Pauli Spin Matrices Questions

1. Show that the Pauli spin matrices satisfy the following: $$$$ $\begin{matrix}&\sigma_x^2=\sigma_y^2=\sigma_z^2=1\\&\sigma_x\sigma_y=-\sigma_y\sigma_x=i\sigma_z\\&\sigma_y\sigma_z=-\sigma_z\sigma_y=i\sigma_x\\&\sigma_z\sigma_x=-\sigma_x\sigma_z=i\sigma_y\\\end{matrix}$

2. Write down Pauli's spin matrices. Express $J_x, J_y, J_z$ in the terms of Pauli's spin matrices. [Mains 2014]

3. Write down the eigen value and spin state of an electron for the spin operator S. $$$$ (i) Show that the spin states of electron are orthogonal to each other. $$$$ (ii) State the spin angular momentum commutation relations $$$$ (iii) Give the explicit forms of all the Pauli spin matrices. [Mains 2001]

4. Show that the Pauli Spin Matrices obey the following relation: $$$$ i) $Tr\left(\sigma_x\right)=Tr\left(\sigma_y\right)$ $$$$ ii) $\det\left(\sigma_y\right)=\det\left(\sigma_z\right)$ $$$$ iii) The eigenvalues of $\sigma_z$ and $\sigma_x$ are the same. $$$$ iv. Write down the y-component of the spin angular momentum matrix corresponding to an antineutrino. [Mains 2009]

5. Show that the Pauli matrices anti-commute.

6. Write down any two properties of Pauli spin matrices, after defining them through suitable expression. [Mains 2001]