# Guessing the Form of Spin Along x in Terms of z

## The Prompt

“Write down what you think $\vert + \rangle_{x}$ is in the $z$-orientation basis.”

## Context

This SWBQ has multiple purposes. We want students to be able to write a mathematical statement based on their experimental observations from SPINS Lab 1. However, they are not yet ready to do so precisely. Having this be open-ended and ill-defined pushes students to see if they can write a simple statement that makes sense based on their observations. Though this SWBQ is intended to be included in a lecture between activities in the Stern-Gerlach sequence, it can be reasonably incorporated in lectures after students have been exposed to both Dirac notation and the standard $z$-oriented basis.

## Wrap Up

Since students know half of the particles prepared as $\vert+ \rangle$ end up as $\vert+ \rangle_{x}$, and the other half as $\vert- \rangle_{x}$, they may write: $| +\rangle = \frac{1}{2}\vert+ \rangle_{x} + \frac{1}{2} \vert- \rangle_{x}$, which is quite a reasonable first guess. Furthermore, students will have different ways of writing their answer down, leading to a possible discussion of different notations, or they may realize that there are multiple possible solutions that could make sense. If students struggle to write something down, challenge them to come up with at least two possible ways to write $\vert+ \rangle _{x}$ that are consistent with their observations. Knowing that there are multiple answers helps generate multiple answers and free up their thinking.

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