Just sharing my QM clicker questions… feel free to use. [Edit: I’ve added some commentary.]

**Day 1: Magnetic Moments and Introduction to QM**

Question 1.1: Classical Stern-Gerlach thought Experiment. I

I used this question before a mini-lecture on magnetic moments, the magnetic potential energy associated with dipoles in a magnetic field, and then derived the magnetic force from the gradient of the potential.

In hindsight, I wish I began with (or also discussed), clicker questions about charged particles moving through magnetic fields (because students are more familiar with this). On a HW problem, many students did not clearly understand that this magnetic force was the result of a neutral atom having a magnetic moment in a magnetic field gradient, rather than a charged particle experiencing a magnetic force due to its motion through a magnetic field.

Question 1.2:

After deriving the relation ship between, magnetic moment and force, I wanted them to think about how only the z-component matters for the deflection. After this, we talked explicitly about the SG experiment, and how the result is surprising because we only see two possible values, rather than a continuous distribution of values.

After the Stern-Gerlach experiment discussion, we returned to question of “Why do particles or atoms even have magnetic moment? After deriving an expression that relates angular momentum and magnetic moment, I asked the following question

1.3: Classical Magnetic Moments

I wish I had spent more time (or pushed to the HW), discussion about why with the silver atoms we are measuring the intrinsic magnetic moment of the electron.

**Day 2: More SG Experiments.**

Question 2.1:

This question was the warm-up, serving to remind them about the SG experiment, and also challenging them on their interpretations of the results.

Question C should be revised to say, “straight up or down”. The goal here is to make sure that students know that the device only measures the z-component. Answers D is an good “classical” interpretation for them to be at this point.This question gets returned to at the end of the day, after we discuss other SG experiments.

Question 2.2: Students were supposed to have read through these experiments. Instead of lecture through them, we did clicker questions. We did lots of “how are you making sense of this?” Lots of “classical” interpretations naturally arise, such as “randomizing”, “filtering”, … these aren’t bad ideas. Later we’ll want to push them toward QM ideas about incompatible measurements, uncertainty relationships, etc, but now is not the time. I need them to “sense-make” with the results.

After each experiment, I introduce the formalism of dirac notation and how we “encode” states and the experimental results using bra, kets, inner products, etc.

Next time, I want to introduce the “frequency vs angular momentum” graphs now rather than later. And informally “notice” things about those distributions.

Question 2.3:

After introducing the formalism, I want to give students a chance to practice.

**Day 3 Clicker Questions:**

Question Set 3.1: Sense-making about Off-Axis Measurements

These questions are after a HW problem where students looked at QM spin states that are prepared “off-axis”. Good warm-up for the day.

Question Set 3.2: “Seeing” Uncertainty and Expectation Value

In previous week, we didn’t talk in class a lot about expectation value and uncertainty. Students had calculated them for the HW, but I wanted to help students understand what they were even calculating. These ranking tasks were great for the job.

The rest of the day was spent doing example problems and introducing rotation operators.

**Day 4: Reviewing Trouble Spots and Representations of Operators**

Question Set 4.1: Overall Phases vs. Relative Phases

Students struggled on a HW assignment about relative phases, and it was clear we needed to discuss it more. So we spent the beginning of the day hashing it out.

The second of these three questions is pretty challenging.

Question Set 4.2: Rotation Operators and Phases

One kind of clicker question I try to use is “getting started”, “or checking for understanding” in derivations I do in class. I try not to lecture too much, but certain derivations are useful to go over in class. My way of engaging students is ask questions along the way, that force them to think about what we are doing.

These questions are doable if you understand what the operators are doing, what the notation means, and can visualize rotations in a right-handed coordinate system.

Question Set 4.3: Representations in Other Bases

The day ended with a review of representing bra/kets in column and row vectors. An upcoming HW problem asked students to do similar work, so I wanted to make sure we discussed it. Prior to this we had only explicitly represented states in the Sz-basis.

We spent the rest of the day introducing operator representations.

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