• Introduce Stern-Gerlach experiments through simulation
• Using statistical analysis to calculate probabilities
• Investigate the effect of making successive quantum measurements
• Gathering evidence for a model of two state spin systems
• Introduce and practice using Dirac notation to describe spin states

1. Run Stern-Gerlach simulations
2. Perform a statistical analysis to find probabilities (with confidence intervals)
3. Model spins systems

None

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

We usually prepare students with some discussion of magnetic moments and the results you would classically expect from a Stern-Gerlach experiment. This includes some discussion of the Stern-Gerlach apparatus.

• The main purposes of this experiment is to get students comfortable using the software. Encourage students to play with the different settings and run different numbers of experiments.
• Students may be confused by the Dirac notation $|+\rangle$ and $|-\rangle$ (we typically do this experiment before extensively discussing this notation).

• Many students fail to understand statistics very well, and often fall victim to the “gambler's fallacy”. This activity is used to illustrate to students that the probability of a single measurement result is (in this case) 50/50 for each particle going through the apparatus. The distributions will not always be exactly even!
• Students should be encouraged to read the statistic appendix. The should come away with the idea that the uncertainty scales with $\frac{1}{\sqrt{N}}$. One problem with this early experiment is that students get lazy (the results are 0,1, or 1/2).

• The purpose of this experiment is to illustrate that quantum measurements are repeatable, and to prepare for the quantum “spookiness” of Experiment 4.
• Students should come to realize that by sending a particle through a Stern-Gerlach device, the particle can be prepared in a particular state to be used in subsequent measurements.
• The instruction introduce students to the Dirac notation for probabilities as $|\langle out|in \rangle|^2$. Again, we do not do too much introduction to Dirac notation, so this is presented as a statement - we wait to discuss projections afterwards. The notation is introduced to help students practice with the notation and help them to fill in the data table in Experiment 4.

• The purpose of this experiment is to illustrate the spookiness of quantum experiment and to collect evidence upon which to build our model of two state systems (specifically, that measurements in the x, y, and z directions are not orthogonal). These data will be used to determine how to write $|\pm\rangle_x$ and $|\pm\rangle_y$
• Students should be specifically directed to fill out the table with probabilities not ratios of counts (# of particles out)/(# of particles in)

By the end of the lab, students should:

1. be familiar with the phenomena of making Stern-Gerlach measurements, what happens when you make several measurements of a single particle, and how to prepare a particle's state using a Stern-Gerlach experiment.
2. know how to calculate probabilities of measurement outcomes from the number of particles measured at each possible output.
3. understand how to express confidence in their calculated probabilities using statistical tools.

One point of difficulty in this lab is distinguishing between a superposition and a mixture of states. This is a very productive conversation to have with the students.

We use the results of these experiments to develop a model of two state quantum systems in Hilbert space from a phenomenological perspective.

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:

• Probabilities in the z-direction for a Spin-$\frac{1}{2}$ System
• Probabilities for Different Spin-$\frac{1}{2}$ Stern Gerlach Analyzers
• Dice Rolling Lab