• The fourth postulate of quantum mechanics
• Probabilities
• Stern-Gerlach devices
• Quantization of intrinsic angular momentum

Estimated Time: 15 minutes

Small groups of students must experimentally find the probability that a particle in an initial state will be measured as another state. That is, they must experimentally find

$$P_{out} \; = \; \vert\langle out \vert in \rangle\vert^{2}$$

where the $\vert in \rangle$ or $\vert out \rangle$ states can be $\vert + \rangle$, $\vert - \rangle$, $\vert + \rangle_{x}$, $\vert - \rangle_{x}$, $\vert + \rangle_{y}$, or $\vert - \rangle_{y}$.

• Background knowledge on how the Stern-Gerlach device physically separates particles.
• The first four postulates of quantum mechanics.

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

First, have the entire class as a whole choose the two Stern-Gerlach devices to both have z-orientation. Have them connect the $\vert + \rangle$ port to the second z-oriented analyzer, and ask them what happens. Students should find that if a $\vert + \rangle$ state particle is passed through a second z-oriented analyzer, it will still come out in the $\vert + \rangle$ state. Provide students with the handout for the activity and have them measure the probabilities for all combinations of x,y, and z analyzers experimentally. Emphasize to the students that the probability they are calculating is only the probability of a particle leaving the first analyzer hitting detector out of the second analyzer, not the probability that a particle leaving the oven hits the detector. Also make sure they fill out the worksheet with probabilities, not the mathematical expressions for the probabilities.

Field any questions students have about the results from the probabilities. Were any of the results unexpected for them?

This activity is the second part of SPINS Lab 1. It is designed to follow Probabilities in the z-direction for a Spin-$\frac{1}{2}$ System and precede the Dice Rolling Lab.

The full lab is designed to be completed in a two hour lab block, while these individual activities are designed to be integrated into a normal lecture. We prefer to use the integrated activities, rather than the lab, but both are effective methods.