Part 1 – Creating an Electric Field
Tutorial - Spin and Angular Momentum
Goals this week:
1. Developing intuition about measurement, bra-ket notation, and finite dimensional systems. (LG Math/physics connection, interpretation, building on earlier work/coherence of the class)
2. Measurement of non-commuting operators, in particular Lx or Sx. (LG Symmetry)
Tutorial Summary: Building on Singh's article of student difficulties, we wanted students to practice interpreting coefficients in an expansion, to build their intuition about the meaning of the radial and spherical harmonic wave functions. We introduce questions where there is degeneracy (so e.g. they need to think about collapsing a wave function into a degenerate subspace), and introduce combined spatial/spin wave functions for the first time in the class here. We address sequential measurements and incompatible observables in several contexts. The Tutorial ends with a somewhat unusual "nature of measurement" page, to ground the formal notation and ideas of this Tutorial in something a little more concrete.
[pic] Reflection after administering the tutorial
Specific comments and difficulties observed:
This Tutorial is very long, some groups spent more than 2 hours on it. Something needs to be trimmed, although the students were excited by it and participated actively.
Page 1:
The time dependence - most know the answer, but I found it hard to get them to explain how they knew. Nobody invokes our "postulate of quantum mechanics" (that Prob(measuring e-value λn)= |||2) without it being dragged out of them
The last part, what does "Ĥ ψ" mean, did not trip up too many people, perhaps we have addressed this issue enough in past Tutorials!
P2: The probabilities and outcomes of L2 and Lx likewise posed not a single problem that I saw. The time dependence question for Lz started generating SOME puzzling, although they almost all got the right answer, but again could not even begin to justify it. It might be worth adding another question, about a measurement (like a particular range of radius?) where time dependence DOES enter...?
P3: The normalization gave a few minutes of pause to many groups, but all that I saw got it on their own.
There was some debate about what I mean when I ask if the STATE of the particle is well-defined (I'm thinking, yes, I can write it down!) But that was ok - the instructors all had agreed that we just wanted them to realize that you can have a well-defined wave function even if you don't have well-defined outcomes of a variety of measurements. In the end, this didn't seem to be a barrier, just took some discussing. A few students were a little ticked about the ambiguity (and some correctly argued that there's still an overall phase we don't know, so it ISN'T completely well-defined after all, so what do we mean?...)
In part F, there was some discussion about what “commutating operators” really means (and many students no longer remember what L2 commutes with), one group was really hung up on this for a long time - does measuring L2 "muck up" Lz? Can it? In this case, you DO change the expectation value of Lz by measuring L2 first. So this bothered them... It was a good discussion, I thought, but needed some outside guidance.
The last part is a *great* question: what can you get if you measure Lx on the Y(2,1) state? Every single group was stumped, and argued. Some came up with good arguments, most ended up in consensus with the right answer, but some of those were not coming up with *clear* arguments. There is LOTS of confusion about what "m" means when you change to an Lx basis. One student insisted that since m=0 or 1 in the initial state, it must still be 0 or 1 when you measure Lx (interpreting "m" as now becoming the x-component eigenvalue).
P. 4 II, Spin: AFTER that long discussion about Lx in the previous problem, we still had huge discussion about Sx in this problem! So, I was happy that we had followed the one up with the other. Here, we asked for the probabilities, and the "symmetry argument" was popular with some students, while others wanted to follow a more mathematical path (finding eigenvalues of the Sx matrix which had just been given at the end of class). There is huge confusion about the idea of an x-eigenstate being written in the z-basis, I could see many students getting it, then losing it, over and over.
Part C Says you measure Sx and get -ħ/2, what is the "state of the electron". There were a LOT of groups who argued that it should be 50/50, thus (1/√2, 1/√2), and here I really had to push the groups which had been using symmetry before to now get mathematical. (I pushed by asking "ok, what if I had gotten + ħ /2, what would be the state? These groups all said "same thing". So I asked if it made sense to them that two states with DIFFERENT Sx eigenvalues could be the SAME STATE.) But this was a hard part for many. In the end, all groups WANTED to go through the "matrix math".
P.5 Mixed states. This was their first exposure to product spin-space wave functions, and my intent had been that this Tutorial would have made this easy and natural, and indeed it seemed to be. I saw no real issues on this part. So, perhaps we might call it a success - although it took an hour and a half, this page was easy :-)
P.6. Part IV. This part requires more discussion, even for faculty. If students went through it without an instructor nearby, they blitzed through, thought it was simple, they didn't understand what I was getting at. So we might need to think harder about the "setup". The first question is not so much about "Quantum mechanics" as"nature of science". I WANTED answer c, and ONLY c, to be "obvious". Many students wanted to claim "the last e- has no definite spin state until you measure it". Many who said this, said so because after 1000 measurements, we still cannot be SURE. So you can't say it's definite. (I asked them if physicists are sure of anything, then? I asked them if we can claim the sun will rise tomorrow? I asked them if they would be willing to publish a paper which says that this electron has no spin state - as compared to, would they be willing to publish a paper which says that this electron is very likely in a spin up state?" ) For some groups who still argued with me that "we can't know", I proposed given them coins from a sack. Flip them - you get heads. And again, and again. How many flips will it take before they are willing to BET good money that the NEXT one will be heads again? Will they NEVER make that bet? Why not? After 20? After 80? After 800? Remember, they randomized them before pulling them out, there's no "conspiracy" at work here. (Our math major said she would simply NEVER believe she could say anything about the next coin, period. I emphasized that the problem asks what a "practical physicist" would say)
We had to go through this argument with every single group, before they got my idea - I'm asking what statement (if any) you feel you can DEFEND in an experimental paper.
If you can think up a plausible "preparation mechanism" which is consistent with the experimental details, but inconsistent with the statement, then you cannot defend it, and should not circle it. So on this basis, my answers are
1-c, 2-b, 3-c, 4-d (see below.) 5-d
(For scenario 4, I can think of cases where the correct answer is c. If for instance the preparer randomly rotates their S-G before sending it along, and records the outcome. But in an EPR scenario then "a" might very well be true. Ditto for case 5. The point is not to find answers that COULD be true, but ones which you can DEFEND on the basis of this experiment alone)
When we got to the last ones, we accepted "it has a definite spin state, we just don't know what it is". But for most groups (who wanted to stay), we talked about EPR and entangled states. I tried to convince them that in fact, although everything we've learned SO FAR in this course would allow you to cheerfully defend "c", that I can prepare states where the SINGLE electron you receive does not have a definite spin state. This was largely just for fun - it's the end of the term, I wanted to talk about EPR and spooky action at a distance. This got great attention, they loved it, though I'm not sure what they will take away.
Summary/big issues to look for todays:
Idea of time dependence has "gone" for many
The "postulate of QM" we invoked in class (that the probability of a measurement n is given by | |2) is not at their fingers, they do not how to figure out the probability of measuring an eigenvalue.
Questions about Lx or Sx threw most for a loop. They struggled with "symmetry arguments". Students do not understand in what ways "z" is special, and in what ways it is NOT. Students do not understand how you would DECIDE just about anything about x-components, quantitatively. Some of these questions could be answered easily by a quick calculation, but many are refusing to do that, they want to "intuit" the answer.
The idea that a spinor in z-basis can represent an eigenfunction of Sx really blew away a LOT of students. Students are still not comfortable with the idea that you can "know the state" without "knowing the z-component" for instance.
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