: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.