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## Probabilities for Different Spin-1 Stern Gerlach Analyzers: Instructor's Guide

This activity can also be part of a larger integrated laboratory. See the Spins Lab 3 activity page.

### Main Ideas

- Fourth postulate of quantum mechanics
- Projection
- Writing states in different bases

### Students' Task

*Estimated Time: 15 minutes*

### Prerequisite Knowledge

- Experience with the first four postulates of quantum mechanics.
- Previous experience with spin-$\frac{1}{2}$ systems is useful.
- Thorough understanding of bra-ket notation is essential.
- Awareness of how a wave state collapses after making a measurement.

### Props/Equipment

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

### Activity: Introduction

Before performing this activity, students should already have experience finding Probabilities for Different Spin-$\frac{1}{2}$ Stern Gerlach Analyzers. Since students have likely not worked with the spin-1 case yet, introduce the system by telling students that the spin-1 system is more challenging than the spin-$\frac{1}{2}$ system because each Stern-Gerlach device has three exit ports. Introduce the class to the proper state notation for the z-basis (that is, $\vert 1 \rangle$, $\vert 0 \rangle$, and $\vert -1 \rangle$ ). If you wish, this is also a good time to introduce the spin operators for the spin-1 system if operators have already been discussed. Let the students take the data and fill out the table on the activity handout.

### Activity: Student Conversations

**Theory is independent of experiment:**this is not a verification lab, so it is helpful to do the experiment first, but make it clear that the theoretical values do not depend on the experimental values they just obtained.- $\mathbf{\vert\bra{out}\ket{in}\vert^2}$: Some students have difficulty reading right to left and identifying which is the in-state and which is the out-state.
- $\mathbf{\vert a\vert^2=aa^*}$: When doing the square of the norm, a lot of students still think of it as a magnitude and try to use the Pythagorean Theorem instead of thinking of the square of the norm as a complex number times its conjugate.
- $\mathbf{{}_y\bra{-1}\ket{1}_y}$: some students try to write both in terms of the z-basis, without realizing that they can do this directly.

### Activity: Wrap-up

Bring the class back together ask the students about any results that they were not expecting. Be sure to note how the probabilities for receiving the states $\vert 1 \rangle_{x}$, $\vert 0 \rangle_{x}$, and $\vert -1 \rangle_{x}$ from the input state $\vert 1 \rangle$ or $\vert -1 \rangle$ are not split into perfect thirds (same for receiving any y states). Also discuss how for the initial state $\vert 0 \rangle$, the probability for receiving $\vert 1 \rangle_{x}$ or $\vert -1 \rangle_{x}$ appears to be one-half from the experiment and that the probability for receiving $\vert 0 \rangle_{x}$ is zero.

These probability results will certainly have an impact on what the representations for $\vert 1 \rangle_{x}$, $\vert 0 \rangle_{x}$, $\vert -1 \rangle_{x}$, $\vert 1 \rangle_{y}$, $\vert 0 \rangle_{y}$, and $\vert -1 \rangle_{y}$ will look like in the z-basis. Having the students find these states in the z-basis makes for an excellent homework exercise.