:orphan: .. _TCCM_pp_exercises: In the text full units are used for clarity, but in practice one can employ SI-based atomic units, setting :math:`\hbar=m_{e}=e=4\pi\varepsilon_{0}=1`. Pauli spin matrices =================== The Pauli spin matrices .. math:: \sigma_{x}=\left[\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right];\quad\sigma_{y}=\left[\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right];\quad\sigma_{z}=\left[\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right] are representation matrices of of the operator :math:`2\mathbf{s}` in the basis :math:`\left\{ \left|\alpha\right\rangle ,\left|\beta\right\rangle \right\}` of spin-1/2 functions (*spin-up* and *spin-down*, respectively), that is :math:`2\mathbf{s}\rightarrow\hbar\boldsymbol{\sigma}`. These spin functions obey the following relations .. math:: \begin{array}{lclclcl} \hat{s}_{z}\left|\alpha\right\rangle & = & +\frac{1}{2}\hbar\left|\alpha\right\rangle & & \hat{s}_{z}\left|\beta\right\rangle & = & -\frac{1}{2}\hbar\left|\beta\right\rangle \\ \hat{s}_{+}\left|\alpha\right\rangle & = & 0 & & \hat{s}_{+}\left|\beta\right\rangle & = & \hbar\left|\alpha\right\rangle \\ \hat{s}_{-}\left|\alpha\right\rangle & = & \hbar\left|\beta\right\rangle & & \hat{s}_{-}\left|\beta\right\rangle & = & 0 \end{array} where appears the ladder operators :math:`s_{\pm}=s_{x}\pm is_{y}`. The spin-1/2 functions are furthermore orthonormal. Let us see how the Pauli :math:`\sigma_{z}` matrix is generated: * Element (1,1): :math:`\left\langle \alpha\left|2s_{z}\right|\alpha\right\rangle /\hbar=\left\langle \alpha\left|\right.\alpha\right\rangle =1` * Element (1,2): :math:`\left\langle \alpha\left|2s_{z}\right|\beta\right\rangle /\hbar=\left\langle \alpha\left|\right.\beta\right\rangle =0` * Element (2,1): :math:`\left\langle \beta\left|2s_{z}\right|\alpha\right\rangle \hbar=\left\langle \beta\left|\right.\alpha\right\rangle =0` * Element (2,2): :math:`\left\langle \beta\left|2s_{z}\right|\beta\right\rangle /\hbar=-\left\langle \beta\left|\right.\beta\right\rangle =-1` 1. In the same manner construct representation matrices for the ladder operators :math:`s_{+}` and :math:`s_{-}` and from this :math:`\sigma_{x}` and :math:`\sigma_{y}`. 2. Demonstrate, for general vector operators :math:`\boldsymbol{A}` and :math:`\boldsymbol{B}`, the Dirac identity .. math :: \left(\boldsymbol{\sigma}\cdot\boldsymbol{A}\right)\left(\boldsymbol{\sigma} \cdot\boldsymbol{B}\right)=\boldsymbol{A}\cdot\boldsymbol{B}+i\boldsymbol{\sigma} \cdot\left(\boldsymbol{A}\times\boldsymbol{B}\right) 3. Use the Dirac identity to calculate * :math:`\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)` * :math:`\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{\pi}}\right)\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{\pi}}\right)`, where :math:`\hat{\boldsymbol{\pi}}` is mechanical momentum :math:`\hat{\boldsymbol{\pi}}=\hat{\boldsymbol{p}}+e\boldsymbol{A}` and :math:`\boldsymbol{A}` is a vector potential. Show how this expression is simplified in Coulomb gauge: :math:`\boldsymbol{\nabla}\cdot\boldsymbol{A}=0`. * Show that :math:`\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)V_{eN}\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)=\boldsymbol{p}V_{eN}\cdot\boldsymbol{p}+\hbar\left(\frac{Ze^{2}}{2\pi\varepsilon_{0}r^{3}}\right)\hat{\boldsymbol{s}}\cdot\hat{\boldsymbol{\ell}}`, where :math:`V_{eN}` is the electron-nucleus interaction :math:`V_{en}=-\frac{Ze^{2}}{4\pi\varepsilon_{0}r}` in an atom