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Homework Help: Fine structure, hydrogen atom, principal quantum number 3

  1. Nov 13, 2011 #1


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    1. The problem statement, all variables and given/known data
    The level n=3 for atoms with 1 electron have the states [itex]3s_{1/2}[/itex], [itex]3p_{1/2}[/itex], [itex]3p_{3/2}[/itex], [itex]3d_{3/2}[/itex], [itex]3d_{5/2}[/itex]. 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
    [itex]E_{n,j}=E_n \left [1+\frac{Z^2 \alpha ^2}{n} \left ( \frac{1}{j+1/2}-\frac{3}{4n} \right ) \right ][/itex]. 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: [itex]E_{n,j}=E_n - \frac{Z^4 \alpha ^2}{n^3}E_0 \left ( \frac{2}{2j+1} -\frac{3}{4n} \right )[/itex].

    3. The attempt at a solution
    I reach, using the book/wikipedia formula that for the states with j=5/2, [itex]\Delta E \approx -2.24 \times 10 ^{-6}eV[/itex].
    For the states with j=3/2, [itex]\Delta E \approx -6.70 \times 10 ^{-6}eV[/itex].
    For the states with j=1/2, [itex]\Delta E \approx -2.46 \times 10 ^{-5}eV[/itex].
    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.
  2. jcsd
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