The Hydrogen atom Fine structure

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The Hydrogen atom Fine structure

The Hydrogen atom

Fine structure


Relativistic effect – the Klein-Gordon

equation

the fine structure constant


• s=1/2

The spin of the electron


The Dirac equation

and act only on the spin

The solution for hydrogen


The total angular momentum

The energy depends on j!

• Second-order approximation


Stationary perturbational method


The perturbational treatment of the

relativistic effect

If the Dirac equation is expanded in terms of v 2 /c 2 , one

obtains 3 correction terms:

• the relativistic correction of the kinetic energy

• the Darwin-term (only for l=0)

• the spin-orbit interaction term


Perturbation theory for a degenerate level

Is desirable that the perturbation LS to be diagonal

This is valid in the representation |lsjm j >,

because L 2 , S 2 , J 2 and J z commute with LS.

We write the scalar product as


The diagonal matrix element is

It can be calculated analytically

Finally


Adding these contributions, the terms depending on l

cancel out.


The difference between the two splitted levels (the spinorbit

splitting)


Lamb shift – quantum field theory


The Lamb shift experiment


Slight changes in the electromagnetic force


Hyperfine splitting

The magnetic momentum of the nucleus

The interaction with the magnetic moment of the

electron (for l=0)

The splitting

• For the ground state


Isotope shift (different reduced mass)

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