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

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

Estimated Time: 15 minutes

• 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.

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

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.

• 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.

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.