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## Analyzing a Spin-1 Interferometer: Instructor's Guide

This activity can also be part of a larger integrated laboratory. See the Spins Lab 3 activity page.

### Main Ideas

- Probabilities
- Projection
- Postulates 1-5 of quantum mechanics

### Students' Task

*Estimated Time: 30 minutes*

### Prerequisite Knowledge

- Experience with postulates 1-5 of quantum mechanics is important.
- Familiarity with the expected results of of the spin-one Stern-Gerlach experimental results is useful.
- Previous experience with projection operators in the spin-$\frac{1}{2}$ system is valuable.

### Props/Equipment

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

### Activity: Introduction

If students have previously Analyzed a spin-$\frac{1}{2}$ Interferometer, little introduction is needed for this activity. Remind students that this interferometer is more complex because there are three outbound ports that will result in more possible combinations of states into the third Stern-Gerlach device. Have the students turn the state of the oven to “RANDOM”; if they forget to do this, the data will be useless and they must start over. Then, ask the students to begin filling out the table in the handout.

Before filling out the theoretical columns in the handout, it is recommended that students have experience with The Projection Operator & Wave State Collapsing. Make sure that students know how to write the $\vert 1 \rangle$, $\vert 0 \rangle$, and $\vert -1 \rangle$ states for x and y-orientation in terms of the z-basis. If students have already seen the projection operator in the context of a spin-$\frac{1}{2}$ system, little more introduction should be necessary.

### Activity: Student Conversations

### Activity: Wrap-up

After students have taken their data using the Spins software, bring the class back together to see if they have any questions. Just as in the case with the spin-$\frac{1}{2}$ interferometer, the results from this experiment are certainly not classically expected. Ask the class as a whole how different the resulting states are for each combination of two beams in the x-oriented Stern-Gerlach device, and how would they predict these results mathematically? The use of projection operators is the key to answering this question; remind students that projections of an initial state $\vert \psi \rangle$ onto a new state $\vert \psi ' \rangle$ is performed using projection operators like so:

$$\vert\psi ' \rangle=\frac {P_{\vert\psi ' \rangle} \vert\psi \rangle}{\sqrt{\langle \psi|P_{\vert\psi ' \rangle}\vert\psi \rangle}} \; \; , $$

where the projection operator is written as

$$P_{\vert\psi ' \rangle}=\vert \psi ' \ \rangle \langle \psi ' \vert \; \; .$$

Performing these projections can become time-consuming; having students compute these projections to fill out the theoretical section of the handout makes for a great take-home exercise or homework assignment.

### Extensions

This activity is the second activity contained in SPINS Lab 3 . This activity is designed to be presented in the midst of lectures, but if you have a 2 hour block of time dedicated to labs, the above lab is a better choice. The preceding activity, also contained in SPINS Lab 3 is Finding Unknown States Leaving the Oven in a Spin-1 System