Effects of applied static magnetic and electric fields

Effects of applied static magnetic and electric fields 

We now derive the additional terms in the Hamiltonian which arise from the application of an external magnetic field. The magnetic vector potential will be a sum of the contributions from the electrons (Af) and from the additional term 

With the substitution of (3.162), the contribution from the scalar product of Af and Pi in the second term of equation (3.140) is given by 

This term represents the interaction of the external magnetic field with the magnetic moment arising from the orbital motion of the electrons. Next we investigate the third term in (3.140) and find 

This term represents the coupling of the spin magnetic dipole moment with the applied field, and was derived previously for a single electron in equation (3.101). Turning to the fourth term in (3.140) we note that in the field free case we retained only the leading term (i.e. Pf). However, the term linear in Af, although formally of order c 3, has been found to contribute measurably to the Zeeman effect in some cases. The resulting term 

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