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===== Expectation Value and Standard Deviation(Lecture, 20 minutes) =====
{{courses:lecture:splec:spins_unit_operators_and_measurements.ppt|}} Pages 26-33

  * It is of value to start the segment on expectation value and standard deviation with a solid grounding in the ideas of measurement - these ideas are confusing for students and somewhat unsettling, so having them contrast what is consistent with their classical understanding gives more concreteness to the ideas
  * We first discussed both an optical and a mechanical method of classical measurement and showed that both cases are limited to disturbing the system at the small scale
  * This is a good opportunity to discuss how measurements of sub-atomic particles are made and show students tracking chambers
  * The argument is made that all measurement yields perturbation to the system, but unlike classical systems, taking more measurements can not reduce our uncertainty to as small as we want - so we need to learn the appropriate statistics for describing spin systems
  * The expectation value is introduced and mathematically defined.  The equation is shown to be analogous to one you would use for finding the average value obtained from rolling dice
  * Students are asked to find the expectation value of Sz for a system prepared in |+> (z-basis)
  * When they find the value hbar/2, they are asked to make sense of this - because this is the only value you will ever get from this measurement it should be the expectation value
  * Students are asked to do this for |+>x, and find 0, again making sense of it from the fact that half the time they will get -hbar/2 and half the time +hbar/2, averaging to zero
  * Standard deviation is defined - this equation is tricky for the students - we spent a considerable amount of time breaking down each term
  * We make sense of the equation by expanding it out and considering each term
  * Students use the equation to find the standard deviation for the same two examples: Sz for a |+> and for a |+>x state, and sense is made from their findings
  * To further emphasize this, the expectation value and standard deviation are shown graphically for some examples

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