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)