Quantum Fundamentals 2022
Consider the arbitrary Pauli matrix \(\sigma_n=\hat n\cdot\vec
\sigma\) where \(\hat n\) is the unit vector pointing in an arbitrary
direction.

Find the eigenvalues and normalized eigenvectors for \(\sigma_n\).
The answer is:
\[
\begin{pmatrix}
\cos\frac{\theta}{2}e^{-i\phi/2}\\{} \sin\frac{\theta}{2}e^{i\phi/2}\\
\end{pmatrix}
\begin{pmatrix}
-\sin\frac{\theta}{2}e^{-i\phi/2}\\{} \cos\frac{\theta}{2}e^{i\phi/2}\\
\end{pmatrix}
\]
It is not sufficient to show that this answer is correct by plugging
into the eigenvalue equation. Rather, you should do all the steps
of finding the eigenvalues and eigenvectors as if you don't know the
answer. Hint: \(\sin\theta=\sqrt{1-\cos^2\theta}\).

Show that the eigenvectors from part (a) above are orthogonal.

Simplify your results from part (a) above by considering the three separate special cases: \(\hat n=\hat\imath\), \(\hat
n=\hat\jmath\), \(\hat n=\hat k\). In this way, find the eigenvectors and eigenvalues of \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\).