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## Spins Lab 2: Instructor's Guide

### Main Ideas

- using Stern-Gerlach data to determine the state vector of an unknown prepared state.
- the importance of relative phase in the terms of the quantum state vector; the arbitrary overall phase of quantum state vectors
- uniform magnetic fields cause spin precession
- investigate the quantum interferometer and state coherence

### Students' Task

To run Stern-Gerlach experiments and use the results to determine the unknown prepared states.

### Prerequisites

- Spins Lab 1 (for familiarity with the program and introduction to two state systems)

### Props/Equipment

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

### Activity: Introduction

- Lecture about writing spin state vectors as a linear combination of basis states and there exists a relative phase between the two terms of the state vector.

### Activity: Student Conversations

##### Section 1

- In this experiment, students must takes data and use the results to determine the quantum state that led to those results.
- The first two cases are (trivially) easy. The third case become significantly more difficult computationally.
- 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 $ |+\rangle_n = \cos{\frac{\theta}{2}}|+\rangle+\sin{\frac{\theta}{2}}|+\rangle$ and then align the detector to verify their results.
- The phase angle $\phi$ for Unknown 3 & 4 is chosen to be greater than $\pi$ so that students must wrestle with the fact that inverse trig functions on their calculators do not tell the whole story.

##### Section 2

- Students continue to practice finding unknown states.
- This activity is meant to be a lead in to Lab 4 where they investigate what happens to the spin state when a particle passes through a uniform magnetic field.
- You can pool together the results from several groups of students to have a discussion about what the magnetic field does to the spin state, but this is also done in Lab 4. (The field rotates the phase by 10 degrees for each digit on the field icon)

##### Section 3

- Students simulate a quantum interferometer
- Students are asked to use the projection postulate:

$P|+\rangle = |+\rangle_{x} \,_{x}\langle+|+\rangle + |-\rangle_{x} \,_{x}\langle-|+\rangle = \,_{x}\langle+|+\rangle|+\rangle_{x} + \,_{x}\langle-|+\rangle|-\rangle_{x}$

- Students are often frustrated about doing a complicated calculation in an unfamiliar notation for such an simple case, but this practice will help for the more difficult spin-1 interferometer.

##### Section 4

- Students simulate a quantum interferometer with the watch feature turned on (so that the wavefunction collapses)
- This is akin to a Heisenberg microscope where information in obtained about which path the particle takes through a double slit interferometer. In that case, fringes are destroyed. Here, the coherence is destroyed and the probability of the last measurement in 50/50.

### Activity: Wrap-up

Little wrap-up is typically needed, though students need a lot of help doing the calculations for Experiment 1. The main take home messages are:

- States can be written as a linear combination of basis states, and the relative phases between terms in imporant.
- States can be determined from Stern-Gerlach experiments
- The projection operator can be used to determine the output of a Stern-Gerlach experiment.
- States maintain coherence until a measurement causes the collapse of the wavefunction.

### 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: