## Stern-Gerlach Simulations

These labs all use a java program called SPINS that simulates Stern-Gerlach experiments with spin-$\frac{1}{2}$ and spin-$1$ particles. The software runs on all platforms and can be downloaded from the SPINS home page.

There are two possibilities on how to order this sequence. The first option, integrated activities, can be fit into a typical class schedule for upper-division students who do not have separate lab times. This sequence of short activities can be interspersed with lectures and allow for activity wrap ups which can motivate the next topic of discussion. The other sequence uses four lab activities which would work well in a class with two hour lab blocks set aside from the lecture portion of the course.

### Activities: Integrated Activities

The following list of activities is pulled from the labs listed below. They are shorter individual activities that allow for wrap-ups after each step, rather than at the end. This order is a better option for those that don’t have a block of lab hours set aside.

• Probabilities in the $z$-direction for a Spin-$\frac{1}{2}$ System (Estimated time: 10 minutes): Students are asked to fire a beam of 10, 100, 1000, and 10000 particles into a single $z$-oriented Stern-Gerlach device multiple times. The groups must convince themselves probabilistically and observationally that the probability of a particle being spin-up in the $z$-direction is $\frac{1}{2}$ and the same for spin-down.
• Probabilities for Different Spin-$\frac{1}{2}$ Stern-Gerlach Analyzers (Estimated time: 15 minutes): Students are placed into small groups and asked to experimentally measure the probability that a $|+\rangle$ particle with $z$-orientation will be in the $|+\rangle$ or $|−\rangle$ state using two $z$-oriented Stern-Gerlach devices. Students will then complete the handout provided by performing similar probability experiments with all possible combinations of two $x$, $y$, or $z$-oriented Stern-Gerlach devices.
• Dice Rolling Lab (Estimated time: 25 minutes): Students are placed into small groups and given a die. The group's task is to determine if the die is fair or “loaded”. Fairness of the die will be experimentally determined by calculating the probability of rolling each number on the die. Groups will have the opportunity to use multiple representations to present experimental data.
• Finding Unknown States Leaving the Oven in a Spin-$\frac{1}{2}$ System (Estimated time: 30 minutes): Students are placed into small groups and asked to find the probability that an unknown quantum wave state $|\psi\rangle$ will leave an $x$,$y$, or $z$-oriented Stern-Gerlach analyzer in either the spin-up or spin-down state. The probabilities can then be used to find the coefficients of the unknown wave state. This process is repeated for all four unknown wave states on the Spins program, with each unknown wave state more challenging to find than the last.
• Analyzing a Spin-$\frac{1}{2}$ Interferometer (Estimated time: 1 hour): Students are placed into small groups and asked to find several probability results from a Spin-$\frac{1}{2}$ interferometer using the Spins software. Then, the small groups will use projection operators to determine the mathematical representations of the states exiting from the final Stern-Gerlach device. These quantum states will also be used to complete the theoretical probabilities on the activity worksheet.
• Finding Unknown States Leaving the Oven in a Spin-$1$ System (Estimated time: 20 minutes): Students are placed into small groups and asked to calculate the probability that an unknown state $|\psi\rangle$ will be found in the new state $|\epsilon\rangle$, where $\epsilon$ can be spin $1$, $0$, or -$1$ in the $x$, $y$, or $z$-direction. Using these probabilities, students will find what each unknown state $|\psi\rangle$ is in the $z$-basis.
• Analyzing a Spin-$1$ Interferometer (Estimated time: 30 minutes): Students are placed into small groups and must find the final probabilities that result from different beam configurations of a Spin-$1$ interferometer. The small groups will also practice using projection operators in a spin-$1$ system to find the states in a spin-$1$ system resulting from the combination of several Stern-Gerlach output beams.
• Determining how a $\vec{B}$ Field Changes a Spin-$\frac{1}{2}$ Particle's State (Estimated time: 30 minutes): For this activity, students are placed into small groups and asked to use the SPINS program to find how a magnet in the program affects the state of an incoming particle. Students will calculate the probability that the $|\psi\rangle$ made by the magnet will be measured in the $|\phi\rangle$ state, where $\phi$ can be any one of spin-up or spin-down states with $x$, $y$, or $z$-orientation.

### Activities: Lab Activities

• Spins Lab 1 (Estimated time: 2 hours): This lab is an introduction to successive Stern-Gerlach spin $\frac{1}{2}$ measurements. The randomness of measurements is demonstrated and students use statistical analysis to deduce probabilities from measurements. The results demonstrate the orthonormality of the $S_{z}$ basis kets and are used to deduce the $S_{x}$ and $S_{y}$ basis kets in terms of the $S_{z}$ kets.
• Spins Lab 2 (Estimated time: 2 hours): Students take data and then use the results to deduce the quantum state vectors that led to those results. Students build an interferometer and then use the projection postulate to calculate the expected results in interferometer. Finally students perform which-path experiments to see the perturbative effect of measurement.
• Spins Lab 4 (Estimated time: 2 hours): Students study spin precession in a uniform magnetic field. Students are asked to design an experiment to figure out how the magnet affects spins. They must take data, analyze, develop hypothesis, test hypothesis, and determine scale for magnetic field parameter displayed on screen.

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