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## Expectation Value and Uncertainty: Instructor's Guide

### Main Ideas

Classical worksheet:

- Review classical probability, average, and standard deviation.
- Use theoretical probabilities to define
*weighted average*.

Quantum worksheet:

- Practice finding expectation values and uncertainties for quantum systems.
- Observe special cases for uncertainty.
- Contrast quantum mechanical and experimental uncertainty

### Students' Task

*Estimated Time: *

- Classical: 30 minutes
- Quantum: 30 minutes

### Prerequisite Knowledge

Equations for average and standard deviation from a data set.

### Props/Equipment

- Tabletop Whiteboard with markers
- Ordinary six-sided dice (two per group)
- Computer with Excel
- Handouts for each student

### Activity: Introduction

Students start with a quick dice-rolling experiment designed to give practice taking data, and finding average and standard deviation, using a system that is likely to be unfamiliar and that does not have equally distributed probabilities. This exercise is intended to provide students a productive analogy for both the equations and the concepts underlying quantum mechanical expectation values and uncertainties.

### Activity: Student Conversations

Do you expect every other groups to get the same average and standard deviation? Does your answer change if each group does a larger and larger number of experiments?

If the average and standard deviation for two systems are the same, does that necessarily mean the systems have the same underlying probabilities?

Are the probabilities of each outcome of the classical dice experiment equal? How does your answer affect how you calculate averages and standard deviations?

What does it mean if a system has a standard deviation or an uncertainty of zero? What kind of quantum state does this correspond to? (Challenge: if the uncertainty in *S _{z}* is zero for a certain state, what do you expect for the uncertainty in a different quantity like

*S*? How would you calculate this uncertainty?)

_{x}
What is the largest possible value for the uncertainty in *S _{z}* for a spin-1/2 system?

### Activity: Wrap-up

Classical: Have each group report their average and standard deviation for their first experiment (6 trials). This is a good place to discuss probabilistic variation and to ask students how they expect the results to differ for larger numbers of trials. It is also crucial that students see a “weighted average” method used to calculate averages *and* standard deviations using theoretical probabilities before moving on to the quantum version of the worksheet.

Quantum: If each group completed a different initial quantum state, have groups report out one by one. These reports will bring up excellent opportunities for whole-class discussions of the student conversations questions above.