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## Analyzing a Spin-$\frac{1}{2}$ Interferometer: Instructor's Guide

This activity can also be part of a larger integrated laboratory. See the Spins Lab 2 activity page.

### Main Ideas

• Postulates 1-4 of Quantum Mechanics
• Probabilities
• Projection operators
• Projections with bra-ket notation

Estimated Time: 1 hour

### Prerequisite Knowledge

• Postulates 1-4 of quantum mechanics are necessary.
• General familiarity with the expected results of the Stern-Gerlach experiment.
• An introduction to projection operators and their use in bra-ket notation is important for the theoretical calculations.

### Activity: Introduction

The experimental portion of this activity needs little introduction. Before the students take data, however, make sure that the state preparation device for the Spins Program is set to the “RANDOM” state; the data will be useless if a group forgets to do this!

For the theoretical portion of the activity, it is recommended that students have previously been introduced to the projection operator(we recommend that students have previously seen the The Projection Operator & Wave State Collapse lecture). After seeing the classically unexpected results of the experimental data, students can now test that the projection operator will actually predict these results.

### Activity: Wrap-up

Experimental: Bring the class back together and discuss the results of this experiment. Ask students to explain what's happening in this experiment in words (perhaps in terms of a sock sorter or some other alternate representation). This is a good time to perform the How Making Measurements Affects Results in Quantum Systems lecture. Students should leave seeing that these results were not classically expected at all, and that a new formalism is needed to mathematically predict these results.

Theoretical: At this point, students should already know how to represent the $\vert + \rangle _{x}$ and $\vert - \rangle _{x}$ states in the z-basis. Re-write on the board that the projection of an initial state onto another second state is defined as

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

Now, since we now know that the projection operator is written as

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

Have the students confirm that this method of projection works by having all of the groups calculate one of the boxes in the theoretical section of the handout. Make sure the students use bra-ket notation for this calculation! Once the groups have finished, perform the projection in bra-ket notation on the board for the class to see.

The calculation that students will be less likely to know how to perform will be when both the + and - state in the x-orientation leave the x-oriented Stern-Gerlach device. Ask if anybody can think of how to represent this projection operator. It should be shown that the projection operator will be represented in bra-ket notation as

$$P_{\vert + \rangle _{x},\vert - \rangle_{x}}=\vert + \rangle_{x} {_{x} \langle + \vert} \, + \, \vert - \rangle _{x} {_{x} \langle - \vert} \; \; .$$

Be sure to note that performing the outer products for this matrix and computing the sum will return the identity matrix.

Show the class that using this projector will simply return the initial state $\vert + \rangle$ or $\vert - \rangle$. This result is expected because the projection operator $P_{\vert + \rangle _{x},\vert - \rangle_{x}}$ is just the identity matrix.

We can now see that our bra-ket representation for projection operators predicts what states will result from a particular measurement, and these state representations can be used to find the theoretical probabilities in the handout.

### Extensions

This activity is the second activity contained in SPINS Lab 2 . 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 2 is Finding Unknown States Leaving the Oven in a Spin-1/2 System

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