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 \(\vec{u}\) and \(\vec{v}\) as:

where \(\vec{\sigma}\) are the Pauli spin matrices and \(I_{2}\) is a \(2 \times 2\) unit matrix. Note that \(\vec{u}\) and \(\vec{v}\) 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:

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 \(\vec{u} = \vec{v}\) the kinematical momentum operator \(\vec{\pi} = \vec{p} +e\vec{A}\) (where \(\vec{A}\) 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 \(c \rightarrow \infty\).

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

The following hints may be useful:

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

elimininate the small-component \(\psi^S\) from the upper component using the magnetic balance condition:

use \(\vec{A} \cdot \vec{p} = \vec{p} \cdot \vec{A} - (\vec{p} \cdot \vec{A})\)

and remember that in Coulomb gauge we have

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

**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.