In the text full units are used for clarity, but in practice one can employ SI-based atomic units, setting $$\hbar=m_{e}=e=4\pi\varepsilon_{0}=1$$.

# Pauli spin matricesΒΆ

The Pauli spin matrices

$\begin{split}\sigma_{x}=\left[\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right];\quad\sigma_{y}=\left[\begin{array}{cc} 0 & -i\\ i & 0 \end{array}\right];\quad\sigma_{z}=\left[\begin{array}{cc} 1 & 0\\ 0 & -1 \end{array}\right]\end{split}$

are representation matrices of of the operator $$2\mathbf{s}$$ in the basis $$\left\{ \left|\alpha\right\rangle ,\left|\beta\right\rangle \right\}$$ of spin-1/2 functions (spin-up and spin-down, respectively), that is $$2\mathbf{s}\rightarrow\hbar\boldsymbol{\sigma}$$. These spin functions obey the following relations

$\begin{split}\begin{array}{lclclcl} \hat{s}_{z}\left|\alpha\right\rangle & = & +\frac{1}{2}\hbar\left|\alpha\right\rangle & & \hat{s}_{z}\left|\beta\right\rangle & = & -\frac{1}{2}\hbar\left|\beta\right\rangle \\ \hat{s}_{+}\left|\alpha\right\rangle & = & 0 & & \hat{s}_{+}\left|\beta\right\rangle & = & \hbar\left|\alpha\right\rangle \\ \hat{s}_{-}\left|\alpha\right\rangle & = & \hbar\left|\beta\right\rangle & & \hat{s}_{-}\left|\beta\right\rangle & = & 0 \end{array}\end{split}$

where appears the ladder operators $$s_{\pm}=s_{x}\pm is_{y}$$. The spin-1/2 functions are furthermore orthonormal. Let us see how the Pauli $$\sigma_{z}$$ matrix is generated:

• Element (1,1): $$\left\langle \alpha\left|2s_{z}\right|\alpha\right\rangle /\hbar=\left\langle \alpha\left|\right.\alpha\right\rangle =1$$

• Element (1,2): $$\left\langle \alpha\left|2s_{z}\right|\beta\right\rangle /\hbar=\left\langle \alpha\left|\right.\beta\right\rangle =0$$

• Element (2,1): $$\left\langle \beta\left|2s_{z}\right|\alpha\right\rangle \hbar=\left\langle \beta\left|\right.\alpha\right\rangle =0$$

• Element (2,2): $$\left\langle \beta\left|2s_{z}\right|\beta\right\rangle /\hbar=-\left\langle \beta\left|\right.\beta\right\rangle =-1$$

1. In the same manner construct representation matrices for the ladder operators $$s_{+}$$ and $$s_{-}$$ and from this $$\sigma_{x}$$ and $$\sigma_{y}$$.

2. Demonstrate, for general vector operators $$\boldsymbol{A}$$ and $$\boldsymbol{B}$$, the Dirac identity

$\left(\boldsymbol{\sigma}\cdot\boldsymbol{A}\right)\left(\boldsymbol{\sigma} \cdot\boldsymbol{B}\right)=\boldsymbol{A}\cdot\boldsymbol{B}+i\boldsymbol{\sigma} \cdot\left(\boldsymbol{A}\times\boldsymbol{B}\right)$
1. Use the Dirac identity to calculate

• $$\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)$$

• $$\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{\pi}}\right)\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{\pi}}\right)$$, where $$\hat{\boldsymbol{\pi}}$$ is mechanical momentum $$\hat{\boldsymbol{\pi}}=\hat{\boldsymbol{p}}+e\boldsymbol{A}$$ and $$\boldsymbol{A}$$ is a vector potential. Show how this expression is simplified in Coulomb gauge: $$\boldsymbol{\nabla}\cdot\boldsymbol{A}=0$$.

• Show that $$\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)V_{eN}\left(\boldsymbol{\sigma}\cdot\hat{\boldsymbol{p}}\right)=\boldsymbol{p}V_{eN}\cdot\boldsymbol{p}+\hbar\left(\frac{Ze^{2}}{2\pi\varepsilon_{0}r^{3}}\right)\hat{\boldsymbol{s}}\cdot\hat{\boldsymbol{\ell}}$$, where $$V_{eN}$$ is the electron-nucleus interaction $$V_{en}=-\frac{Ze^{2}}{4\pi\varepsilon_{0}r}$$ in an atom