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:

\[(\vec{\sigma} \cdot \vec{u})(\vec{\sigma} \cdot \vec{v}) = \vec{u} \cdot \vec{v} I_{2} + i \vec{\sigma} \cdot (\vec{u} \times \vec{v})\]

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:

\[ \begin{align}\begin{aligned}\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)\\\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = I_{2}\\\sigma_i \sigma_j = \delta_{ij} I_{2} + i \epsilon_{ijk} \sigma_k\end{aligned}\end{align} \]

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 \(c \rightarrow \infty\).

\[\left[ \frac{\vec{p}^2}{2m_e} + \frac{e^2 \vec{A}^2}{2m_e} + \frac{e}{2m_e}(\mathbf{l} + 2 \mathbf{s}) \cdot \vec{B} + V \right] \psi^L = i \hbar \frac{\partial}{\partial t} \psi^L\]

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

\[\begin{split}i \hbar \frac{\partial}{\partial t} \left( \begin{array}{c} \psi^L \\ \psi^S \end{array} \right) = c \left( \begin{array}{c} (\vec{\sigma} \cdot \vec{\pi}) \psi^S \\ (\vec{\sigma} \cdot \vec{\pi}) \psi^L \end{array} \right) + m_ec^2 \left( \begin{array}{c} \psi^L \\ -\psi^S \end{array} \right) + V \left( \begin{array}{c} \psi^L \\ \psi^S \end{array} \right)\end{split}\]

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

\[V \rightarrow V - m_ec^2\]

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

\[\psi^S \approx \frac{\vec{\sigma} \cdot \vec{\pi}}{2m_ec} \psi^L\]

  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

\[\vec{\nabla}\cdot\vec{A} = 0\]

  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.

Literature and further reading