combine the Relativistic Kinetic Energy, Spin-Orbit Interaction, and Darwin fine structure correction terms into a single formula for the energy shift in the Hydrogen atom. The formula must depend only on j = l +/- 1/2, but not l, and must be valid for all l, including l = 0.

## Homework Equations

The above corrections are given as:

https://mywebspace.wisc.edu/dpfahey/web/PF01.bmp [Broken]

## The Attempt at a Solution

Well, [itex] \Delta {E}_{n,total} = \Delta {E}_{n,kin} + \Delta {E}_{n,so} + \Delta {E}_{n,D}[/itex]

Where [itex]<S \cdot L> = \left[j(j + 1) - l(l + 1) - s(s +1) \right ] [/itex]

So, presumably, we just add the given corrections, and collect/eliminate like terms. I began doing this until I became confused by stipulation of dependence on j only, and not l.

So my (simple) question is: If the formula will depend on j, and j depends on l, then how will the formula not depend on l?

Also, how will the resultant formula be good for all l, as one of the correction terms does not allow for l = 0?

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