:orphan:

.. _day_1_exercises:

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 :cite:`Dirac1928` , 
which can be written for two arbitrary vector operators :math:`\vec{u}` and :math:`\vec{v}` as:

.. math::

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

.. math::

  \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

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

* **Problem 2**: derive a final expression inserting for :math:`\vec{u} = \vec{v}` the kinematical momentum operator :math:`\vec{\pi} = \vec{p} +e\vec{A}` (where :math:`\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 :math:`c \rightarrow \infty`.

.. math::

  \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:

.. math::

  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)

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

.. math::

  V \rightarrow V - m_ec^2

2. elimininate the small-component :math:`\psi^S`  from the upper component using the magnetic balance condition:

.. math::

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


3. use :math:`\vec{A} \cdot \vec{p} = \vec{p} \cdot \vec{A} - (\vec{p} \cdot \vec{A})`

   and remember that in Coulomb gauge we have

.. math::

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

4. for a constant and homogeneous magnetic field :math:`\vec{B} = \vec{\nabla}\times\vec{A}`, we may write :math:`\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
==============================

* :cite:`Dyall2007`, Chapter 4.

* :cite:`Reiher2009`, Chapter 5.