No Timelog Experiment

I arbitrarily chose today to mark the beginning of my sabbatical. In my mind, the most significant change I will be making is that I will NOT keep a timelog for the next twelve months. I had been keeping one since I started my tenure-track faculty position a long time ago; I even provided a ten-year snapshot with brief analysis in this blog.

 

Many faculty members maintain a flexible schedule through most of their careers. Sometimes work is done on evenings and weekends; sometimes one runs a non-work-related errand in the middle of a weekday. In that sense, their life is like that of a college student: Show up for classes and meetings, and do other work when you’re not in class or in a meeting, with more frenzied activity as a deadline approaches. Like some students, professors can also be procrastinators and constantly doing things at the last-minute. On top of that, there’s research. It never ends. There’s always something to read, something to think about, an experiment to run, and it constantly occupies one’s mind-space.

 

I’m an atypical professor with regard to time. From the beginning of my career, I made a clean division between work and non-work. I treated work as a 9-to-5 job (more like 7-to-4 in my actual case). When at work, I focused on work. When away from work, I did no work and did not check my work email on evenings and weekends. Studies suggest that most professors work significantly more than 40 hours per week; one that includes charts and breakdown of time usage claims 60 hours per week on average for tenure-line faculty although this is self-reported data. My average is 42 hours over more than two decades of data, and while my data is also self-reported, I meticulously logged my time use daily. But this also means, that when I’m at work, I have to be efficient to get things done. There’s no time for procrastination or distraction, and overall, this has made me a disciplined worker.

 

Keeping a timelog is very useful when you have a self-directed job with many different things you want to accomplish on top of teaching obligations and open-ended research projects. To make sure I was maintaining the balance I wanted, I regularly looked at how my time was being spent and make adjustments. This provides stricture and discipline, but I wonder if over the years it has reduced my flexibility and squelched my creativity. Perhaps I have even become risk-averse intellectually; honestly, I’m not sure. Hence, I am embarking on my no-timelog experiment. I will, over the next year, not place strictures on when I work and how I work, allow some of my work and non-work to mix, and observe what happens. My pitch to myself is that I am aiming for a wholistic sabbatical.

 

I did write a sabbatical proposal which was approved, so I do have some professional goals to accomplish. I have prior research I’d like to write up, new research projects I’d like to think about, classes I’m excited to overhaul, and new classes I’d like to design. Maybe I will learn a language; maybe I will do more traveling; maybe I will learn to use some new tech tools; and I will certainly do more reading! I would also like to be rested, refreshed and rejuvenated when I return to my regular daily job for the Fall 2027 semester. But one step at a time, and this morning’s blog post is just the first step on the first day of the next twelve months!

Need for Speed

This morning I watched two pre-recorded webinars (on A.I. tools) at 1.5x speed. I did slow down to 1.25x speed at certain portions where I anticipated something I wanted to learn. Why did I speed up the videos? Often there’s dross and filler I am less interested in, so it feels efficient to let those pass by at a higher speed where presumably I don’t do as much cognitive work. I did pause the videos on a couple of occasions to write a note on something I should try out.

 

Post-pandemic, students sometimes ask if my lectures can be recorded (not by them, but by the “system” setup in the classroom). The purported reason for asking is so they can review the lecture again later. The answer thus far has been No. I strongly suspect that if I provided recorded lectures, attendance would drop. While attendance in my G-Chem classes is overall quite good (over 90% most days), the small number of students who consistently miss class on a regular basis do poorly in the course. (Attendance in P-Chem is usually close to 100%, and students fret if they miss class, because when it happens they very quickly realize they have fallen behind.)

 

And in the age of A.I., if I provide recorded lectures, I’m certain the students would watch it at accelerated speeds, and then ask A.I. to provide a written summary. This is unlikely to help their learning although it might give them the illusion of learning. But this isn’t to say that you cannot learn from videos. There are many examples where video is excellent at helping you learn how to do something that physically requires motor skills – if I need to make a minor repair or if I wanted to learn a new cooking technique. But does that extend to learning scientific concepts?

 

With a brief but lazy search, I came across “The effects of lecture speed and note-taking on memory and educational material” (Chen et al, Applied Cognitive Psychology 2024, DOI: 10.1002/acp.4166). Here’s the abstract which also summarizes the work, but the paper is worth reading in full. It’s a small study and thus its extensibility is less clear at the moment, but the results are interesting, and I hope there is further study in these areas. The pre-recorded video materials were on the Paleozoic Era and Microeconomics, and there was a post-test. Interestingly, watching at 2x speed results in a small drop in test results. Taking notes helped.

 

I’m not going to discuss the article details as I’m unlikely to implement video recordings of my classes, but it made me think about the speed of my lectures. My course teaching evaluations have students regularly saying that I go fast. This is true and there are reasons behind it. I won’t hash those out here, but I will say that I tell students how to approach my classes and that pre-reading before class is important to getting the most out of class in-person. While I do pause my speech when I observe the majority of the students writing furiously what I have written on the board, it’s possible I need to provide longer pauses so they can digest the material. The natural punctuating rhythm of in-class student questions helps in this regard. What I don’t know is the quality of the student notes, whether they are taking notes while cognitively engaging the material or just passively copying stuff down. I can’t peer into their minds. For the students who come to drop-in (office) hours, I do get to see the quality of their notes, however this is a small subset and it’s usually the stronger students who stop by. I have had limited success getting the students who need the most help to visit my office and ask me questions.

 

One positive aspect of the speed in my classes is that practically no students are on their phone or surfing the web if they use a tablet to take notes (hardly anyone uses a laptop to take notes during class). The pace means there is less time and opportunity to be distracted, at least visibly so. I can’t peer into their minds so there might be passive, mechanical, note-taking with the mind wandering elsewhere. Students look like they are paying attention, and the lecture classes are interactive. When I break students into groups to work on something and walk around, students do work on my assigned questions, (because they might randomly be called to answer to the whole class) and the time is tightly controlled – they have to work relatively quickly and productively.

 

Actually, knowing the material and learning it well results in a fluency when it comes to taking exams. Thus, my exams are written with time being relatively tight. The strongest students will finish with time to spare; the weakest students will also finish early because they simply didn’t know how to do a problem; but the majority of students might finish with barely time to spare. I’ve read the literature about the supposed “harm” of timed tests, and I’m overall not convinced by the arguments (although I have made modifications to my pedagogical approach after reading some reasonably raised points).

 

I suspect that short-form incessant video watching has changed the way the student today “consumes” information in my classes, certainly differently from the students I had in the previous generation when I started teaching at the university two dozen years ago. Extensive gen-A.I. use among students in the last couple of years has further changed things, possibly drastically. All this keeps my life as an educator interesting as I learn to adapt with the evolving technology that students use and consume. Acceleration, the need for speed, seems to be the new name of the ongoing game.

LMS Liabilities

Many digital pixels have gone into dissecting the Canvas shutdown debacle ten days ago. What certainly emerges is that Canvas did a poor job communicating with its constituents. Whether or not they paid a ransom, or whether the hacked data is contained, or whether they have fixed their vulnerabilities, is unclear. I don’t think my campus was directly targeted; my students and I did not see a ransomware notice, only that the system was down. It was accessible again the same evening, so it was down for maybe six to eight hours. I had a P-Chem problem set due the following morning. The majority of students uploaded it to Canvas, a handful sent me a pdf directly via email, and one additional student turned in a hard copy.

 

I had not experienced using a Learning Management System (LMS) as a student. In my first couple of years teaching, everything was done hard copy just like when I was a student. I don’t recall when my campus started using WebCT, but I do remember teaching myself HTML and building a simple rudimentary website that provided my syllabus, basic information, and downloadable materials. It had a simple password protection scheme. It lived on one of the university servers that hosted faculty home pages. Few faculty had or used them. I still maintain my text-only webpages making minor updates every semester. Very low maintenance. I never bothered to learn Wordpress or other fancier tools as they showed up over the years.

 

WebCT looked clunky and I resisted using it, preferring my low-maintenance website. Blackboard bought WebCT. I used it a couple of times when I was team-teaching with colleagues who used it. The university continued to push (or “encourage”) more faculty to utilize Blackboard so that students would have a more “uniform” experience. I was one of the rogue faculty members who did not for a variety of reasons. At one point I vaguely recall being told that the university was shutting down the faculty webpage host server, but that did not come to pass. I think there were enough rogue users like me who hacked together their own sites that made sufficiently reasonable arguments. I did lose the battle over using Pine for my email and was forced to switch to my university’s Gmail.

 

Then Covid-19 came. I was on sabbatical in Spring 2020 so I did not have to make the hard pivot that my colleagues scrambled to do. I had time to think about how I was going to teach the next academic year online. My clunky website did not support video but the Zoom integration in Blackboard did. I spent part of Summer 2020 setting up all my classes in Blackboard, not just material delivery, but a Discussion Board, a wiki, online quizzes, and assignments for uploaded exams and other materials. Blackboard wasn’t too bad, and after we went back to in-person classes, I retained using Blackboard instead of my simple HTML website for managing class materials. P-Chem homework could be submitted online which worked well for students retaining their original problem set for annotations. Quizzes and exams reverted to in-person. I stopped using the Discussion Board when ChatGPT arrived.

 

Two and a half years ago, our campus made a hard switch to campus. I had winter break to make the adjustments. It was annoying because the Blackboard import did not work well for setting up my Canvas shells so I put up everything again from scratch. I don’t use the majority of the LMS features, and certainly not the gradebook, so the Canvas outage ten days ago did not affect my classes significantly. The way I set up the G-Chem online homework system and eTextbook as single embedded links to access, rather than a full Canvas integration, meant that there was an alternative way the students could access both directly from the publisher website. In a more extended outage, I could have revived the use of my HTML website and everything would have still worked.

 

Covid-19 made me think about how to set up my courses to pivot quickly without too much hassle. Although I hope it doesn’t happen before I retire, I expect a reasonable probability of another pandemic forcing us to move online. The possibility of zoonotic virus spread is very likely given how we humans live today and how biology works. But with the latest Canvas debacle, it now makes me think about how I might pivot quickly should another hack occur. I think there is an even higher probability that LMS vendors will be breached by hackers; not a question of if but when. Something will go down at a crucial time. I hope the education industry has learned something from all of this, but I’m honestly not sure if anything is going to change. In the meantime, I’ve downloaded all my Canvas materials and will rework how they are organized in case a quick pivot is needed in a future year. I just finished giving my last final exam and I go on sabbatical next academic year so I have time to think about the reorganization.

All About Maxwell

I am halfway through The Man Who Changed Everything, a biography of James Clerk Maxwell, written by Basil Mahon. As a physical chemist, I know something about Maxwell’s scientific achievements. The Maxwell-Boltzmann distribution shows up in multiple places because chemistry is about the movement of zillions of tiny particles, meaning you have to apply statistical methods to bridge the microscopic world to macroscopic phenomena that we large lumbering humans observe. Maxwell’s 1873 paper titled “Molecules” is marvelous, showcasing his lucid writing and insightful though; I have on occasion assigned it to first-year undergraduates along with light annotations to help them read along. I mentioned this in my very first blog post!

 

What I didn’t know much about, and am delighted to learn from Mahon’s book, is who James Clerk Maxwell was as a person. I learned about his rural upbringing that made him initially stick out as a weirdo in a more urban school, and how his geniality, generosity and genius eventually won over his classmates. While his mother had passed away in his early life, he had a loving extended family, and in his early twenties, he devoted much of his time to caring for his ailing father. He was beloved by his friends, an occasional prankster, liked to exercise, and wrote poetry. Given his fame for conjuring mathematical relationships of physical phenomena, I was surprised to learn that he frequently made lots of math mistakes in his derivations, but his scientific intuition was brilliant and almost always on the mark. A famous contemporary scientist said: “He is a genius, but one has to check his calculations.”

 

Maxwell was also a devout Christian but did not get sucked in to the many debates pitting science against religion. In declining to join an eminent society discussing such matters, he replied: “I think that the results which each man arrives at in his attempts to harmonise his science with this Christianity ought not to be regarded as having any significance except to the man himself, and to him only for a time, and should not receive the stamp of a society. For it is in the nature of science, especially those branches of science which are spreading into unknown regions, to be completely changing.” Maxwell’s views on the relationship between theory and experiment in science are also immensely quotable, for example: “I have no reason to believe that the human intellect is able to weave a system of physics out of its own resources without experimental labour. Whenever the attempt has been made it has resulted in an unnatural and self-contradictory mass of rubbish.”

 

I confess that I never took a physics course in college or beyond. How I became a professor who teaches physical chemistry still amazes me. My weak physics background, and perhaps lack of effort to improve my mediocre mathematical ability, means that I don’t really understand Maxwell’s famous equations although I do have the gist of its broader impact. I was heartened to read that Maxwell made great effort to find physical analogies to explain seemingly mysterious phenomena such as lines of force. Even now, I find it challenging to think through the lens of a field approach, and I use Maxwell’s ideas of fluid flow as a crutch to think about flux. Maxwell’s analogy of potential difference and hydrostatic pressure is also helpful; I use it when I teach electrochemistry in General Chemistry. (In fact, I will use it in my class tomorrow morning!)

 

What jumped out at me in reading the account of Maxwell’s struggle to derive a mathematical framework for Faraday’s lines of force was the ability to bring together insights from one area of physics to solve another. The jumping off point was a discovery by William Thomson (later Lord Kelvin) who found that the equations for the strength of electrostatic force looked similar to those describing the rate of steady heat flow. This seems odd: why would static equations resemble dynamic ones? But Maxwell made it work by imagining the flow of an “ideal” weightless incompressible fluid through pipes. I’m presently covering kinetics in my Physical Chemistry, and was looking ahead at my lecture on molecular collision theory. With Maxwell in my mind, one of the equations looked suspiciously familiar. I flipped back to a lecture I had given in the second week of the semester on the Lennard-Jones potential energy curve (for two-body molecular interactions), and sure enough, the mathematical expression for the static temporary dipole attraction looked analogous to the rate equation in collision theory. Wow!

 

I was impressed to read about the breadth of problems Maxwell tackled. His work on optics and colour vision culminating in his famous colour triangle is brilliant. He even devised spectacles for those with red-green colour-blindness. I did not know that Maxwell won a prestigious award for deriving mathematical equations to describe the conditions of stability of Saturn’s rings. When tackling the possibility that the rings are a fluid rather than a solid, he showed they would break up into smaller entities. But how would a hodgepodge of particles maintain an orbit? Maxwell showed that such rings vibrate in different ways and could be stable at low enough average densities. When he considered multiple rings, “he found that some arrangements were stable but others were not: for certain ratios of the radii the vibrations would build up and destroy the rings.” This sounds like the remarkable Bohr orbits of quantum mechanics where the electron orbiting the nucleus is treated as a standing wave to be stable.

 

Another surprising thing I learned was that despite his lucid and clear writing, Maxwell’s success in classroom teaching was mixed. Mahon writes: “For all his talents, he never mastered the technical part of teaching. He would prepare a lesson beautifully, do fine for a time while he stuck to his script, and then fly into analogies and metaphors which were intended to help the students but more often than not mystified them. He was not expert on the blackboard, where he made algebraic slips which took time to find and correct. And yet the students liked him and some found him truly inspiring… It seems paradoxical [for] such a fine scientific writer… as he believed fervently in the value of good education… Appreciating that people learn in different ways, he may have tried too hard to bring in helpful illustrations and analogies, confusing his audience with a welter of rapidly changing images… And perhaps he was too much of an idealist. All good teachers aim, as he did, to teach people to think for themselves, but most also recognize that all some students want is to gain a second-hand smattering of the subject so they can pass exams, and make a specific effort to help them succeed in this limited ambition. Maxwell never did.”

 

Those are sobering words for me as an educator who is also very excited about imparting chemistry to my students. I certainly try to give metaphors and analogies which I hope are helpful. Given my theoretical bent (a product of both my training and my interests), I have noticed that I now spend more time trying to impress upon my students the key frameworks on which my discipline builds its foundations. And I do this unprompted; it’s not in my lecture notes. It’s almost as if, like Maxwell, I can’t help myself. I feel compelled to make those connections to the broader edifice of how chemists think about the world. One progresses from novice to expert by first glimpsing and then progressively seeing more clearly the abstract categories that undergird chemical knowledge. I pontificate more than I used to. When I first started teaching, I couldn’t see some of the hidden frameworks; my focus was getting the students through the material in a systematic way that allowed them to (hopefully) provide them the basics to solve chemical problems on an exam to prove they understood what I was trying to teach. I am still aware that the majority of students in my classes are interested in the “second-hand smattering of the subject so they can pass exams”, and make efforts to help them along, but I also want to truly inspire the minority to see the beauty and depth of chemistry. Maxwell cannot help me resolve this tension, but I am inspired by his efforts. I look forward to sinking my teeth into the second half of his biography!

Biochemistry Mishmash

I am slowly working my way through The Natural Selection of the Chemical Elements by Williams and Frausto da Silva. It’s not the easiest book to read, but it approaches issues of biochemistry from an inorganic and evolutionary lens that I find helpful. I used one of their books in a class three years ago because our library had a digital copy.

 

Today’s post is a mishmash of thoughts sparked by my reading of chapters 11-13 touching on the evolutionary organization of cells and the roles of different chemical substances. Since I study the chemical origins of life, I filter what I’m learning through that particular lens. From that perspective, the book’s contents are idiosyncratic and generates more questions than it answers. But it gives me much to mull about.

 

Since the authors have a background in inorganic chemistry, the function of metal ions features prominently. The big change to the chemical environment is a redox shift from reducing to oxidizing conditions. We have plenty of O₂ in our atmosphere today, but this was not so on the Hadean Earth. The progressive oxidation led to a decrease in the availability of some substances, particularly Fe(II) and sulfides, but led to the increase in others, with newcomers such as Zn and Cu becoming available, alongside a shift to complexity, symbiosis, and eventual multicellularity.

 

The final paragraph of chapter 13 begins: “The conclusion we have reached is that multicellular development was bound to increase in complexity as newly available elements were incorporated but could only do so by coexistence with simpler forms. Complexity is eventually self-defeating and the escape from this dilemma is only possible with an ecosystem of the simple and the complex.” Biochemistry is a tinker, so the first sentence is not surprising. There is a mishmash of systems layered upon more primitive ones, palimpsests sometimes peeking through. The second sentence is provocative. Is it true? I don’t know. But we do know that complex systems open the possibility of catastrophic system failure.

 

Things that jumped out of me:

(1) The evolution of life is all about kinetic traps. “Energized” molecules quickly dissipate energy in their thermodynamic progress towards the equilibrium state. But to get a system going that allows for control, kinetic traps are essential, as is the evolution of catalysts. Before central control emerged, persistence is about being trapped long enough or often enough.

(2) Vesicles and other compartments within the cell have chemical environments that can be very different from the cytoplasm and use messenger systems similar to the “outside” of a cell, calcium-based systems for example.

(3) The rates of phosphate versus thioester hydrolysis can vary greatly over different pH and temperature. This may be a clue to a takeover of energy transduction from a thioester world to the modern one primarily using phosphate esters.

(4) The assertion the DNA codes qualitatively for proteins, but not quantitatively, is interesting. The quantitative aspects that require control and regulation were previously “set” by more primitive cells. Since I think a primitive metabolism is prior to nucleic acid coded information, this makes sense to me.

 

Things I need to ponder more: Why is negative feedback more prevalent in the evolution of living systems? Does it arise because living systems are thermodynamically semi-closed? I regularly tell my students in G-Chem II and P-Chem II that we study equilibrium thermodynamics because we can construct a model and its accompanying equations for closed systems. I contrast this to the non-equilibrium thermodynamics of open systems and use life as an example of staying alive by avoiding the equilibrium state. But an enclosed cell that tries to maintain some level of homeostasis along with growth and repair isn’t completely open. It’s very finicky about what goes in and what goes out, and what concentrations are maintained inside.

 

The authors also reminded me about the distinction between control and regulation: “Control acts at the level of metabolism and one part of it is concerned with the use of proteins including catalysts but not with their productions… Regulation acts at the level of gene and… was little altered from that in anaerobes by the development of multicellular organism…” From my slant, this suggests that control precedes regulation, at least on a local level. A protometabolic system evolves to control matter and energy. How? By tinkering! Why? I don’t know but it reminds me of the dictum: “What persists, persists. What does not, does not.”

Discovery Learning: ADOM Edition

This past weekend, I notched my fifth ADOM win with a Drakeling Elementalist who is now #2 on the high-score list. In preparation for today’s blog post, I also replayed a Tutorial game on a new installation to remind myself what tips the system provides to brand new players.

 

What I’ve been musing about is “Discovery Learning”, a buzz-phrase that leverages (in this case simple-minded) “common sense” thinking. In the extreme version, there is no formal schooling for kids. Let them explore and discover the world and learn “naturally” from nature. Natural – good! Artificial – Bad!

 

I don’t disagree that much can be improved about the seemingly artificial settings of today’s classrooms especially for kids who have lots of energy and are bouncing off the walls. But I don’t think Discovery Learning and doing away with formal schooling is the answer. It could work well for some people after they’ve had a decent foundation (acquirable in diverse ways). The media likes highlighting the college dropout who went on to found a tech company and become ridiculously wealthy. They don’t tell you about the tens of thousands of other dropouts who did not become billionaires or even millionaires.

 

In ADOM, the world of Ancardia has rules. The basics are provided in the manual, and I found the tutorial much clearer now that I have 70-80 games under my belt compared to the very first time when I was floundering around. Because I had experience with old-school CRPGs, ADOM wasn’t impenetrable, but many of the rules are “hidden”. I actually did okay getting to my first mid-game character within a dozen games purely through Discovery Learning. Characters die early and often in ADOM. I could sink hundreds or thousands of hours into the game and learn more nuances about staying alive and making further progress towards the end goal, or I could learn from experts who have already traversed the path. I chose the latter, and my enjoyment of the game increased by more efficiently getting over many of the otherwise frustrating barriers that would have killed dozens of characters.

 

The “natural” environment of ADOM is brutal. You might even say they are hostile to learning efficiently. While the tutorial gives you a “warning” when you first enter the Small Cave, you have no idea what that really means. And until you notice or understand how the hostile monsters are generated, you’ll bang your head against the wall trying to get through. You have no sense of how the difficulty level scales. You encounter monsters you know nothing about. You might get cursed or doomed and not realize why it happened and what it means for your character. You don’t know what talents or attributes are helpful and how they might be trained naturally. There is plenty that is hidden unless you know exactly what to look for. I’m not sure how many games it would have taken me to figure out that dropping a potion of water on a co-aligned altar blesses it, and that when you dip a scroll of identify into holy water, you can then read it (if your literacy is high enough) to identify all your items in a single swoop.

 

As a chemistry professor, the natural sciences and math are the areas I am most familiar with. Learning math or chemistry efficiently is very unnatural. If you had to figure it out from scratch, it might take you several lifetimes. (Also, failure is not always productive.) The accumulation of human knowledge has taken lifetimes – small bits of info passed down from teacher to student. The apprenticeship model has been true for a long, long time. It’s far better than having to discover everything from scratch through trial and error, but this one-on-one learning is inefficient and very expensive on a larger scale. I don’t like having forty students in G-Chem; I think I do a better job when I have five or ten or twenty. (At least it’s not four hundred.) But I recognize the efficiency of teaching a group of students. They can also help and encourage each other, which is a plus in my opinion.

 

Becoming an expert requires depth of knowledge and acquiring abstract schemas in long-term memory. Without books and teachers and some very effortful thinking on my part, I would not have the expertise that I now have in chemistry. I can’t imagine getting there through pure discovery. Of course, here I’m caricaturing Discovery Learning, and an advocate would say that no one is promoting pure “throw you into the deep end of the pool and you sink or swim”. They’d say the learning has to be guided. I don’t disagree. But the same advocates caricature current classroom practices, especially what is known as “explicit teaching” as inferior to discovery approaches, or “lecturing” as an artifice and therefore worse than a more “natural” approach. In reality, one balances multiple aspects when considering pedagogical strategies.

 

My current ADOM character is a level 13 gnome druid. I just made it to the High Mountain Village although I was not able to retrieve the waterproof blanket on the way because I understand how the Small Cave works. The fun in ADOM is that the dungeon layouts (and the game is a dungeon-crawler) are randomly procedurally generated, so each game feels quite different. Your character’s inherent skill set provides even more variation. I think this is my third druid (the previous two did not make it past level 10 before succumbing), and I’ve learned how to balance spellcasting with traditional weapons. I also now know that most animals are generated friendly, and switching my alignment to Lawful means that I have a reasonably good chance of completing the Rolf Quest and getting the ring of the master cat, provided I don’t die in the Pyramid or somewhere else. The balance of some discovery and some guide-reading, in my case, has led to maximum enjoyment. I still do bits of both when I encounter something rare (statues and artifacts) or exploring a different aspect of the game, and I wouldn’t do this any other way.

Square Integrable

I am reading about the extraordinary math and science contributions of John von Neumann in Ananyo Bhattacharya’s book The Man from the Future. I definitely get the feeling that von Neumann was indeed a rare genius. I also got the feeling that maybe I should have persevered in learning more math when I was younger. If so, not only would I have a better appreciation of von Neumann’s achievements, I would also be able to tackle some interesting problems in my research that require mathematically modeling beyond my current abilities. Feynman’s quote notwithstanding, I would like to better understand quantum mechanics since I use it heavily in my research.

 

Today’s blog post is about Chapter 3 of Bhattacharya’s engaging book. The chapter is titled “The Quantum Evangelist” and leverages the author’s physics background. While I know a number of facts about the history of the development of quantum mechanics, I learned a lot more about von Neumann’s contributions and the context surrounding his work. Reading this chapter gave me a better idea of the conceptual differences between Heisenberg’s matrix mechanics and Schrodinger’s wave mechanics. The connections to set theory in mathematics (and Hilbert’s program of systematization) helped clarify the context. Quoting the author: “An atom has an infinite number of orbits… so Heisenberg’s matrices must also be of infinite size to represent all possible transitions between them. The members of such a matrix can… be lined up with a list of the counting numbers – they are ‘countably’ infinite. Schrodinger’s formulation, on the other hand, yielded wave functions describing… an uncountably infinite number of possibilities. An electron that is not bound to an atom… could be literally anywhere.”

 

I now have a better appreciation of Dirac’s “ingenious trick to merge the ‘discrete’ space of Heisenberg’s matrices and the ‘continuous’ space of Schrodinger’s waves” with the delta function. Bhattacharya describes it as a “salami slicer, cutting up the wavefunction into ultra-thin slivers in space”. While Hilbert space still feels fuzzy to me and I don’t quite comprehend it, I can dimly see where square-integrable functions come from. When I teach quantum chemistry, I tell students about this important property and its practical uses along with Born’s probability postulate, I had never talked about their mathematical basis (because I didn’t understand it myself).

 

Where does von Neumann come into the story? Given his mathematical talents, he realized that square integrable functions “can be represented by an infinite series of orthogonal functions, sets of mathematical independent functions that can be added together to make any other… How much of each function is required is indicated by their coefficients... [which] were exactly the elements that appear in the state matrix.” In my class, I invoke orthogonality from a consequence of Hermitian operators. I discuss the importance of having linearly independent functions and spaces (e.g. Cartesian space or polar coordinates) conceptually but my students still struggle to think about it. Linear algebra is not a pre-requisite for my class and most students haven’t taken it. Neither have I for that matter. Until reading this chapter, I had not realized the connection between square integrable wavefunctions and orthogonality. In my class, when we get to multi-electron multi-atom systems, I introduce students to manipulating linear combinations of functions that sum up (invoking the principle of superposition) to get better results when solving the Schrodinger equation. They learn that the sum of the squares of the coefficients must add up to one, but I hadn’t made the connection to square-integrability.

 

There is plenty more in the chapter about the weirder aspects of quantum mechanics, wavefunction collapse, hidden variable theory, pilot waves, Bell inequalities, and Many Worlds. But what really stood out to me was where square integrable functions come from (as part of Hilbert space) and how they connected to orthogonal component wavefunctions. All these connections were a revelation to me, and I’d been teaching for a quarter of a century! How little I know. How much more to learn. This reminds me that I should get back to Beyond Weird by Philip Ball.