The students must determine if the instructor's claim is true, and why or why not, by using the density matrix to calculate the probability of finding a sample coming out of the Stern-Gerlach oven in either the spin up or spin down state with x, y, or z orientation.

Estimated Time: 15 minutes

1. Students should be able to compute the outer product of the general quantum state vector $$\vert\psi \rangle\,=\, \left(\begin{array}{c} sin\left(\frac{\theta}{2}\right)\\ cos\left(\frac{\theta}{2}\right)e^{i\phi}\\ \end{array}\right) \; \; . $$

and find that the outer product of any quantum state vector with itself is Hermitian.

1. When students take the determinant of the 2×2 matrix yielded from the outer product they should find that in any case, the determinant of the outer product of $\vert\psi \rangle$ with itself is zero and the trace of the outer product is one.

• Tabletop Whiteboard with markers

This activity is best done following the “Practice with Outer Product Matrix Properties” activity.

Prompt the students with an introduction similar to the following: The trace of any density matrix must be 1 and the determinant must be 0. But, if we relaxed these limitations, we could likely get other useful terms. My claim to all of you is that the oven in the Stern-Gerlach Experiment is a superposition of the density matrices $\vert + \rangle \langle + \vert$ and $\vert - \rangle \langle - \vert$. If the states $\vert + \rangle$ and $\vert - \rangle$ are not treated differently in Stern-Gerlach experiment, and we need the total trace to be equal to one, the only choice of coefficients left will give the expression

$$Oven \; \dot = \; \frac{1}{2}\vert + \rangle \langle + \vert \: + \: \frac{1}{2}\vert - \rangle \langle - \vert \; \; . $$

Have the students test this claim by using the operator to calculate different probabilities.