Periodic Systems: Spring-2026
HW 1: Due Day 3

1. Cyclic Translation of 3 Distinguishable Beads Consider 3 (distinguishable beads) arranged clockwise on a ring.

   1. Construct a matrix operator that translates (moves) each bead clockwise one “bead slot” along the ring.

   2. Write down an eigenvalue equation for this operator.

   3. Find eigenvalues and eigenvectors of this operator.
2. Adding Operators with a Common Set of Eigenvectors

   1. Consider linear operators \(\hat{A}\) and \(\hat{B}\) share a complete set of eigenvectors but with different eigenvalues.

      \begin{align*} \hat{A}\left|{ab}\right\rangle &= a\left|{ab}\right\rangle \\ \hat{B}\left|{ab}\right\rangle &= b\left|{ab}\right\rangle \end{align*}

      Now consider an operator \(\hat{C}\) that is a linear combination of \(\hat{A}\) and \(\hat{B}\).

      Show that \(\left|{ab}\right\rangle \) is also an eigenvector of \(\hat{C}\) and find it's eigenvalues.

3. $\hat{S^\uparrow}$ and $\hat{S^\downarrow}$ Consider the operators: \begin{align*} \hat{S^\uparrow} &\doteq \begin{bmatrix} 0&1&0 \\ 0&0&1 \\ 1&0&0\end{bmatrix}\\[12pt] \hat{S^\downarrow} &=\begin{bmatrix} 0&0&1\\1&0&0\\0&1&0 \end{bmatrix} \end{align*}

   1. Show that \(\hat{S^\uparrow}\) and \(\hat{S^\downarrow}\) are inverses of each other.

   2. Show that the eigenvectors of \(\hat{S^\downarrow}\) are also eigenvectors of \(\hat{S^\uparrow}\).

   3. Find the three eigenvalues of the operator (\(\hat{S^\uparrow}+\hat{S^\downarrow}\)) and show that at least one of them checks out.