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## Spins Lab 3: Instructor's Guide

### Main Ideas

• Modeling Spin-1 systems in a similar way as was done for Spin-1/2 systems
• Determining state vectors from Stern-Gerlach simulations
• Using the projection postulate to analyze the Spin-1 version of an interferometer

1. Run Stern-Gerlach simulation with Spin-1 particles
2. Use simulated experimental results to determine unknown prepared states.
3. Use the projection operator to analyze a quantum interferometer.

### Prerequisites

• Spins Labs 1 & 2

### Activity: Introduction

• Brief lecture about Spin-1 systems as an example of a three state system.
• Students need to be given the Spin-1 state vectors $|1\rangle_x$, $|0\rangle_x$, $|-1\rangle_x$, $|1\rangle_y$, $|0\rangle_y$, and $|-1\rangle_y$ written in terms of the basis kets $|1\rangle$, $|0\rangle$, and $|-1\rangle$.

### Activity: Student Conversations

##### Section 1
• Students “measure” the probabilities of successive spin measurements and compare their results to theoretical predictions made by taking the norm squared of the appropriate projections.
• This is similar to Experiment 4 of Lab 1
##### Section 2
• Students are asked to determine the state vectors of four unknown spin-1 particles (as was done in Section 1 of Lab 2).
• Students can be guided to use the amplitude and phase notation for complex numbers since this makes it easy to associate states with the general state $|1\rangle_n = \frac{1+\cos{\theta}}{2}\,e^{-i\phi}|1\rangle + \frac{\sin{\theta}}{2}|0\rangle + \frac{1-\cos{\theta}}{2}\,e^{i\phi}|-1\rangle$ (etc) and then align the detector to verify their results.
• Some students have trouble doing algebra with complex numbers, and will have difficulty finding the relative phases. It helps to adopt the convention that the coefficient of $|1\rangle$ is real and positive.
• Note that Unknown #3 does not correspond to a eigenstate in a rotated system.
##### Section 3
• This is the Spin-1 interferometer. Students should use the projection postulate to calculate theoretical probabilities for comparison with the experiment. This experiment is similar to Experiment 3 of Lab 2 (Spin-1/2 case).

### Extensions

This lab has been broken up into parts in order to be better integrated into a classroom setting. If you currently have a 2 hour lab block set aside, this lab may be the best choice. If not, we have found that the smaller activities often work better.

This lab contains the following small activities:

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