eigenvectorsQuantum Fundamentals 2023
The operator \(\hat{S}_x\) for spin-1 may be written as:
defined by:
\[\hat{S}_x=\frac{\hbar}{\sqrt{2}}
\begin{pmatrix}
0&1&0\\ 1&0&1 \\ 0&1&0 \\
\end{pmatrix}
\]

Find the eigenvalues and eigenvectors of this matrix. Write the eigenvectors as both matrices and kets.

Confirm that the eigenstates you found give probabilities that match your expectation from the Spins simulation for spin-1 particles.

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).

Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.