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Determining how a Uniform $\vec{B}$ Field Changes a Particle's State: Instructor's Guide

Main Ideas

• Magnetic moment
• Unknown states
• General quantum state vector

Students' Task

Estimated Time: 30 minutes

Students are placed into small groups and asked to find how changing the strength of a magnet affects an incoming quantum state. In this case, the incoming state is $\vert \psi \rangle$.

Prerequisite Knowledge

• Experience with the first four postulates of quantum mechanics.
• Knowledge of how to represent the general quantum state vector is useful.
• Understanding of the magnetic dipole moment of particles and how the magnetic dipole moment is related to the intrinsic angluar momentum.

Activity: Introduction

Before performing this activity, students should be shown the general quantum state vectors. This can be accomplished by starting with the general spin operator

$$S_{\hat{n}}=\vec{S} \cdot \hat{n} \; \; ,$$

where

$$\vec{S}\, = \, S_{x}\hat{i} + S_{y}\hat{j} + S_{z}\hat{k} \; \;$$

and

$$\hat{n}\, = \, sin\,\theta\, cos\,\phi\,\hat{i} + sin\,\theta\, sin\, \phi\,\hat{j} + cos\,\theta\, \hat{k} \; \; .$$

Since the eigenvalues of $S_{\hat{n}}$ are $\pm \frac{hbar}{2}$, the operator and the eigenvalues can be used to find the eigenvectors of the generalized spin operator. These eigenvectors are

$$\vert + \rangle_{\hat{n}}=cos\, \frac{\theta}{2}\vert + \rangle\, + \, sin\, \frac{\theta}{2}e^{i\phi} \vert - \rangle \; \; ,$$

and

$$\vert - \rangle_{\hat{n}}=sin\, \frac{\theta}{2}\vert + \rangle\, - \, cos\, \frac{\theta}{2}e^{i\phi} \vert - \rangle \; \; .$$

Now, have the small groups run the experiment detailed in the handout. Students should notice that when the magnet placed between the z-oriented devices is turned on, the probabilities measured on the second device begin to change. Students must pick a single magnetic field strength, find the new unknown state produced by exposure to the magnet, and discover how each unit of strength added to the magnet affects the state of the incoming particles. Make sure each group uses a different strength of magnetic field.

Activity: Wrap-up

Bring the class back together and collect some of the results found. Ask the class what it appears is happening to the quantum state fired through the magnet. If you wish, find the unknown state resulting from a magnet of strength “01” on the program display and show that each digit added onto the magnetic field's strength will increase the phase angle $\phi$ by 10 degrees.

Extensions

This activity is a part of SPINS Lab 4 . 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 likely a better choice.

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