Show that the $S^2$ operator can be written in terms of the pauli spin matrices
Ask the students to find $S^2$ from their knowledge of Sx, Sy, and Sz
Discuss what operators $S^2$ can commute with and therefore what we can know about whether we can simultaneously know $S^2$ and Sx, Sy or Sz
Have students find the eigenvectors and eigenvalues of $S^2$
Have students find the expectation value of $S^2$ and deduce the 'length' of the spin vector S
Show the graphical representation of the spin vector, S based on the idea that its total length is root(3)/2 but we only ever measure 1/2 via any projection onto the x, y or z axis
We spent time trying to reason about this physical model and representation and what aspects of it were useful and what were not. A student pointed out that this can not be directly analogous to a classical effect because the x, y and z axis choices are arbitrary so in principle we should be able to rotate our system to measure the full value of root(3)/2 but that is never possible
We generalized this to other spin systems by defining s and m (the spin quantum number and projection)