Quantum Fundamentals 2022
The Pauli spin matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are
defined by:
\[\sigma_x=
\begin{pmatrix}
0&1\\ 1&0\\
\end{pmatrix}
\hspace{2em}
\sigma_y=
\begin{pmatrix}
0&-i\\ i&0\\
\end{pmatrix}
\hspace{2em}
\sigma_z=
\begin{pmatrix}
1&0\\ 0&-1\\
\end{pmatrix}
\]
These matrices are related to angular momentum in
quantum mechanics.
By drawing pictures, convince yourself that the arbitrary unit
vector \(\hat n\) can be written as:
\[\hat n=\sin\theta\cos\phi\, \hat x +\sin\theta\sin\phi\,\hat y+\cos\theta\,\hat z\]
where \(\theta\) and \(\phi\) are the parameters used to describe
spherical coordinates.
Find the entries of the matrix \(\hat n\cdot\vec \sigma\) where the
“matrix-valued-vector” \(\vec \sigma\) is given in terms of the
Pauli spin matrices by
\[\vec\sigma=\sigma_x\, \hat x + \sigma_y\, \hat y+\sigma_z\, \hat z\]
and \(\hat n\) is given in part (a) above.