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

• Postulate 4 of quantum mechanics
• Overall vs. relative phases
• Solving for the coefficients of a quantum state vector

Estimated Time: 20 minutes (Without wrap-up)

• Experience in the first four postulates of quantum mechanics is essential.
• The ability to use complex numbers in polar and rectangular form.
• Know the definition of “overall phase” and “relative phase” for quantum states.

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

Before performing this activity, it is advised that students have experience with Finding Unknown States Leaving the Oven in a Spin-$\frac{1}{2}$ System . Place the students into small groups. Working with the spin-1 system in the Spins software, have the groups change the setting of the oven from “RANDOM” to “UNKNOWN #1”. Have the class fill out the handout by calculating the probability that the unknown state will be in the $\vert 1 \rangle_{x,y,z}$, $\vert 0 \rangle_{x,y,z}$, or $\vert -1 \rangle_{x,y,z}$ states.

• What would the form of a correct answer look like?: asking this question to help students who are struggling in getting started. It is helpful to write it as $\ket{\psi_i}=\_\_\ket{1}+\_\_\ket{0}+\_\_\ket{-1}$
  • Name the think you don't know: encourage them to choose variables, since otherwise they can't appear in equations.
  • What kind of a beast is it?: many students want to just write $\ket{\psi_i}=a\ket{1}+b\ket{0}+c\ket{-1}$, without taking explicitly into account that the coefficients can be complex. It is helpful to guide students to use the amplitude and phase notation for complex numbers since this makes the algebra easier and also makes it easier to associate states with the general state $\ket{1}_n = \frac{1+\cos{\theta}}{2}\,e^{-i\phi}\ket{1} + \frac{\sin{\theta}}{2}\ket{0} + \frac{1-\cos{\theta}}{2}\,e^{i\phi}\ket{-1}$ (etc) and then align the detector to verify their results.
  • Some students have trouble doing algebra with complex numbers, and will have difficulty finding the relative phases. It helps to adopt the convention that the coefficient of $|1\rangle$ is real and positive.
• How many pieces of information do you have?: sometimes students are able to find the real parts of the coefficients (using the probabilities in the z-direction), but are unsure of how to find the phase(s). This question can help them begin to think outside of the z-direction.
• Everything in the same basis: Some students recognize that the coefficients of $\ket{\psi}$ in the z-basis are different than the coefficient of $\ket{\psi}$ in the y-basis (or the x-basis). They then assume that when they project $\ket{\psi}$ onto a vector in one of the other bases, the coefficients change. This is particularly a challenge for students who tend to write $\ket{\psi}$ in terms of the basis they are projecting onto, as opposed to writing that vector in the same basis as $\ket{\psi}$.
• $\mathbf{\vert a\vert^2=aa^*}$: 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.
• Think strategically: encourage students to think about which projections will give them the things they want with the least fuss.
• Note: Unknown #3 does not correspond to a eigenstate in a rotated system.

After students have taken their data, bring the groups back together to discuss how these probabilities will be used to find the unknown quantum state in terms of the z-basis. Remind students that the unknown quantum state vector can be represented by

$$\vert \phi \rangle\, = \, a \vert 1 \rangle \, + \, be^{i\phi}\vert 0 \rangle \, + \, ce^{i\delta} \vert -1 \rangle \; \; . $$

Now, solving for an unknown state in the spin-1 system is more challenging because there is now the $ce^{i\delta}$ term, but the same strategy used for solving the spin-$\frac{1}{2}$ unknown state can be utilized. Taking the inner product of the unknown quantum state with one of the basis state vectors $\vert 1 \rangle_{x,y,z}$, $\vert 0 \rangle_{x,y,z}$, or $\vert -1 \rangle_{x,y,z}$ and then plugging in the inner product into the fourth postulate of quantum mechanics for the $\langle out \vert in \rangle$ term, where the fourth postulate looks like

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

will lead us to information about the real coefficients and the phases angles.

This exercise makes a good homework problem for students; they already have the tools to perform the calculations, and a chance for them to compute the quantum states independently is useful for their independent growth.

This activity is the first activity contained in SPINS Lab 3 . This activity is designed to be presented in the midst of lectures, but if you have a 2 hour block of time dedicated to labs, the above lab is a better choice. The following activity, also contained in SPINS Lab 3 is Analyzing a Spin-1 Interferometer