spins_unit_operators_and_measurements.ppt, pages 13-21, 24, 40-41

Lecture (60 minutes)

  • In Winter 2010, the previous activity was broken up into smaller chunks - this segment starts with having students find normalized vectors that are orthogonal to the given vectors. They then find the outer products for combinations of the given vectors, the transformation each outer product causes, and the determinant of each outer product. This helps them think about the outer product in a general form and practice the mathematics before putting specific meaning to it for the spin system
  • Students are then introduced to the projection operator and a geometric interpretation of it
  • At this point in the lecture, during Winter 2010, I brought students back to the postulates and what understanding we had built thus far. Students struggle with quantum measurement and I wanted to give them a strong grounding in what they were solid with before moving on.
  • Students are asked to write Sz, Sx and Sy as a linear combination of projection operators, then they make sense of their finding by considering what transformation the spin operator corresponds to
  • Students are prompted to think about operators as linear transformations and to apply this to understanding Sx, Sy and Sz, as well as the projection operators
  • This is tied into the fact that Hermetian Operators always have real eigenvalues and orthogonal eigenfunctions that form a complete basis set
  • The projection operator is then shown to relate to to the ket the atom is in after the measurement, but the state found with the projection operator isn't normalized so the formula from the quantum postulates is introduced
  • It has been found that students have a difficult time understanding the relationship between operators, measurements, and eigenvalue equations - the following activity was devised to assist with this and shown to decrease student difficulty
  1. Students count off by 9 forming groups with 2-3 students each
  2. Each group does 2 calculations, for example group 1 calculates Sx|+> and Sx|→x, while group 2 calculates Sx |→ and |+>y
  3. The work is distributed in such a way that each group does two calculations and each calculation is done by two different groups
  4. The groups are asked to write their results on the board in the form of a table with Sx, Sy and Sz forming the row headings and each of our kets (in each basis) forming the column headings
  5. Students then double check the table entries (since each calculation was done by more than one group)
  6. It may be necessary to ask students to simplify some of the table entries so they are a single element in whatever basis is makes that happen
  7. Once the table is complete, students are asked to look for patterns in the table - the eigenvalue terms are typically quickly pointed out by students - so they can be asked 'what does it mean to be an eigenvalue' - they should realize this only occurs when Si acts on the i-kets
  8. Students can then be asked 'what does it mean to be in a basis' - because they can observe that if an operator acts on a ket in a given basis it returns a multiple of a different ket in that basis
  9. Other observations can be made from the table, but these two are critical
  • Students are introduced to spin projection in a general direction using the unit vector n


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