Quantum Fundamentals: Winter-2025 Eigenvectors Practice: Due Day 12 Tu 2/25 Math Bits
Eigenvectors of Pauli Matrices
Find the eigenvalues and normalized eigenvectors of the Pauli
matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) (see the Spins Reference Sheet posted on the course website).
Eigen Practice
Find the eigenvectors and eigenvalues of the matrices from
the Linear Transformations small group activity from Tuesday's
class. Keep working until you are fluent. Make up some \(2\times 2\) and \(3\times 3\) matrices of your
own if you need more practice.
Diagonalization
Let
\[|\alpha\rangle \doteq \frac{1}{\sqrt{2}}
\begin{pmatrix}
1\\ 1
\end{pmatrix}
\qquad \rm{and} \qquad
|\beta\rangle \doteq \frac{1}{\sqrt{2}}
\begin{pmatrix}
1\\ -1
\end{pmatrix}\]
Show that \(\left|{\alpha}\right\rangle \) and \(\left|{\beta}\right\rangle \) are orthonormal.
(If a pair of vectors is orthonormal, that suggests that
they might make a good basis.)
Consider the matrix
\[C\doteq
\begin{pmatrix}
3 & 1 \\ 1 & 3
\end{pmatrix}
\]
Show that the vectors
\(|\alpha\rangle\) and
\(|\beta\rangle\) are
eigenvectors of C and find the eigenvalues.
(Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)
A operator is always represented by a diagonal matrix if it is written in terms of
the basis of its own eigenvectors. What does this mean? Find the matrix elements for a
new matrix \(E\) that
corresponds to \(C\) expanded in the basis of its eigenvectors, i.e. calculate \(\langle\alpha|C|\alpha\rangle\),
\(\langle\alpha|C|\beta\rangle\), \(\langle\beta|C|\alpha\rangle\) and
\(\langle\beta|C|\beta\rangle\)
and arrange them into a sensible matrix \(E\). Explain why you arranged the matrix
elements in the order that you did.
Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?
Eigen Spin Challenge
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 \(\sigma_z\).