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