The Hydrogen atom Fine structure
The Hydrogen atom Fine structure
The Hydrogen atom Fine structure
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<strong>The</strong> <strong>Hydrogen</strong> <strong>atom</strong><br />
<strong>Fine</strong> <strong>structure</strong>
Relativistic effect – the Klein-Gordon<br />
equation<br />
the fine <strong>structure</strong> constant
• s=1/2<br />
<strong>The</strong> spin of the electron
<strong>The</strong> Dirac equation<br />
and act only on the spin<br />
<strong>The</strong> solution for hydrogen
• <strong>The</strong> total angular momentum<br />
• <strong>The</strong> energy depends on j!<br />
• Second-order approximation
Stationary perturbational method
<strong>The</strong> perturbational treatment of the<br />
relativistic effect<br />
If the Dirac equation is expanded in terms of v 2 /c 2 , one<br />
obtains 3 correction terms:<br />
• the relativistic correction of the kinetic energy<br />
• the Darwin-term (only for l=0)<br />
• the spin-orbit interaction term
Perturbation theory for a degenerate level<br />
Is desirable that the perturbation LS to be diagonal<br />
This is valid in the representation |lsjm j >,<br />
because L 2 , S 2 , J 2 and J z commute with LS.<br />
We write the scalar product as
<strong>The</strong> diagonal matrix element is<br />
It can be calculated analytically<br />
Finally
Adding these contributions, the terms depending on l<br />
cancel out.
<strong>The</strong> difference between the two splitted levels (the spinorbit<br />
splitting)
Lamb shift – quantum field theory
<strong>The</strong> Lamb shift experiment
Slight changes in the electromagnetic force
Hyperfine splitting<br />
• <strong>The</strong> magnetic momentum of the nucleus<br />
• <strong>The</strong> interaction with the magnetic moment of the<br />
electron (for l=0)<br />
• <strong>The</strong> splitting<br />
• For the ground state
Isotope shift (different reduced mass)