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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 level Is desirable that the perturbation LS to be diagonal This 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 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)