On Teaching Quantum Mechanics

Many years later, as I wrapped up the Quantum Mechanics course one more time, I remembered that distant afternoon when my Department Head took me to see the semester schedule for the first year Master’s Programme. I was to teach PE1 — An Introduction to Quantum Mechanics. This was to be followed up by a continuation course in the Second year (not by me) that covered the applications: the General Theory of Angular Momentum, Perturbation and Scattering Theory and Relativistic Quantum Mechanics. I was only a week into my new job, my first, in a small, young University and an even smaller Physics Department, fresh from a Post-Doctoral fellowship at a National Research Facility and full of ideas on how things must be done. Teaching Quantum Mechanics was going to be, as my Head confidently predicted, “a piece of cake”. Having spent a better part of my budding research career, summing up different types of perturbation diagrams, I was not inclined to disagree.

A quick look at the course content only increased my confidence. Wave-particle duality, Heisenberg Uncertainty Relations, Schroedinger Equation, Eigenfunctions, Eigenvalues — all on which I had firm and clear opinions. The list went on to include simple one and three dimensional potentials, rotational angular momentum and stopped at linear vector space description of quantum states and dynamics. All this had to be completed in 36 hours, with roughly four hours every week for lectures and one for tutorials, beginning tomorrow. I was supposed to hand out three quizzes, one every month, and two tests, one in the middle of the course and one at the end. An external examiner would set the questions for the final examination. I remember going home that night, thinking of tricky quizzes (What units is the wavefunction measured in?) that would, no doubt, floor everybody.

The next day did nothing to diminish my bravado. The class (about 18 of them, mostly boys) seemed as curious about their new teacher, as I was eager to know them. I quickly ran through the course outline. I told them about the quizzes and the exams they were expected to take. I exhorted them to work hard and hand in their assignments on time. I promised to reward them with good grades if they did, and then threatened to punish them if they didn’t. The class seemed a bit tense, but interested. I quickly got down to give, what I thought was, a neat little survey of Classical Physics. I told them about Galileo’s experiments, about Newtonian particle dynamics and about Maxwell’s fields, all following precise mathematical laws that could be used to predict the future. I wrote all the classical laws that I knew, on the board with a breathless flourish. I reminded them about Einstein’s Relativity and how it had made only a few general changes to the Newtonian edifice, but had not lead to any fundamental revolution in our view of the world. I brought them to the beginning of the 20th
century and cautioned them that they were now looking over the edge of the Classical precipice, warning them that I was going to tear down this carefully constructed structure in the next lecture. In the first one hour of my life in a classroom, I had run through a wide gamut of roles. I had begun like a friend, warned them like a stern father, and ended like an erudite scholar. As I left the classroom, I had a warm fuzzy feeling that I had got through to them. I had even managed to use the word “paradigm” twice.

The next day, I strode purposefully into class with the merciless air of a demolition expert. I told them about the photoelectric effect. I drew graphs of black-body measurements that made mockery of our naive ideas of radiation and reminded them that the radiating electron loosing energy was doomed to crash into the nucleus if we continued to persist with our classical naivete. If I was to be believed, the fall of Classical logic made the fall of the Roman Empire look like a mere stumble. Picking my way carefully, but deftly, through the rubble of classical logic lying around me, I pointed to our only hope: abandoning, completely, the comfortable and “real, hard-to-the-touch” characteristics of particles like position, and momentum. I then went on to state the DeBroglie hypothesis and described wave-particle duality, promising that all would be set right, if only we used our language with care. To illustrate the syntax of this new grammar, I described the double-slit experiment, and tracked the path of a photon as it passed through a polarizer. I promised to set things right with this new logic beginning tomorrow and began to wind up the lecture with a clever quotation that had been handed down to me through generations of teachers that would surely have them in splits: “The electron”, I told them “behaved like a particle on Mondays, Wednesdays and Fridays, like a wave on Tuesdays, Thursdays and Saturdays, and on Sundays…”, I paused dramatically, “it is both”. So caught up was I in the clarity of my practiced logic, that I missed the confused silence that followed. I did not notice that nobody had laughed. I was too revved up to care.

That night, going through my notes, I had my first nagging doubts about where this would all lead. Was I promising more than I could deliver? Was I actually going to demonstrate the manner in which the new laws, “explained” the photoelectric effect? Would I be able to “prove” that the electron in a quantum orbit somehow does not radiate and stays in the same orbit? When would I “derive” the existence of the photon and the manner in which it explained the black-body results. How would I justify the use of words like position and momentum? Hadn’t I just debunked them as unreal? Could I describe an experiment that measured the wavefunction so that it could lose some of its nebulous quality and appear a bit more real to them. After so much tom-tomming about wave-particle duality, shouldn’t I refer to the wavefunction as the wave-particle function? How could I convince a group of unsuspecting innocents, that r and p which were till now, the dependent variables, had been suddenly liberated by the quantum revolution and begun to appear inside a bracket as independent ones? Why was I going on and on about momentum anyway? Wasn’t velocity a legal expression any longer? The sudden panic that seized me quickly lost its sharpness as I turned the pages of my notes. The neat and unambiguous algebra of the sharp and clearly defined potential wells that followed somehow seemed reassuring. But the seed of doubt had been planted. Could I pull it off?

The next day, I laid down the Uncertainty principle. I made some crude (and I knew patently wrong) analogies with trying to grab a marble that had gone down the gap of a sofa, and how widening that gap to get it out, would only take it further down. After this proof by analogy, I got down to proof by intimidation: I pulled out the free particle Schroedinger Equation and wrote down the wavefunction of a free particle, as a Fourier transform. Indulging in some rampant ad-hockery, I constructed wave-packets and for no apparent reason, started a discussion on the relation between group and phase velocity. I was on firm ground now. Finally it was algebra and symbol-manipulation time! I transformed a few functions with an air of practiced efficiency, showed how the widths were related and justified the uncertainty principle. This had taken me to the end of the first week. Except for a strange incident during the tutorial at the end, I had survived the week. During that tutorial I thought I heard a student’s voice ask: “Why Fourier, why not the Laplace transform?”. I turned round in surprise, only to find all heads down writing busily. It must have been my own mind up to its usual tricks. The strain of pretending to be logical was beginning to tell.

It was soon after this that the course began quickly going downhill. I was losing a little bit of conviction and two students, every week. Things would liven up during the time I did the algebra and the mathematical manipulations, and for those brief moments, we would all pretend to be rigorous and logical. But things would soon become unbearable when I did some explaining, trying desperately to keep the students with me. I had now begun to avoid eye contact with the class and was increasingly beginning to feel like a shifty-eyed purveyor of stolen antiques, desperate to make a deal. My arguments seemed to rest on the back of improbable lies. Not one lie. Many lies. And all of them disconnected, rambling and worse still, pointless. The disappointment at my failure to hold their attention turned to dismay, when I spoke to my colleagues, one of whom was teaching a course on Classical Mechanics and the other on Electromagnetic Theory. The first told me that he hadn’t come to the Hamiltonian yet and the other told me that the class knew nothing about radiation, black-body or otherwise, because as he put it “he was still stuck with Statics”. I realized with a sinking feeling that I had with great fanfare, claimed to destroy a logical structure that my colleagues were yet to build.

My entries into the classroom became less dramatic, more furtive. My explanations began to sound less like logical expositions and more like pleas to their conscience. Without the support of a central core, my course lacking direction, had begun to drift and collapse. Every new topic that I took up seemed to hang up there, unrelated to anything done earlier and leading to nothing further. My humiliation was complete when a student came to meet me after class one day. I had just handed out one of my assignments, this one on 3-dimensional potentials. “Sir!” he said confidently, “the problems have got mixed up, Sir”. I quickly went over the sheet but couldn’t find anything wrong. “Where?” I asked him. “Here, this one. You have asked us to calculate the angular momentum of a free particle, Sir!”

“Yes, what’s wrong with that?”

“Only hydrogen atoms have angular momentum Sir, not free particles. Free particles only have linear momentum”

His answer left me gaping at him. What pedagogic blunder of mine had turned this right-thinking, reasonably intelligent teenager into a freak of logic? He had clearly confused context with content, and I was responsible. By the time I recovered, the student had left. With the weight of my pedagogic sins sitting heavily on me, my lectures began to plummet. For the entire week when I talked about Vector Spaces, I had just two or three students in Class. They were not even the same ones each day. As a result, my lectures began to attain a surreal quality. I contrived to teach orthogonality to students who weren’t there when I talked about the scalar product, and completeness to students who weren’t there for the orthogonality class. When the course ended, I was as relieved as my class. The external examiner for the final examination from a neighboring University delivered the death blow. He asked them to write a “short note” on Wave-particle duality. My first course had been an unmitigated disaster, and the scars would take a long time to heal.

The next semester, I taught a longish (56 hours) course on Statistical Mechanics. I spent the vacation between semesters writing up extremely detailed notes. When I look at those notes now, they resemble more, a script of a film rather than notes for a course. There are desperate exhortations in the margins to “Repeat this twice, or they might forget”; “Tell them this is only an analogy or they might take you literally”; “Force them to read Landau and Lifshitz” and sometimes, even, “Check the blackboard and wipe it, if full”. My tender mind, scarred by the experience of my first course had now begun to take to the most unreasonable of precautions. I stated nothing that I could not prove. If there was a result I needed from an earlier lecture, or even an earlier course that they might have taken, I re-proved it in full. I was unwilling to assume anything, and my lectures began to resemble tight, but long legal arguments. Vulgar reductionism unfortunately, takes time, and my lectures began to take longer than planned. By the time I came to the end, I had to drop about 1/5th of the topics I was required to complete. The semester after that, I was scheduled to teach that dreaded Quantum Mechanics course again. When my colleague agreed to let me swap my course with his (Classical Mechanics, 56 lectures), I felt a great relief sweep over me.

It was almost two and half years later, that I was able to take a reasonably dispassionate view of the debacle of my first course. Why had I failed to get across? I thought myself to be reasonably articulate and secretly prided myself on my ability to explain tricky points to my friends during my student days. I enjoyed teaching, which is why I had left a research institution to join a University, against the well-meaning advice of my peers. Yet somehow the discomfort that I felt during the course was unnerving. The course, I felt, did not have a core around which it should have developed. There was no central principle from which everything else flowed towards a tangible goal. And if there was one, I had not been able to find it. Classical Mechanics had one: The program there was simple, or at least it appeared to be. Given the forces find the motion. Electrodynamics had one too. Given the charges and currents, find the fields. Everything in these courses was somehow connected or could be reduced to one central aim. A clear program does wonders to motivate and focus the learning mind and drive it towards understanding, and I was forced to admit, that my course had none. Having used up the considerable good-will of my colleagues, I had avoided teaching this course during the last five terms. Now I could not afford to duck this responsibility any longer, and I resolved to give it one more try in the next semester. I spent the better part of my vacation thinking about my options in the classroom. I dug into my own days as a Masters student, when I had the fortunate benefit of being taught by some of the best teachers in this country. I spoke to my colleagues and badgered them to tell me their own experiences. And slowly, but surely, the pedagogical puzzle began to fall into place.

First, I had to make adjustments for a fact I had always known: the historical development of Quantum Mechanics had little, if not nothing, to do with its logical development and structure. As I remember telling a friend then — Quantum Mechanics was not the same as the history of Quantum Mechanics. In a metaphorical sense, Planck had quantised the Electromagnetic Field, much before Dirac had written down its Hamiltonian, or even before Schroedinger had conjectured its dynamics. I remembered my own feelings, while reading Max Jammer’s classic on the Conceptual Development of Quantum Mechanics which my guiding teacher had urged me to read during my days as a research student. The logical cauterwaulings, the broadest of generalizations drawn from the flimsiest of evidences, the “proofs” — more wishful thinking rather than rigorous reason, had given me an acute sense of vertigo. I realized that this ran counter to the pedagogical development of almost all the Classical Physics which closely followed its historical development. The evolutionary biologists have a beautiful phrase to describe this: “Ontogeny recapitulates Phylogeny”. Higher animals in their embryonic development, are believed to rapidly pass through a series of stages that represent, in proper sequence, the adult forms of their evolutionary ancestors. A human embryo for example, first develops gill slits, like a fish and later a three chambered heart like a reptile, and still later a mammalian tail. This provides a neat parallel for the manner in which we organize our education; Pedagogy, at least in a Classical education, recapitulates history.

While this is an efficient route to take in, teaching Classical Physics, its utility in making sense of a major paradigm shift like Quantum Mechanics is, at best, dubious. If I was to make any sense to a new learner, I had to sacrifice the historical narrative, and strive for an internally consistent logical one. To carry it off successfully, I would have to warn my class right on day one, that this was what I was going to do. I had to tell them, up front, that there they would be able to make sense of its historical development only after they had understood its logical structure. There would be a time when wave-particle duality and the uncertainty principle would be discussed, but that would be around the end of the course, not the beginning. My course would have a beginning, a middle and an end, but not necessarily in that order. In the beginning would come the axiomatic rules of the game: the Postulates of Quantum Mechanics. I recalled a remark by Kenneth Wilson that while “…there has been great success in calculating Quantum Mechanics, there has been very little success in understanding it”. If a Nobel Laureate could say that in public, I could perhaps be forgiven my reductionist crimes.

A second fact came to me from a different direction when I stood the question on its head. The problem was not: Why is Quantum Mechanics difficult to teach, but rather: Why is Classical Mechanics, apparently, so easy to teach? Most of us, would agree that the Classical Mechanics course at the Masters level, has a significantly richer and more complex mathematical structure than an introductory course on Quantum Mechanics. Yet 8 out of 10 students seem to take the complexity of generalized coordinates, Legendre Transformations and even infinitesimal contact transformations and generators, in their conceptual stride. The reasons for this lie, I suspect, not so much in the fact that Classical Mechanics deals with the “real” world, but more in the manner in which instruction in this subject is organized. We first teach the student to merely “describe” the motion without seeking the cause for it. The dynamics of motion is clearly separated from and is preceded by a thorough indoctrination into its kinematics. The initial emphasis, beginning with the school curriculum is clearly on the kinematical rather than the dynamical. The student is given sufficient drill with calculating distances given the velocity, the velocity given acceleration, maximum heights and ranges of projectiles and periods of circular orbits — problems that emphasize the methods by which gross measurable characteristics can be extracted from a general description of its motion. It is only much later that we go into the cause of the motion, and equip the student to solve the dynamical equations to arrive more rigorously, at an orbit whose general characteristics, he had known all along. I wondered whether a similar strategy would work in a course on Quantum Mechanics.

To this end, I decided to begin the course with the kinematical aspect, imparting skills that would enable them to extract all possible information about a system given its wavefunction. This, the first section, would therefore have to include a discussion on operator correspondences, eigenspectra, and complete set expansions. The idea was to teach the student to obtain the probability distributions and averages of any observable whose representation could be expressed in position space. This required my class to learn two skills:

1. To obtain the operator corresponding of an observable;
2. To obtain the distribution given the operator and relate it to the measurement process.

I reckoned that this program would take me around two-thirds of the 36 hours alloted. But at any given time, I would have a clear sight of my, admittedly narrow, goals. I set up the operators for a series of observables, one by one and uncovered their eigenspectra, starting with position going on to momentum and angular momentum and finally to the Hamiltonian of various systems. I discovered with considerable surprise that obvious aspects of the quantum postulates that have become second nature and therefore self evident to us, need to be specifically and loudly emphasized to a student who is exposed to these postulates for the first time. I discovered for example that students easily miss the fact that while observables like position, momentum and angular momentum can be diagonalized and tabulated once and for all, an observable like the energy was specific to each system and had to diagonalized anew each time in a kinematical programme. Except for this one difference all these operators could be treated equally, as far as their kinematics is concerned. In each case,I would give them toy time-dependent wavefunctions and ask them to extract distributions and averages, emphasizing each time, the difference between a wavefunction and an eigenfunction, a distinction that might be a truism to a seasoned user of the quantum postulates, not to a student. Specific systems, treated in a conventional Masters programme — particle in a box, potential wells, and the simple 3-dimensional potentials, were all treated with the simple kinematical question in mind: If I were to make a measurement, what possible values would I get and with what probabilities?

The natural culmination of the kinematical postulates lies in the description of the act of measurement. This extremely important topic has unfortunately, been missing from most of the Master’s curricula and its absence is one of the major reasons, why most new learners of Quantum Mechanics tend to perceive the course as rambling and leading nowhere. A quick survey of introductory texts on this subject will throw up a startling fact: the last text that described the peculiarities of making a quantum measurement in a clear, unambiguous and emphatic manner, giving it the importance it deserved, was David Bohm’s. It was first published in 1954! Not to put too fine a point on it, the last text that clearly explained why particles leave a track in a bubble chamber was Schiff’s. That too was published in the early fifties. Spending two to three hours to discuss this extremely important aspect can be a richly rewarding experience.

It is only after the, postulatory basis is laid down, practiced and applied in a variety of situations, that I would begin to discuss the dynamics: the Schroedinger Equation and some simple time-dependent solutions, that can be obtained in closed form — the spreading of a free particle wave-packet (usually done in an ad-hoc manner during the initial part of a conventional text), the large n, classical solutions of the harmonic potential and the two level problem. The continuity equation would be discussed here. Again, the organization here is guided and focused by a search for the answer to a single specific question: the initial value problem. How does the distribution of observable possibilities change from time to time.

After students have absorbed the postulates and learnt to manipulate operators to extract the distributions with an understanding of the measurement process, a discussion of tricky concepts like the wave-particle duality, the uncertainty principle, the double slit experiment and the passage of a photon through the polariser, quickly reduce to a series of pedagogically comfortable truisms. Which is what they should be. This brings me to the conceptual end of the course, with only the section on linear spaces and the task of writing the postulates in a representation independent way, left to do. Before I get down to that, I have almost always (with one exception) found time (about three, but usually two lectures) in a course, to atone for my reductionist sins and return to spend some time on its historical development. I have found the Nobel lectures by Heisenberg, Schroedinger and Dirac to be rich resource material for these classes, richer sometimes than Jammer’s classic.

I have now taught this course thrice in the last six years, after that initial embarrassing and humbling experience, with increasing confidence. Has it made any difference? I think yes. The class, I suspect has a better idea of the overall plans and goals of that course and the motivation for studying it. At least, its been a long time since somebody told me that angular momentum was the sole preserve of the hydrogen atom. And yes, a request for a “short note on Wave particle duality” by misguided examiners, often brings forth a reasonably well argued answer from my class.