1.3.2 Spin-orbit Coupling

For an electron in a static electric field A=0, V=0, you should take E → E-eA0= E-V(r)=E′+mc2-V(r), and then you have the following stationary Dirac equation

If you consider the case of the non-relativistic motion in a week field so that|E′-V|≪mc2is valid, the second equation of(1.51)gives rise to

Substituting it into the first equation of (1.51), you get an equation con-taining the two-component function φ only

In deriving the second line of the above equation, you need to employ the operator identity(1.36). Eq. (1.53)can be rearranged as

Here you need carefully deal with the consistency of the normalization condition because ρ=Ψ†Ψ=φ†φ+χ†χ=φ†(1+)φ up to the order of v2/c2. Now, instead of φ, you need to use another function

with

so that the normalization is consistently fulfilled

Multiplying(1.54)from left with Λ-1and calculating the relations

and

you will obtain, up to v2/c2order, the equation

with the following Hamiltonian

where the prime in E was omitted.

Now you perceive that the first two terms in (1.58) are just the non-relativistic Hamiltonian in Schrödinger equation. The next three terms are relativistic corrections of order v2/c2. The last term was first introduced by Darwin, which determines an additional energy contributed by the in-teraction between an electron in s-state and the nucleus. This is because in Coulomb field V=-Ze/r and∇2(1/r)=-4πδ(r), the Darwin term is given by Zπeħ2δ(r)/(2m2c2). The third term is a correction to the kinetic energy operator arising from the change of electron mass with velocity, i.e., E=m[1+p2/(mc)2]1/2c2=[1+p2/(2m2c2)-p4/(8m4c4)+. . . ]mc2. It was L. H. Thomas who firstly perceived the relativistic correction in 1926. The fourth term is called spin-orbit interaction operator, about which you can understand more clearly if considering a centrally symmetric potential field V=V(r). In this case, the fourth term becomes

Thus it represents the spin-orbit coupling which is zero for s-state of elec-tron in atoms. It is the effect of spin-orbit coupling that causes the so-called fine structure of atomic spectra.