Students are reminded that in some measurement sequences there was loss of information about the initial state while in other sequences the initial state remained known, this motivates the ideas that some operators relate differently from than others
Students are reminded of the mathematical definition of commutivity and it is applied to matrices, with the reminder that they generally do not commute
The property that commuting operators have the same eigenvalues is derived
This idea is tied back to sequences of measurements and the idea that for some properties you can know the eigenvalues associated with multiple operators - they are simultaneous observables, but not for other operators
Do an example calculation showing that Sz and Sx do not commute and have students find the commutation relations for the other spin matrices
Relate commutation to the uncertainty principle and tie this back to the idea of the projection operator and the representation of the spin vector S
Use Sx, Sy and Sz to write the spin vector S in terms of unit vectors in spherical coordinates