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===== The $S$ and $S^2$ vectors (Lecture, 15 minutes) =====
{{courses:lecture:splec:spins_unit_operators_and_measurements.ppt|}} Pages 42-45

  * 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)


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