Spin-Orbit Coupling

This is the most interesting term as it involves the electron spin. Furthermore, this type of interaction has found a wide-ranging interest in other areas of physics, for example in the context of spin-electronics (`spin-transistor') in condensed matter systems.

The general derivation of spin-orbit coupling from the Dirac equation for an electron of mass $m$ and charge $-e<0$ in an external electrical field ${\bf E}({\bf r})= -{\bf\nabla}\Phi({\bf r})$ yields

$$\hat{H}_{\rm SO} = \frac{e\hbar}{4m^2c^2}{\bf\sigma} [ {\bf E}({\bf r})\times {\bf p}],$$ (3.6)

where ${\bf p}=m{\bf v}$ is the momentum operator and ${\bf\sigma}$ is the vector of the Pauli spin matrices,

$$\hat{\sigma}_{x}\equiv \begin{pmatrix}0 & 1 1 & 0 \end{pmatrix... ...hat{\sigma}_{z}\equiv \left( \begin{matrix}1 & 0 0 & -1 \end{matrix}\right).$$ (3.7)