2) [20 pts] Spin angular momentum operators.
a) Demonstrate that the 2 x 2 spin matrices $\vec{S} = (\hbar/2)\vec{\sigma}$ with $\vec{\sigma}$ the three Pauli matrices,
which operate on spin-1/2 particles, satisfy the correct angular momentum commutation
relations,
$[\hat{S}_x, \hat{S}_y] = i\hbar\hat{S}_z$, $[\hat{S}_y, \hat{S}_z] = i\hbar\hat{S}_x$, $[\hat{S}_z, \hat{S}_x] = i\hbar\hat{S}_y$.
(4)
b) Consider the spin-1 case. There are three eigenvectors; let them be called
$|s = 1, m_s = 1\rangle = (1,0,0)$, $|s = 1, m_s = 0\rangle = (0,1,0)$, $|s = 1, m_s = -1\rangle = (0,0,1)$, (5)
and construct the matrices for $\hat{S}_x$, $\hat{S}_y$ and $\hat{S}_z$ for this spin. Hint: as with the spin-1/2 case,
$\hat{S}_z$ can be deduced from the eigenvectors, and $\hat{S}_x$ and $\hat{S}_y$ can be deduced from the action of
$\hat{S}_+$ and $\hat{S}_-$.
$\hat{S}_\pm|s, m_s\rangle = \hbar\sqrt{s(s+1) - m_s(m_s \pm 1)}|s, m_s \pm 1\rangle$
(6)
c) Verify that the matrices you constructed in part b) also satisfy the correct angular mo-
mentum commutation relations.
