Tuesday, May 23, 2017

How should undergraduate quantum theory be taught?

Some of my colleagues and I have started an interesting discussion about how to teach quantum theory to undergraduates. We have courses in the second, third, and fourth years. The three courses have independently evolved, depending on who teaches each. Some material gets repeated and other "important" topics get left out. One concern is that students seem to not "learn" what is in the curriculum for the previous year. The goal is to have a cohesive curriculum. This might be facilitated by using the same text for both the second and third-year courses.
This has stimulated me to raise some questions and give my tentative answers. I hope the post will stimulate lots of comments.

The problem that students don’t seem to learn what they should have in pre-requisite courses is true not just for quantum. I encounter second-year students who can’t do calculus and fourth-year (honours) students who can’t sketch a graph of a function or put numbers in a formula and get the correct answer with meaningful units. As I have argued before, basic skills are more important than detailed technical knowledge of specific subjects. Such skills include relating theory to experiment and making order of magnitude estimates.

Yet, given the following should we be surprised?
At UQ typical lecture attendance probably runs at 30-50 per cent for most courses. About five per cent watch the video. [University policy is that all lectures are automatically recorded]. The rest are going to watch it next week… Only about 25 per cent of the total enrolment in my second-year class are engaged enough to be using clickers in lectures. Exams are arguably relatively easy, similar to previous years, usually involve choosing questions/topics, and a mark of only 40-50 per cent is required to pass the course.
I do not think curriculum reform is going to solve this problem.

Having the same textbook for 2nd and 3rd year does have advantages. This is what we do for PHYS2020 Thermodynamics and PHYS3030 Statistical Mechanics. But, some second years do struggle with it... which is not necessarily a bad thing. The book is Introduction to Thermal Physics, by Schroeder.

Another question is what approach do you take for quantum: Schrodinger or Heisenberg, i.e. wave or matrix mechanics? The mathematics of the former is differential equations, that of the latter is linear algebra. Obviously, at some point you teach both, but what do you start with. It is interesting that the Feynman lectures really start with and develop the matrix approach, beating the two level system to death...
At what point do you solve the harmonic oscillator with creation and annihilation operators?
When do you introduce Dirac notation?

I would be hesitant about using Dirac notation throughout the second year course. I think this is too abstract for many of our current students. They also need to learn and master basic ideas/techniques about wave mechanics: particle in a box, hydrogen atom, atomic orbitals, … and connecting theory to experiment... and orders of magnitude estimates for quantum phenomena.

What might be a good text to use?

Twenty years ago (wow!) I taught second (?) year quantum at UNSW. The text I used is by Sara McMurry. It is very well written. I would still recommend it as it has a good mix of experiment and theory, old and new topics, wave and matrix mechanics….
It also had some nice computer simulations. But it is out of print, which really surprises and disappoints me.

Related to this there is a discussion on a Caltech blog about what topics should be in undergraduate courses on modern physics. Currently, most "modern" physics courses actually cover few discoveries beyond about 1930! Thus, what topics should be added? To do this one has to cut out some topics. People may find the discussion interesting (or frustrating…). I disagree with most of the discussion, even find it a little bizarre. Many of the comments seem to be from people pushing their own current research topic. For example, I know it is Caltech, but including density matrix renormalisation group (DMRG), does seem a little advanced and specialised...
There is no discussion of one of the great triumphs of "modern" physics, biophysics! I actually think every undergraduate should take a course in it.

What do you cut out?
I actually think the more the better, if the result is covering a few topics in a greater depth that develops skills, creates a greater understanding of foundations, that all leads to a greater love of the subject and a desire and ability to learn more.
In teaching fourth year condensed matter [roughly Ashcroft and Mermin] it is always a struggle to cut stuff out. Sometimes we don't even talk about semiconductor devices. This year I cut out transport theory and the Boltzmann equation so we could have more time for superconductivity. This is all debatable... But I hope that the students learned enough so that they if they need to they have the background they need to easily learn these topics.

A key issue that will divide people concerns the ultimate goal of a physics undergraduate education. Here are three extreme views.

A. It should prepare people to do a PhD with the instructor.
Thus all the background knowledge needed should be covered, including the relevant specialised and advanced topics.

B. It should prepare people to do a physics PhD (usually in theory) at one of the best institutions in the world.
Thus, everyone should have a curriculum like Caltech.

C. It should give a general education that students will enjoy and will develop skills and knowledge that may be helpful when they become high school teachers or software engineers.

What about Academic Freedom?
This means different things to different people. In some ways I think that the teacher should have a lot of freedom to add and subtract topics, to pitch the course at the level they want, and to choose the text. I don't think department chairs or colleagues should be telling them what they "have" to do. Obviously, teachers need to listen to others and take their views into account, particularly if they are more experienced. But people should be given the freedom to make mistakes. There are risks. But I think they are worth them in order to maintain faculty morale, foster creativity, maintaining standards, and honouring the important tradition of academic freedom. Furthermore, it is very important that faculty are not told by administrators, parents, or politicians what they should or should not be doing. Here, we should bear a thought for our colleagues in the humanities and social sciences, particularly in the USA, who are under increasing pressure to act in certain ways.

I welcome comments on any of the above.
My colleagues would particularly like to hear any text recommendations. Books by Griffiths, Shankar, Sakurai, and Townsend have been mentioned as possibilities.


  1. Since you have posted University of Wisconsin article by D Moynihan.. Here is some more
    The Tenure Apocalypse
    another one
    The End of research in Wisconsin

    1. Thanks for the links. They are good (but depressing) articles.

  2. Regarding students who enter a course without "basic skills" like dimensional analysis, calculus, or plotting a graph... what can you do except say in the first lecture that, here are some elementary abilities you need to do the course, if you don't have those abilities, you shouldn't be doing it. Or if you're generous, you might tell them, a good remedial introduction can be obtained in a certain book or course.

  3. At the infamous U. of Wisconsin we teach a single, 2 semester course on quantum to advanced undergraduates. The last few years I have been using Townsend, supplemented in the first semester by Susskind's wonderful Quantum Mechanics: the Theoretical Minimum. Both books use the Dirac notation and spin-1/2 from the beginning, emphasizing the basics of measurement, bases, time-evolution before generalizing to continuous systems. I also use Mathematica to keep time spent doing algebra to a minimum and it seems to help the students with their weak linear algebra skills. Teaching the matrix/vector approach first also helps students see the close relationships between functions and vectors. Plus, it is more interesting to the instructor than solving one partial differential equation after the other...

    1. Thad, Thanks for the helpful comment.
      Sorry that Wisconsin is getting famous for the wrong reasons....
      But, I presume your course is to juniors and seniors, and that as sophomores they took a course in "modern" physics that does wave mechanics, one dimensional potentials, the hydrogen atom, with no Dirac notation.

    2. You are correct. Most of the students have had a second course in mechanics and E&M as well.

  4. The PhysEd literature of course is full of this stuff. Most recently, I noticed a literature review, DOI:10.1103/PhysRevPhysEducRes.13.010109, "Insights into teaching quantum mechanics in secondary and lower undergraduate education". This has its failings, as does most everything about teaching QM.
    You ask the question "Schrodinger or Heisenberg, i.e. wave or matrix mechanics?" A heterodox alternative is "signal analysis", using Hilbert spaces. Specifically, you mention "relating theory to experiment": how do you teach how analogue signals out of an apparatus become discrete events become statistics, and are modeled using QM? Many experiments don't have a "discrete events" middle point, in which case how do the statistics that are modeled by QM emerge? How much does it matter whether it's waves or particles or matrices that cause the analogue signals, if we can engineer them? For the axe I'm attempting to grind here, see the 4'26" of https://www.youtube.com/watch?v=frSL-BJTh90, "Quantum Mechanics: Event Thinking", if you will.

  5. I will offer a couple of comments about "curriculum reform" from experience in a different discipline (chemical engineering).

    1. Remember that changes to content should be a zero sum game. You can't add topics without taking topics out.
    2. In my department we typically have multiple instructors teach any given course over the period of a year or two, since we offer courses multiple times per year. We have developed pretty strong community expectations about what is to be covered in a course, leaving some flexibility for each instructor to adapt to their preferences. It is not OK for people to change the book or software choice for a course without getting buy in from their colleagues. This is especially important in a series of courses that are linked together, such as the quantum courses that Ross mentioned. Yes, this diminishes the "academic freedom" of each individual, but hopefully the value to the students is considerable.
    3. A divergent range of opinions among well meaning people seem inevitable when discussing content and curriculum issues. It is important to respect the views of others, and probably also important to not set 100% consensus on all points as the goal.

  6. I'll comment from the standpoint of a masters student. In undergraduate I have taken modern physics, two courses on quantum mechanics. I have also taken the first offered course for graduate quantum mechanics.

    Before continuing, I should note that my pick from the three educational approaches is C, enjoyable and general knowledge with the associated skills.

    This, I would insist, is the most important thing. You cannot learn if you don't pay attention, and paying attention is easiest when you are engaged. Not necessarily fun, but whatever you do must keep you interested.

    As long as the standard of one curriculum for all students continues, there will always be a number of students who drop out, mentally speaking. Things will go too fast, or maybe a tad bit too slow.

    Moreover, curriculum thinking puts the weight on the shoulders of teachers, whereas learning happens on the minds of students. Courses then, ideally, shouldn't be a single progression of topics and must be designed around activities the student will do. This is difficult, I admit. It also is far more than just rocking the boat, basically sinking it and building a new (and shiny!) one.

    But, assuming such a thing is not possible, then the second of the above criteria may still be applicable - activities. Homeworks with simulation software perhaps, or finding a quantum topic the student will be interested in and having them do those projects (either individually or as a team). Of course there may not be enough TAs for such a thing, but it's not possible to get results without spending resources.

    As for the books, I wish we had used David McIntyre's wonderful text (Quantum Mechanics: A Paradigms Approach) in undergraduate. I found it helpful even in graduate QM.

    McIntyre's QM starts with spin systems, with Dirac notation and some matrix algebra and later on introduces the wave function method with the Hydrogen atom. Understanding the notion of quantum states comes rather easily with the Dirac notation, I think. (Pretty much all my friends agreed with this when we were taking grad QM.)

    P.S: Sorry for the long comment, I'm somewhat exasperated about education in general, and it leaks whenever I talk about it.

    P.P.S: As long as there are exams that may be passed without mastery, we (the students) will never feel compelled to master them, unless there's personal interest. Any personal interest without knowledge of the topics, however, is likely to be misguided and may waver rather easily. Can I just say it's a difficult problem and leave it at that?

  7. Book recommendation: UK universities have a heavy bias towards using Alastair IM Rae's "Quantum Mechanics". It starts with wave mechanics, introducing operator formalism and then matrix mechanics in the discussion of angular momentum, does perturbation theory, some aspects of many-body, and finishes with a chapter on measurement + the EPR.

    It was the only book recommended for purchase as part of an Imperial College physics degree, circa 2002. It was certainly useful in that it spanned such a broad range, that it could be used to help place different QM lecture courses (using the lecturers preferred notation) into one cohesive body of knowledge.

  8. "Exams are arguably relatively easy, similar to previous years, usually involve choosing questions/topics, and a mark of only 40-50 per cent is required to pass the course."

    Isn't this the root cause of the problem, as properly recognized by Mr. Akbaba? (Who quite adequately discusses a range of issues.)
    If one is not forced to master a subject, the result is that for a large portion of the population there is no incentive to reach mastery.

    The grading systems of exams is I think fundamentally flawed; often one gets a fraction of the total for each (sub-)question properly answered, and the sum of these fractions reaches the total maximum (e.g. a "10" on the scale of 1-10).
    This results in people passing with 55% of the answers (which are an -imperfect- gauge for knowledge). That should not be acceptable.
    If one does a driving exam and in 45% of the actions one breaks the law, it's quite probable someone gets killed in the first year of driving...

    I also agree with Prof. Sholl's #1. And his #3 is a statement that is useful far beyond the current discussion...!

  9. "Furthermore, it is very important that faculty are not told by administrators, parents, or politicians what they should or should not be doing" In relevance to your above comment some interesting tweets from academics.

    Here is classic tweet on this from Prof Dan Singleton Chem Dept Texas A&M.

    Dan Singleton‏ @dasingleton May 15

    I just blocked my Department Head. Good idea or bad idea?
    Then a reply from Prof Zimmerman chem dept U &I
    Steve Zimmerman‏ @steveczimmerman May 16
    Replying to @dasingleton
    Didn't know you could do that. Beats my aluminum foil hat and shield. (seems reasonable but I don't know circumstances).
    Then Prof Dan replies again.
    Replying to @steveczimmerman
    I heard indirectly he didn't like one of my tweets criticizing a teaching evaluation form. Figure better out of sight out of mind.
    Then Prof Zimmermann replies
    Steve Zimmerman‏ @steveczimmerman May 16
    Makes sense. Surprised a Dept Head has time to be monitoring faculty tweets. I was lucky to find time for bathroom breaks
    I stumbled across Prof Dans twitter for this sweet tweet which is very relevant to modern academia.
    Dan Singleton‏ @dasingleton Apr 19
    "Do not underestimate the old-timers of science. They had intuition. We have calculations. We are diminished"
    Yes, Mr. Akbaba , I agree with you.on the following statement.
    "I'm somewhat exasperated about education in general".

  10. I agree with Mr. Akbaba and pcs about the grading system. It is fundamentally flawed. It becomes more of a problem, in my opinion, when a prof takes off a letter grade for a math error on an exam. For example, on an exam in my undergrad QM class, a common mistake on a particular problem when doing an integral was to get a negative sign. This answer was completely incorrect resulting in the grade on the exam dropping by a whole letter (e.g. A to B). That's a bit extreme for a simple mistake and way too much pressure to put on students.

    There is also another inherent problem with professors interest in teaching. I've had grad classes where the professor obviously does not care about teaching (incomplete lecture notes, no interest in answering student questions, etc.). One possible fix is to hire 2 or 3 types of professors; ones that teach, ones that research (and train students to do research), and ones that do both. Each take unique skill sets.

    To get back to your question, ultimately the best approach would be C. One complaint among my undergrad peers was that our education did not prepare us for employment. (Although a PhD only prepares you for a postdoc...)

  11. I second the recommendation of Susskind's QM book from the Theoretical Minimum series. It's great for getting the big picture and avoiding getting bogged down in calculational details.

    For a more traditional textbook, I'm a big fan of Shankar's book. The whole thing might be too much for an undergraduate course, but with a careful choice of chapters it works fine.

  12. Hi Ross, you pitched a lot of questions. The discussion regarding why do students perform badly, how to incorporate academic freedom, and other like that are subject to so many different opinions that I do not think my thoughts would matter. Pragmatically I'll suggest some approaches and textbooks, which may be more concrete.

    I am personally in favor of doing matrix mechanics first, that's because I think students get confused with using x and \partial_x as operators from the get go. In the past I have taught a second year optional intro to QM. I have used T. F. Jordan "Quantum Mechanics in Simple Matrix Form" for the first half of the course, and then selected Lectures from Feynman vol3. Since you only teach matrix operations the students will not learn particle in a box, but tackle all finite dimensional problems, as well as creation operators for harmonic oscillator and hydrogen atom, as well as experiment outcomes. It is very unorthodox, and I'm not saying is a good fit for UQ, but it is possible and had good feedback.

    For 3rd and 4th years an alternative to Townsend or Griffiths is D. Park's "Introduction to the Quantum Theory". It has the basic material, but with lot of applications that can be chosen by each lecturer.

    Since you mention Sakurai, an alternative that I prefer at that level is M. Le Bellac's "Quantum Physics". Same math level, but with more detail, extra attention to entanglement, and lots of connection to experimental setups (plus a chapter on open quantum systems and decoherence).

    As a minor note, I don't think there's anything wrong with stopping a QM course in 1930. By that date we could explain Alpha Decay, both Valence Theory and Molecular Orbital for bonding, Thomas-Fermi Model, both Bloch's Theorem and Tight-Binding, Electromagnetic Field Quantization and Dirac's Equation. Most physicist I know (faculty included) would be hard pressed to explain all these topics.

  13. RFeynman says I really don't no how to teach after so many yrs. One feels by saying that I don't know how to teach ..... is making students at level playing fields with him, i.e making students comfortable. Feynman surely understood students/audience very well. This is a short utube on his excellent approach.

    The are four given in 1979 at Uni Auckland. Just type in the video section of google all four will be there. Each of RFeynman lectures runs for more than 60 mins.

  14. The edition 3.7 of Opera Magistris seems quite good reference book for introducing quantum physics. Here is the link:


    Caution it's 6,200 pages and 340 MB PDF...

  15. I think ultimately any course design has to be realistic about the students that you’re teaching.

    The marking criticism from pcs I don’t think is fair, and I don’t agree with the analogy. There are more learning outcomes to a degree than specific knowledge in a topic. What do you actually want students to take away from doing a course? If your goal is for them to perfectly replicate the QM that they’re taught, then I think you’re doing a disservice to the majority of students. You aren’t trying to teach someone how to perfectly drive, because you shouldn’t expect all of them to drive (doing a PhD/ research).

    At UQ in 3rd year there are ~70 (paying) students in QM per year. I think more than 5 of them going on to do a PhD using QM is wishful thinking. Shouldn’t your course additionally assess more general concepts? The 1-10 scale is only bad if all you’re trying to assess is a list of ‘topics covered’. I would like to imagine that a QM course is additionally a conduit to more general scientific skills overaching an entire physics degree. Most exams at UQ (and I imagine most universities) are not structured just on ‘topics covered’. They are tiered. If you only obtain 50% on a final, you are at least demonstrating some additional learning outcome that is not just this list of topics. You could argue about the mertis of having an education focused on physics when most will never use the specific knowledge; and how this might dilute actual learning outcomes you want from students going into a PhD. The reality is that almost all of your students are never going to touch QM ever again, and I think your teaching should in some way reflect that.

    Also, I think the ‘B’ view raises an interesting point. Not all universities are MIT/Cambridge/Caltech. Their curriculum is trying to distinguish the top 10% of UQ students (or similar universities), whereas UQs curriculum should distinguish all of them. A consequence is that repeating concepts might be an extremely good idea. It’s important to keep students reaching for knowledge almost out of their grasp, but no one will learn anything if it’s a chasm.

    And anyway, the book used or list of topics is always going to be secondary to the quality of the teaching. Through the actual lectures, assignments, and exams.

    1. Hi Henry,

      Thanks for this helpful comment. It is very important to hear from a student who has actually taken these courses and who has also gone on to a Ph.D.

  16. Food for thought: