# Fine structure, hydrogen atom, principal quantum number 3

1. Nov 13, 2011

### fluidistic

1. The problem statement, all variables and given/known data
The level n=3 for atoms with 1 electron have the states $3s_{1/2}$, $3p_{1/2}$, $3p_{3/2}$, $3d_{3/2}$, $3d_{5/2}$. If we ignore the spin-orbit coupling these states are degenerated. Calculate the degeneration due to the the spin-orbit coupling for the levels 3p and 3d for the hydrogen atom.

2. Relevant equations
$E_{n,j}=E_n \left [1+\frac{Z^2 \alpha ^2}{n} \left ( \frac{1}{j+1/2}-\frac{3}{4n} \right ) \right ]$. I've found this formula in books and in wikipedia. My professor gave us a slightly different formula so I'm confused which one to use.
His formula: $E_{n,j}=E_n - \frac{Z^4 \alpha ^2}{n^3}E_0 \left ( \frac{2}{2j+1} -\frac{3}{4n} \right )$.

3. The attempt at a solution
I reach, using the book/wikipedia formula that for the states with j=5/2, $\Delta E \approx -2.24 \times 10 ^{-6}eV$.
For the states with j=3/2, $\Delta E \approx -6.70 \times 10 ^{-6}eV$.
For the states with j=1/2, $\Delta E \approx -2.46 \times 10 ^{-5}eV$.
I have the feeling that my numbers are somehow wrong but I am not sure.
Could someone confirm my results? I tried to check up these values in some book and on the Internet but I didn't find them.
Thanks for any input!
Edit: Hmm I think both formula are equivalent.