In QM, by far the most common thing that students said for what was helpful was clicker questions. Below are upper division students’ responses to why clicker questions are helpful.
Students’ responses touched upon important themes about learning such as active processing (i.e. force me to think), self-assessment (test our understanding), meta-cognition (i.e., see what I know), and proximal feedback (wrong understanding gets corrected in the moment).
“It gives us some time to process what we’ve learned rather than just write it down and hope we understand our notes later.”
” I feel like it helps sink things in we’ve talked about.”
“They usually contain questions that really test our understading of the material. They are also a good tool for us to identify our weak areas and what we need to work on.”
“It is nice to see what I know and/or can deduce from what we are going over in class.
“They tend to be helpful in checking my understanding conceptually of what is going on.”
“Knowing that they’re going to happen forces me to come to class prepared to think, which helps me focus more intently while you are lecturing. Also, having to explain our reasoning has forced me to think deeply”
“Are very helpful because it forces us to take a breather and actually process what you just told us/ what we just wrote down. “
“They are very helpful in that they give me time to stop and make sure I truly understand what I’ve been watching be presented. The class leaves me feeling challenged and engaged. “
“I find it helpful since it forces a question/problem that makes you actually think and apply the material instead of just going into a note taking mode. It also helps if your understanding is wrong, because at that moment it’s corrected. “
Other helpful things
In addition to clicker questions students’ mentioned other things that were helpful, that touch upon important educational themes such as “making connections to prior knowledge”, “relational vs instrumental knowledge”, and “multiple representations”, “multiple encounters”.
Students brought up things like:
– Making connections / analogies to things we already know.
“In class, if we relate a new topic to an old one with which we are more familiar (like between the Schrodinger Equation and N2L) it helps my understanding of what we are doing.”
– Focusing on what is physically / conceptually going on (not just mathematically)
“I am greatly appreciative that you do your best to give us conceptual understandings for the material. Having that conceptual knowledge to start from, it gives me a basis to understand the rest of the material. “
– Emphasizing visualization.
“I find the visual representations very useful for learning (even though its not easy to do for quantum mechanics).”
– Articulating Learning (Goals)
“It does not always happen, but some days you tell us to write down something new that we learned. I have gotten into the habit of doing this at the end of each class, as well as a question or two that I may still have. This has proven to be beneficial.”
“I like that we start the class with a quick summary of the day’s material that will be covered. ”
Feedback is for Me and for Students
Part of why I ask for feedback is to actually get feedback from students’, and to make adjustments. This semester with my QM students I am making adjustments to homework grading, how I articulate to students’ the reasons why we are doing a particular example problem, and how often and when I pause to give them time to write down thoughts from a discussion, among other things.
But asking for feedback, gets them to articulate things about their own learning. It is always the case that students’ (collectively) responses touch upon sooooo many important things about teaching/learning. When people ask me how to get students to “buy-in” to active learning, my responses is always, “give them an opportunity to articulate things about learning” and help them to see how your class to organized to do those very things.
This week we started look at time evolution in quantum mechanics. Since this is a spins-first approach, that we means we first take a look at the spin-1/2 particle in a uniform magnetic field.
Here are clicker questions from our day looking at this:
- This first question was asked after setting up the problem, but before getting into QM. Take in a moment to make sure we understand the classical perspective.
2. This question was asked after having worked through the time evolution, and have an expression for the state as a function of time. Good time to re-emphasize the significance of an overall phase change and the idea of energy eigenstates as being stationary.
3. This question was asked after changing the initial state to be a superposition of energy eigenstates. In talking through this question, we both “cranked out the calculations”, but spent a fair amount of time sense-making about precession in the x-y plane.
This last question, brought us back to physically sense making and to take in the big picture, rather than sense-making about a particular mathematical expression or result.
Teaching QM has certainly made me think back on my own experience being in upper division courses. Specifically, I’ve been wondering recently what were the conditions that contributed to a handful of lectures I watched as an undergrad staying in my brain ever since. Like so much so that I clearly remember that feeling of being in that class, I can see writing on the board, the writing in my notebook, and forever have been able to recreate these particular derivations with little trouble.
Two of them are
- The method for deriving the results of gaussian integrals (from a Calc III class)
- How to derive the Green’s functions for damped SHM, and in the process applying the Residue Theorem as an integration technique (from a Classical mechanics course).
I think part of my answer for why I remember them is that I was so intrigued by both of the methods at the time, that I pondered them over and over and over, and recreated them again and again. The initial condition was certainly whatever it was that made me so intrigued at that moment, but the process of crystalizing that knowledge was not the lecture itself, but the acts of non-stop thinking about them over a long period of time.
This also reminders me in high school, I was obsessed with calculating the moment of inertia of three dimensional objects, under various geometries and mass distributions. I loved setting up the integrals and working them out, especially in spherical coordinates. I would work these out again and again again every time I was sitting in some other boring HS class. This was also something that was just “lectured” to me, but again my learning was immensely active and sustained over a long period of time.
It makes me think that a lot of the reason I did fairly well with math and physics throughout high school and college was that thinking about physics and math was not really school work, but an obsession. Re-deriving interesting things or playing with the math was like doodling, something I did constantly, all the time, anytime a piece of paper was at hand.
But this was also one of the reasons why I was not a “great” student in college. I didn’t do my HW all the time, because I’d be spending my time “doodling” what was interesting to me, rather than what the teacher wanted me to do at that time. While there was significant overlap between my doodles and the course work, this overlap was not so great as to make me a top student. [I am just remember know how much I loved solving normal modes problems].
Anyway, that’s been on my mind.
This question was good one from today. After introducing the raising and lowering operators, we had some “practice” reasoning about which matrix elements would be zero (and thus saving the work of cranking out each calculation).
I don’t spend a crazy amount of time deriving everything little thing in QM class, but I do work through some of our text’s derivations at the board. I try to chose those that I feel either give us an opportunity to grapple with important ideas, emphasize certain techniques, or those that help us keep a coherent narrative through our course of study.
I’ve been trying to use clicker questions at key points during the derivations to keep students engaged and hold them accountable for reasoning. Today we were working through ideas about how commuting operators have simultaneous eigenvalues, and how this leads to fact that QM states must exist with both definite values of angular momentum (magnitude) and one component of angular momentum (but not more than one).
Here are two examples of places where we paused to think about what we can and can’t conclude.
I’m not sure if doing these is optimal, but so far it’s not been awful either.
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.
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.
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.
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.
In class this semester, we used bar magnets oscillating on springs through solenoids to induce alternating currents. In getting ready for lab, I noticed that, in certain configurations, the current induced was sinusoidal (as desired), but in other configurations they induced current was periodic but not very sinusoidal. This got me wanting to model the phenomena a little more closely.
By Faraday’s law, the induced potential (and current) is related to change in magnetic flux through the coil. A big reason for the non-sinusoidal regime it turns out is due to how a magnetic dipole field (on axis ) goes like B ~ 1/z^3, where z is the distance from the center of the dipole. Anyway, since the area isn’t changing, the change in magnetic flux will be driven by the change in magnetic field in the coil as the magnet moves, which goes like 1/z^3.
Using the Chain Rule, we can write, dB/dt = dB/dz* dz/dt. This shows that the change in flux is driven by two factors, the gradient of the B-field and the velocity of the bar magnet.
Assuming the motion of the bar magnet is roughly damped harmonic oscillation, we can write z(t) = A exp(-rt) cos(wt) + C. A is the amplitude of the oscillation, C is the offset equilibrium position of the magnet above the coil.
Making additional assumptions that the damping is small, the velocity will be roughly, dz/dt = wA exp(-rt)sin(wt) [The other term in the product rule will be small]
Anyway, plugging in our expression for z into our expression for the dB/dt, we get V ~ exp(-rt) sin(wt) / (A(cos(wt) +C)^4
To make sense of this, we have to think about large amplitudes vs. small amplitudes (relative to the offset C).
When A is really small relative to offset C, the denominator is roughly constant, so the change in magnetic field is dominated by the numerator and should be roughly sinusoidal. This means that the change in magnetic flux should be in sync with the velocity. High velocity means rapidly changing flux. When the amplitude is significant compared to the the offset, the denominator comes close to zero, and so has a big impact. That is, the magnetic flux is strongly affected by gradient of the magnetic field.
Here is a graph of the induced current vs time (black) and speed vs time (green), based on the models above.
You can see that the model shows that as the amplitude decays, the induce current and speed of the bar magnet get in sync (and both are sinusoidal). When the amplitude is large, the peak current does not correspond with peak velocity.
Actual data for the induced current looks something like this:
There are a lot of assumptions and simplifications of this model, but it gets a lot of the behavior right. Lots more to dig into, but don’t have the time to get distracted right now!