The Hydrogen atom Fine structure

Relativistic effect – the Klein-Gordon equation

the fine structure constant

The spin of the electron

• s=1/2

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 v2/c2, 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 levelIs desirable that the perturbation LS to be diagonalThis is valid in the representation |lsjmj>,because L2, S2, J2 and Jz 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 spin-orbit 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)