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:

\begin{equation*} (\vec{\sigma} \cdot \vec{u})(\vec{\sigma} \cdot \vec{v}) = \vec{u} \cdot \vec{v} I_{2} + i \vec{\sigma} \cdot (\vec{u} \times \vec{v}) \end{equation*}

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:
\begin{equation*} \vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z) \end{equation*}
\begin{equation*} \sigma_x^2 = \sigma_y^2 = \sigma_z^2 = I_{2} \end{equation*}
\begin{equation*} \sigma_i \sigma_j = \delta_{ij} I_{2} + i \epsilon_{ijk} \sigma_k \end{equation*}

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 $$c \rightarrow \infty$$ .

\begin{equation*} \left[ \frac{\vec{p}^2}{2m_e} + \frac{e^2 \vec{A}^2}{2m_e} + \frac{e}{2m_e}(\mathbf{l} + 2 \mathbf{s}) \cdot \vec{B} + V \right] \psi^L = i \hbar \frac{\partial}{\partial t} \psi^L \end{equation*}
• Problem 3: Derive the above equation starting from the one-electron Dirac equation in external magnetic fields written in two-spinor form:
\begin{equation*} i \hbar \frac{\partial}{\partial t} \left( \begin{array}{c} \psi^L \\ \psi^S \end{array} \right) = c \left( \begin{array}{c} (\vec{\sigma} \cdot \vec{\pi}) \psi^S \\ (\vec{\sigma} \cdot \vec{\pi}) \psi^L \end{array} \right) + m_ec^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{equation*}

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
\begin{equation*} V \rightarrow V - m_ec^2 \end{equation*}
1. elimininate the small-component $$\psi^S$$ from the upper component using the magnetic balance condition:
\begin{equation*} \psi^S \approx \frac{\vec{\sigma} \cdot \vec{\pi}}{2m_ec} \psi^L \end{equation*}
1. use $$\vec{A} \cdot \vec{p} = \vec{p} \cdot \vec{A} - (\vec{p} \cdot \vec{A})$$

and remember that in Coulomb gauge we have

\begin{equation*} \vec{\nabla}\cdot\vec{A} = 0 \end{equation*}
1. 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.