The course comprises two pen-and-paper exercises which are based on materials of the morning-session lectures of Day 1.

Dirac’s relation

A relation that is often exploited throughout the course, e.g. to achieve a separation of spin-dependent and independent terms in the Dirac equation (see lecture on Day 2), is Dirac’s relation [Dirac1928] , which can be written for two arbitrary vector operators $\overrightarrow{u}$ and $\overrightarrow{v}$ as:

$$(\overrightarrow{\sigma }\cdot \overrightarrow{u})(\overrightarrow{\sigma }\cdot \overrightarrow{v})=\overrightarrow{u}\cdot \overrightarrow{v}{I}_2+i\overrightarrow{\sigma }\cdot (\overrightarrow{u}\times \overrightarrow{v})$$

where $\overrightarrow{\sigma }$ are the Pauli spin matrices and $I_2$ is a $2\times 2$ unit matrix. Note that $\overrightarrow{u}$ and $\overrightarrow{v}$ do not necessarily commute.

• Problem 1: Verify the relation expanding the left-hand side of Dirac’s relation. Hint: use the relations for the Pauli spin matrices:

$$\begin{array}{r}\begin{array}{c}\overrightarrow{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)\\ {\sigma }_x^2={\sigma }_y^2={\sigma }_z^2=I_2\\ {\sigma }_i{\sigma }_j={\delta }_{ij}{I}_2+i{ϵ}_{ijk}{\sigma }_k\end{array}\end{array}$$

Note that we use the Einstein summation convention; in the final expression above the index $k$ appears twice in the same term, which implies that we sum over it.

• Problem 2: derive a final expression inserting for $\overrightarrow{u}=\overrightarrow{v}$ the kinematical momentum operator $\overrightarrow{\pi }=\overrightarrow{p}+e\overrightarrow{A}$ (where $\overrightarrow{A}$ is an external electromagnetic vector potential).

Two-component Pauli equation (0th order Pauli equation)

From the one-electron Dirac equation in external magnetic fields we may derive the Pauli equation (also known as 0th order Pauli equation) by considering the non-relativistic limit $c\to \mathrm{\infty }$.

$$[\frac{{\overrightarrow{p}}^2}{2m_e}+\frac{e^2{\overrightarrow{A}}^2}{2m_e}+\frac{e}{2m_e}(\mathbf{l}+2\mathbf{s})\cdot \overrightarrow{B}+V]{\psi }^L=i\hslash \frac{\partial }{\partial t}{\psi }^L$$

• Problem 3: Derive the above equation starting from the one-electron Dirac equation in external magnetic fields written in two-spinor form:

$$\begin{array}{r}i\hslash \frac{\partial }{\partial t}\left(\begin{array}{c}{\psi }^L\\ {\psi }^S\end{array}\right)=c\left(\begin{array}{c}(\overrightarrow{\sigma }\cdot \overrightarrow{\pi }){\psi }^S\\ (\overrightarrow{\sigma }\cdot \overrightarrow{\pi }){\psi }^L\end{array}\right)+m_e{c}^2\left(\begin{array}{c}{\psi }^L\\ -{\psi }^S\end{array}\right)+V\left(\begin{array}{c}{\psi }^L\\ {\psi }^S\end{array}\right)\end{array}$$

The following hints may be useful:

1. shift the zero of energy to the non-relativistic limit one. This corresponds to the elimination of rest mass and is achieved by the substitution

$$V\to V-m_e{c}^2$$

2. elimininate the small-component ${\psi }^S$ from the upper component using the magnetic balance condition:

$${\psi }^S\approx \frac{\overrightarrow{\sigma }\cdot \overrightarrow{\pi }}{2m_{e}c}{\psi }^L$$

3. use $\overrightarrow{A}\cdot \overrightarrow{p}=\overrightarrow{p}\cdot \overrightarrow{A}-(\overrightarrow{p}\cdot \overrightarrow{A})$

   and remember that in Coulomb gauge we have

$$\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{A}=0$$

4. for a constant and homogeneous magnetic field $\overrightarrow{B}=\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{A}$, we may write $\overrightarrow{A}=\frac{1}{2}(\overrightarrow{B}\times \overrightarrow{r})$

• Problem 4: define the Bohr magneton and the gyromagnetic ratio g of the electron according to the Pauli Hamiltonian.

Literature and further reading

• [Dyall2007], Chapter 4.

• [Reiher2009], Chapter 5.