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## Probabilities for Different Stern Gerlach Analyzers: Instructor's Guide

### Main Ideas

- The fourth postulate of quantum mechanics
- Probabilities
- Stern-Gerlach devices
- Quantization of intrinsic angular momentum

### Students' Task

*Estimated Time: 15 minutes*

Small groups of students must experimentally find the probability that a particle in an initial state will be measured as another state. That is, they must experimentally find

$$P_{out} \; = \; \vert\langle out \vert in \rangle\vert^{2}$$

where the $\vert in \rangle$ or $\vert out \rangle$ states can be $\vert + \rangle$, $\vert - \rangle$, $\vert + \rangle_{x}$, $\vert - \rangle_{x}$, $\vert + \rangle_{y}$, or $\vert - \rangle_{y}$.

### Prerequisite Knowledge

- Background knowledge on how the Stern-Gerlach device physically separates particles.
- The first four postulates of quantum mechanics.

### Props/Equipment

- Computers with the Spins OSP software
- A handout for each student

### Activity: Introduction

First, have the entire class as a whole choose the two Stern-Gerlach devices to both have z-orientation. Have them connect the $\vert + \rangle$ port to the second z-oriented analyzer, and ask them what happens. Students should find that if a $\vert + \rangle$ state particle is passed through a second z-oriented analyzer, it will still come out in the $\vert + \rangle$ state. Provide students with the handout for the activity and have them measure the probabilities for all combinations of x,y, and z analyzers experimentally. Emphasize to the students that the probability they are calculating is *only* the probability of a particle leaving the first analyzer hitting detector out of the second analyzer, not the probability that a particle leaving the oven hits the detector. Also make sure they fill out the worksheet with probabilities, not the mathematical expressions for the probabilities.

### Activity: Student Conversations

### Activity: Wrap-up

Field any questions students have about the results from the probabilities. Were any of the results unexpected for them?

### Extensions

This activity is the second part of SPINS Lab 1. It is designed to follow Probabilities in the z-direction for a Spin-$\frac{1}{2}$ System and precede the Dice Rolling Lab.

The full lab is designed to be completed in a two hour lab block, while these individual activities are designed to be integrated into a normal lecture. We prefer to use the integrated activities, rather than the lab, but both are effective methods.