Fine-Tuned Grades

A common speculation in physics, armchair or otherwise, is that our universe is fine-tuned. The proposal of a fine-tuned universe hinges on the argument that a tiny change in the fundamental ‘constants’ of physics will lead to a very different universe, one devoid of life or complex molecular structure.

The fine-tuned argument often invokes the anthropic principle; but while the two concepts overlap but are not identical. The anthropic principle has two forms, neither of which I find useful from a scientist point of view. The weak anthropic principle (Brandon Carter), in my opinion, is obvious. “Only in a universe capable of eventually supporting life will there be beings capable of observing and reflecting on the matter.” The strong anthropic principle (John Barrow and Frank Tipler) avers that the universe compels the formation of intelligent life to muse upon it. I don’t think this is testable scientifically, and it remains in the philosophical realm.

Let me try to answer something simpler. Do I fine-tune grades in my classes? Do I design my exams with this fine-tuning in mind, consciously or unconsciously? That’s what I’ve been pondering this week as I’ve been grading final exams. I have some external data and some internal musings in my head, so let’s see where it leads.

Teaching at a liberal arts college with small class sizes means there could be significant variations in student grades depending on the ‘sample’ of students I get in any particular class. I decided to analyze General Chemistry I and II, because I teach these classes most years. The “Regular” sections are typically 40 students and the “Honors” sections are typically 20 students. Some years I teach a smaller 20-student section of GChem1 in the fall semester, where incoming first-year students have often indicated an “Interest” in chemistry or biochemistry, i.e., these students are science or chemistry-inclined, but not in the honors program.

The x-axis in the graph represents relative time in each category, but the years are not necessarily consecutive. Otherwise the lines would jump between categories; I might teach a Regular section one year, an Honors section the next year, and then another Regular section the following year. Hence, I’ve chosen to group the data by category. I’ve omitted a couple of data points – one where my spreadsheet was empty (I must have deleted the data by mistake some years ago), and another where I structured the grade distribution in the class very differently because of a final project. I’ve also included both the mean and median average scores.

On the first day of class, I tell my students that I grade on an absolute scale, i.e., there is no curve in the class. (I also explain why so the class learns something about sample size and the normal distribution.) This means everyone could get an A, a possible but unlikely scenario. Everyone could fail, a possible but very, very unlikely scenario. I band my grades in 15% increments and my plus/minus increments are in the top and bottom 3% of each band. On average 85% of the class grade is based on exams, and the final exam is typically a third of the overall course grade.

The Regular GChem1 mean is close to the C+/B- borderline, with the median in the B- range. When I teach the smaller class of science-inclined students, both mean and median are mostly in the low B range. Regular GChem2 has mean and median scores close to or in the C+ range. The material in GChem2 is more challenging than GChem1. Some students who do well in GChem1 coast on prior knowledge if they’ve had a good high school chemistry class, but then run into difficulty in GChem2. I don’t have as many data points for the Honors sections, but they are typically in the mid-to-high B range with GChem1 slightly higher than GChem2. The exams in the “Interest” group do not differ in difficulty compared to the “Regular”. The exams in the “Honors” group are only slightly longer with marginally higher expectations, but overall not too different.

There are some ups and downs in the average grades, but my broad brush says they are by and large consistent. Early in my teaching career, I might rescale the grade on a particular exam, but these were minor adjustments and happened rarely. No rescaling was needed in recent years. In my first year, I asked my department chair what the averages were and was told that C+/B- was typical in General Chemistry. That likely provided a reference point for my exams in the early years, and then it was just fine-tuning to the present day. Nowadays, I can pretty much sit down and write an exam from scratch in a couple of hours and it will likely yield an average grade close to the average in each category. I still keep up the practice of taking my own exams (usually a day or two after writing them so my memory buffer has cleared) and occasionally fine-tune it further depending on how long the exam took me and how much I had to write. (I write up the Answer Key in full when I take the exam; this key is provided to students when I hand back their graded exams.)

What do my data tell me? Somehow over the years, I have subconsciously imbibed an internal standard to exam-writing that seems to work well on average providing a consistent distribution in student grades, at least for the larger Regular sections.

Is the universe designed by an intelligent being? There are philosophical or theological arguments for and against, but I don’t think it can be proved scientifically either way because we honestly don’t know how or what to measure. (Yes, I’ve read many of the claims and counterclaims closely.) Are my exams designed by an intelligent being? Me, I’d like to think so. But there’s an intuition to the way I write exams, now that I’ve had many years of experience, rather than the consciously carefully calibrated scientist-ideal. In the early years, I fussed a lot over details and took a lot longer. Now it seems fine-tuned in a way, but not analytically. Could my intuition be translated into a computer A.I. that generates and grades exams? If so, would it be intelligent? Will it fine-tune through machine-learning? I don’t know, but I hope I’m not replaced anytime soon.

The 19th Century American Potioneer

I’ve started exploring the history of potions, by that I mean using the search terms pharmacy, druggist, apothecary coupled with history to find articles of interest. As an introductory article, I very much enjoyed Gregory Higby’s Chemistry and the American 19th Century Pharmacist (citation: Bull. Hist. Chem. 2003, 28, 9-17).

Higby focuses on the American drugstore. He lists its four roots: “the traditional apothecary’s shop; doctor’s shops – where physicians prescribed and dispensed; the general store; and the wholesale druggist.” Prior to the 19^th century, physicians dispensed medication formulated by themselves or their own apprentices. The doctor and pharmacist was often one and the same; and in the early 19^th century apothecary, potioneers often had a “room in the back” to make their elixirs and potions.

As the industry grew, the selling of pharmaceutical ‘preparations’ became separated from the actual activity of (chemical) preparation. You could buy your elixirs from the general store, where the storekeeper had no expertise (unlike the apothecary). Lack of regulation meant that you might be sold useless snake oil, or worse, something inadvertently poisonous. Adulteration of drugs with other substances was a huge problem – either through insufficient purification of the efficacious compound, or worse, the active adding of other cheap substances to increase bulk. This unfortunately still happens today. Where there’s money to be made, the con-artists will gather.

According to Higby, “the American Journal of Pharmacy was filled with articles long and short on detecting drug adulterations and sophistications. Pharmacists proudly portrayed themselves as the most reliable monitors on drug quality. The problem came to a head during the Mexican War, when newspapers reported a high number of deaths among soldiers from disease rather than combat. Army physicians blamed poor quality drugs. Congress quickly passed the Drug Importation Act of 1848 after lobbying by the young American Medical Association.”

The problem is that the drug inspectors needed to know what the standards should be. Hence, pharmacy colleges across the country banded together to establish such standards; a byproduct of those meetings was forming the American Pharmaceutical Association, composed primarily of apothecaries and druggists in those early days. Over the years, colleges of pharmacy evolved to prioritize actual chemistry labwork over prior apprenticeship in a drugstore, thanks to the dogged efforts of Albert B. Prescott, an analytical chemist, physician and professor.

One thread I found interesting in Higby’s article was the evolution of the drugstore itself. By the late 19^th century, “more profitable goods such as tobacco and candy” took center-stage, and the “shift of pharmacy practice to the back of the shop gave the soda fountain front and center position. Ironically, it was the pharmacist’s practical chemistry expertise that allowed him to make up flavorings and to handle temperamental carbonated water generators.” Food chemists, here we come! It was pharmacists that invented Dr. Pepper, Coca Cola, and root beer! There was a downside, though. “Unfortunately, the public soon came to see pharmacists more as sellers of chocolate sodas than health care professionals.”

Many 21^st century students in my General Chemistry courses come to college thinking they want to go to medical school. Rarely, do I meet a first-year student considering pharmacy school, at least here in the U.S.; in other countries, you apply to undergraduate pharmacy programs directly. (Personal story: I was accepted into the pharmacy program at several U.K. schools through the UCCA process, but ended up going to a liberal arts college in the U.S. instead. At the point of entering college, I did not know that chemistry would become my main interest.) A number of my students have gone to pharmacy school after completing their undergraduate degree; for many of them, the work-life-time balance of being a pharmacist seemed more attractive than being a physician.

Today’s pharmacist is not involved in the making of drugs and potions. At least for regulated drugs, the synthetic work is done by chemists (in research and development) or by robots for larger-scale production. The closest modern-day equivalent of the 19^th century druggist would be the small-scale manufacture of illegal drugs, sometimes for personal use, and sometimes to make money on the side – think Walter White of Breaking Bad in his early days before he breaks badder. He could have chosen Prescott as his moniker to honor the pharmacist, but being bad-ass, he chose Heisenberg!

Where might today’s students experience the art of the 19^th century pharmacist? In the organic chemistry and analytical chemistry lab courses! I haven’t taught either of those, since I’m a theorist and my colleagues would prefer me not too much meddle with chemicals. (I do teach the first-year general chemistry labs.) Instead, I get to show students how to derive the Heisenberg relations in quantum mechanics and dose their minds with the unreasonable effectiveness of mathematics in physical chemistry. You can see the unseen, predict the future, and tame the chaos. It’s a superior mind trip than taking drugs.

P.S. For ways I am like or not like Walter White, click here.

Science: Parable of the Brickyard

Science is like a brickyard.

How so? If you’re a biochemist, you might be thinking about BioBricks. But that is not what Bernard Forscher’s 1963 letter in Science is about. “Chaos in the Brickyard” is a cautionary tale of how the enterprise of science can veer off-course, missing the forest for the trees – or perhaps the branches, twigs or leaves.

That was 55 years ago. Forscher’s dystopian predictions have come to pass, possibly exceeding anything he might have imagined. Like an Aesop fable, the one-page letter is structured as a parable. You will experience the full impact reading it in its entirety, but to whet your appetite, I will just quote the first sentence or two of each paragraph.

Once upon a time, among the activities and occupations of man there was an activity called scientific research and the performers of this activity were called scientists. In reality, however, these men were builders who constructed edifices, called explanations or laws, by assembling bricks, called facts…

It came to pass that builders realized that they were sorely hampered in their efforts by delays in obtaining bricks. Thus there arose a new skilled trade known as brickmaking, called junior scientists…

And then it came to pass that a misunderstanding spread among the brickmakers… [They] became obsessed with the making of bricks… the ranks of the brickmakers were swelled by augmented training programs and intensive recruitment…

And so it happened that the land became flooded with bricks. It became necessary to organize more and more storage places, called journals, and more and more elaborate systems of bookkeeping to record the inventory… the size and shape [of bricks] was now dictated by changing trends in fashion…

The letter concludes that the builders, those who construct edifices, are “almost” destroyed. The reasons are sad, and seemingly inevitable. Here’s the last sentence of the concluding paragraph.

And saddest of all, sometimes no effort was made even to maintain the distinction between a pile of bricks and a true edifice.

Forscher would agree that bricks are crucial to the edifice, and that the making of bricks is an important task. But there might be something rotten in the core of the guild of builders. Is it that the reward system for how one advances in the guild has been skewed? Is it because the quantitative exercise of counting bricks seems ‘easier’ than a qualitative evaluation of… for lack of a better word, quality? We scientists know how to get at tricky measurements – find an appropriate proxy. But after multiple uses of the proxy, it morphs into its own unit of measurement and establishes itself as a rubric – the acid test, to use an analogy from chemistry.

Assessment, previously qualitative and less data-driven, has now morphed into its own science of building. The builders are fewer, while brickmaking is on the rise. A new exalted activity has arisen. Brickmeasuring. It feeds on data from piles of bricks. I can only speak knowledgeably of higher education, as my entire career thus far is in this realm, but there is more chaos in the brickyard than ever before.

Energy Levels: A Game Idea

A couple of Fridays ago, one of the students in my creativity cluster, proposed the idea of a boardgame as an educational tool to learn about energy levels and atomic spectroscopy. This made me hunt around to see if anyone had made such a game. The closest I came was “Orbital Battleship” from a 2016 article in the Journal of Chemical Education. It borrows the mechanics of classic Battleship, but with opponents (presumably students learning chemistry) fill out electron configurations of an atom and try to guess their opponent’s atom by calling out orbital names – checking if they are occupied or empty. I didn’t find it very appealing after reading the rules because, while it might be fun the first couple of games, I predict that interest will wane very quickly.

After the meeting, my mind was still buzzing with the idea so I quickly sketched together the basic idea of a game. Here’s my photo proof of my original idea – I am hoping it will get developed further. I think it’s better than my previous back-of-the-envelope design that hasn’t yet seen the light of day, although I did make some progress with the atom cards over winter break.

My yet untitled energy levels game uses a modular board of narrow strips, each with varyingly spaced energy levels. Players need to get from a start to an end point by playing color cards. The colors represent the number of levels traversed, and they can be played as absorption or emission to move up and down the modular ‘maze’. Inability to play a card leads to a step back via infra-red decay. There are also special ultraviolet cards that allow a larger jump in energy levels than usual. I presented my idea to the group the following week. They liked the game and suggested ideas to refine the rules. One student asked what the goal of the game would be. I hadn’t thought about the theme yet, but off the top of my head I suggested ‘electron nirvana’. The electron finds its destination outside of the influence of the atom. I suppose it could go the other way, i.e., the electron from afar finds its true place in its quantum atomic universe.

This past Friday, which was our last meeting of the semester (since we’re moving into finals week), my student who had first suggested the idea, presented a prototype with strips of cardboard and paper cards, illustrated with colors! We didn’t actually play through a game but talked about what the next prototype should like as we refined it. An eraser substituted as a player piece (the electron eraser, I suppose). In the excitement, I forgot to take a picture of that first prototype. Argh!

I think this game idea has actual promise, and I now feel motivated to work on it as summer approaches. My students are looking forward to some playtesting. We talked about game balance, timing, card distribution, etc., and it made me realize that I was being creative in two areas where I have a fair bit of knowledge: chemistry and gaming! Hopefully we’ll have an updated prototype sometime in the summer.

P.S. An unrelated blog post about managing energy levels.

P.P.S. Electrons magically apparate when they traverse energy levels. So says Bohr.

P.P.P.S. Could Orbital Azkaban be a game? Hmm... I'll have to think about it.

Qualitative Formative Assessment

This week I read a thirty-year old article by D. Royce Sadler on formative assessment, with a focus on qualitative and multiple-criteria approaches. These are situations where simple standard rubrics are found severely wanting and “warp teaching and assessment”. Creativity and complex learning outcomes, with less well-defined criteria, are among those situations. They are not just limited to qualitative judgments in the fine arts and humanities, but also in “science and mathematics, where students are required to devise experiments, formulate hypotheses or explanations, carry out open-ended field or laboratory investigations, or engage in creative problem solving.” (Citation and abstract are shown below, accessible from JSTOR.)

Sadler defines a qualitative judgment as where the assessor is “both the source and the instrument for the appraisal”. The assessment therefore cannot be reduced to a formula or a rubric that could easily be used to check-off criteria. Unfortunately, the push for measurable or quantitative assessment either ignores these complexities or an unsatisfactory proxy is chosen. We the faculty, subject-matter experts in tertiary education, aim to guide our students into depth and complexity in our fields; and we should push back strongly against the tide that focuses on things that can be counted easily.

There are five characteristics where qualitative judgments apply according to Sadler: (1) when multi-dimensional criteria are required and their inter-relationships are important, (2) when criteria are fuzzy and continuous, (3) when the criteria pool is large and comprises both manifest and latent criteria, (4) when there isn’t a single objective independent measurable standard for comparison, and (5) “the final decision is never made by counting things, making physical measurements, or compounding numbers and looking at the sheer magnitude of the result.” In fact, our simple rubrics are often an attempt to do #5, and Sadler warns specifically against this.

Two things that Sadler then discusses in detail are what may constitute effective feedback from instructor to student (or expert to novice), and how students might practically benefit from the feedback – essentially, learning how to learn, as self-monitoring approaches are honed by the student. I recommend reading the article in detail because Sadler does an excellent job setting up the conceptual framework for his ideas and provides good examples to illustrate the theory. Instead, I want to briefly mention three ancillary things that jumped out at me from the article.

First, it can be useful to think of the community of subject-matter-experts as members of a guild. Members of the guild instinctively recognize quality broadly in our fields, even if it be in different shades. If asked to articulate the criteria that define quality, we might have partial and differing lists from one another (I mentioned above that criteria can be manifest or latent), but we might not be able to agree easily on a simple explicit rubric or checklist to assess that quality. In the guild system, the “novice is, by definition, unable to invoke the implicit criteria for making refined judgments about quality. Knowledge of the criteria is ‘caught’ through experience, not defined… [through] a prolonged engagement… shared with and under the tutelage of a person who is already something of a connoisseur.” This is why many of us scientists at liberal arts colleges consider the undergraduate research experience to be crucial for a high-quality education of our majors, and many of us embrace a guild-tutelage model in our research labs.

Second, I was struck by Sadler’s claim that “some teachers feel threatened by the idea that students should engage openly and cooperatively in making evaluative judgments”. He provides several plausible reasons for this: loss of control, undermining expertise authority, and others. In the sciences, especially at the undergraduate course level, we still gravitate towards the “one right answer”. In Fall 2016, I experimented with take-home exams in my first semester general chemistry course. It had its pros and cons. Reading Sadler’s article made me think of adding a student self-evaluative piece to the take-home exam. After taking the exam individually (closed-book and with time constraints), students must evaluate their own performance, by going through the exam again but may do so collaboratively or looking up resources. I think I’ve come up with a workable scheme to try next semester with a new group of students.

Third, Sadler argues that continuous cumulative assessment for grades may actually subvert or hinder the benefits of formative assessment. Let me explain. In most U.S. college introductory science classes, the bulk of the grade comes from midterm exams (2-4 throughout the term) and a final exam; there might be some homework or quiz grades throughout the semester too. The student’s final grade comes from summing up all these mini-grades. I grew up in a different country with a different educational system, where the final (nation-wide) exam was the end-all. Nothing counted but the final performance. Both systems have their pros and cons, but perhaps there is a way of maximizing the pros and minimizing the cons in a hybrid system. I have some ideas of how this might work, but I think I will float them to some students I’ve had in the past, and see what they think.

If formative assessment interests you, the appropriate bible is Dylan Wiliam’s book. In my opinion, it’s one of the best books out there on this subject. I highly, highly recommend it. The figure below is taken from Greg Ashman’s blog (also recommended) and his book Ouroboros – the sampler chapter on Rubrics is available.

Life After Death: Stiff Version

If anyone can tackle the subject of the myriad things that happen to cadavers with humor and grace, it has to be Mary Roach. Her book Stiff, published back in 2003, is subtitled The Curious Life of Human Cadavers. I only recently discovered Roach’s blend of writing in mid-March when I read Gulp. Based on just that one book, Roach shot up to my top five favorite authors. Since Gulp was her most recent book, I decided to go back to her first book. Stiff does not disappoint. Her humor and daring is exquisitely balanced with the most interesting facts and research you’ve never heard.

I learned about surgeons practicing cosmetic surgery on heads, and how respectfully they (the heads) are treated, and the protectiveness of the chief decapitator. There’s more decapitation if you want to learn about experiments involving head transplants. Early anatomists partnered with body snatchers, and even William Harvey (discoverer of the human blood circulation system) performed dissection on his own father and sister. Some physicians even dug up corpses themselves. Cannibalism, cremation, and composting are more fascinating and complex than I knew. Yes, composting dead bodies is a thing. At least in Sweden. And the argument is environmental. Wouldn’t you prefer an ecological funeral? Or you could donate your body to science – it might help the army test weaponry or car manufacturers test crash damage, among many other uses.

My favorite chapter, though, is titled How to Know if you’re Dead. It turns out there was quite a bit of controversy as to whether the inactivity of the heart or the brain should be used to define when someone is dead. In the U.S., it’s brain death. Amazingly, this legal definition only started in 1967, first in Kansas. And at least in 2016, there was still significant variability in individual institutional policies. I suppose it makes sense that the heart was used as the measure for a long time, since brain activity was more difficult to measure.

That being said, before an effective stethoscope was invented, it was sometimes hard to tell – especially if someone had a very faint pulse. Thus, the fears of being buried alive. I actually learned in a seminar that the last line of Alfred Nobel’s 1895 will (which is famous for endowing the Nobel prizes) reads: “Finally, it is my express wish that following my death my veins shall be opened and when this is done and competent Doctors have confirmed clear signs of death, my remains shall be cremated in a so-called crematorium.”

Besides the heart/brain controversy, Roach also tackles the question of where the soul is located. I learned that while the ancient Egyptians favored the heart, the Babylonians favored the liver. On the other hand, “the Mesopotamians played both sides of the argument, assigning emotion to the liver and intellect to the heart. These guys clearly marched to the beat of a freethinking drummer, for they assigned a further portion of the soul (cunning) to the stomach.” Descartes picked the pineal gland and an Alexandrian anatomist named Strato picked “behind the eyebrows”. Edison apparently thought that there existed “smaller-than-microscope entities that inhabited each and every cell and, upon death, evacuated the premises, floated around awhile, and eventually reassembled to animate a new personality.” Midichlorians, anyone?

The movie 21 Grams takes its title from the premise that the body loses 21 grams of weight upon death – as the soul departs the body. Roach discusses the early experiments of one Dr. Duncan Macdougall who installed a special bed in his office connected to a sensitive weight balance. He was quite the experimenter. After observing a loss of three-quarters of an ounce, the good doctor tried to rule out in scientific fashion other sources – evaporation of moisture, bowel or bladder movement, residual air in the lungs, etc. He then apparently tested these on dogs (it is speculated that he poisoned them) and found no weight change. His conclusion: Dogs have no soul while humans do. The scientific community at present thinks the early experiments were flawed, and that there is no evidence for the “21 grams theory”.

In the same chapter, Roach discusses the curious “phenomenon of heart transplant patients’ claiming to experience memories belonging to their donors”. They’re strange. Very strange. You’ll have to read Roach’s book if anything I’ve said so far sounds interesting. And I’ve only discussed snippets from one out of twelve chapters. Stiff is a breezy read and very engaging. I highly recommend it. Let me close this post with a quote from the author Burkhard Bilger praising the book.

“Mary Roach is the funniest science writer in the country. If that sounds like faint praise – or even an oxymoron – there’s proof to the contrary on almost any page of this book. Stiff tells us where the bodies are, what they’re up to, and the astonishing tales they have to tell. Best of all it manages, somehow, to find humor in cadavers without robbing them of their dignity. Long live the dead.”

Appropriately, I'm looking forward to reading Spook next. Science tackles the afterlife, apparently.

Orbital Azkaban

We’ve been thinking a lot about orbitals in my creative cluster. My students brought up this topic as one that was murky in their minds when they took first semester general chemistry. They had no problem drawing the shapes of orbitals, ordering them according to energy, and writing ground state electron configurations. However, the finer points – thinking about nodes, probability distributions, how all these orbitals are ‘stacked’ (superimposed) on each other, what the electrons are even doing – these they thought were rather confusing. Since they’re sophomores they haven’t taken Quantum Chemistry yet. (They are looking forward to taking Quantum with me next semester, or at least that’s my impression!) Is there anything we can do to help students at the general chemistry level gain a better conceptual understanding of orbitals?

As we started discussing nodes, someone in the group (most likely me) used the analogy of teleportation to describe how an electron ‘travels’ between the two lobes of a p-orbital while avoiding the nodal plane. We talked about how it is difficult to pinpoint exactly where an electron is located – perhaps it disappears and reappears from view. Inevitably Harry Potter makes an appearance as we discuss Apparition as an analogy. My memory is hazy at this point, but possibly the Room of Requirement came up in the context of the Marauder’s Map. Several moments later, one of the students was sketching the Quantum Marauder’s Map on the white board.

We started thinking about how rooms in a castle might represent orbital volumes. Given our limitation of whiteboard-flatland, we first worked in two dimensions. Footprints signify an electron (a quantum wizard!) and the floor-area of the room bounded by the walls represented orbital boundaries. Each color represents the same individual apparating between different ‘allowed’ rooms. Our simple floor plan has no overlap between rooms. Individuals never meet each other. They’re sort of like prison cells. Orbital Azkaban. Voila!

How might we conceive Orbital Azkaban in 3-D? We decided that some rooms must be circular to represent s-orbitals, possibly with concentric rings since the larger s-orbitals have circular nodes within. (Note we’re still working with partial 2-D constructs, as technically these should be spheres rather than circles.) The planar nodes of p-orbitals could be different levels in a multistory building where individuals apparate between levels. You have to jump between levels in quantum world. We then came up with what looks like a multi-level, multi-ring Bundt cake (see the lower right drawing in green), a panopticon prison of sorts, where the guards hover in a nucleus at the center (similar to the Kyln interstellar prison in Guardians of the Galaxy).

But orbitals overlap each other in an actual atom. In these superimposed mess, the ‘walls’ of each cell are different for each prisoner even though they occupy shared space. Electrons also ‘avoid’ each other for both electrostatic and symmetry (Pauli principle) reasons. How does one keep a prisoner within a certain space without actual physical walls? The first thing that jumped to mind was an electronic ‘collar’ akin to dog collars that prevent them from wandering beyond certain confines. A morbid idea when applied to prisoners. To make the idea less repulsive, we started thinking about Orbital Azkaban as a zombie trap (hence the title of bundt-layer-cake structure) except that this prison would not have physical walls. Different types of zombies with different levels of danger could represent different orbitals. The zombie collars confine these creatures to certain spaces although the zombies can teleport into allowed areas.

If I were to return to the Quantum Marauder’s Map picture, what might it look like as we start to conceive of orbitals in the same space?

The 1s orbital is a small circular room confining the prisoner.

The 2s orbital is a larger circular room. The prisoner spends most of the time in the larger outer ring, but occasionally teleports into the inner circle.

The 2p orbital is two adjoining circular rooms that ‘touch’ at a single point. The prisoner spends an equal amount of time in each ‘lobe’, apparating between them.

And here’s what it looks like when the three different prisoners are put together. There are no actual physical walls. And in fact the boundaries represent only a 90% chance where the zombie-prisoner is found. (Orbital boundaries are typically defined in the same way: There is a probability of 0.9 that the electron is within the enclosed volume.)

Each ‘orbital’ can take up to two prisoners. (Only one prisoner is shown in each orbital.) These two prisoners in the same orbital will mildly avoid each other as they move around in their respective areas/volumes. This represents the electrostatic repulsion between negatively-charged particles. Prisoners from different orbitals with the same gender will avoid each other in a different but more significant way representing the Pauli Exclusion Principle. (An electron may be either spin up or spin down. I’m going to represent it by a gender binary for simplicity’s sake even if it might be less PC.) It’s a little tricky to describe what exactly this avoidance means without going into the mathematics. Technically, the overlap integral (calculus!) of the two orbitals must go to zero. The Pauli avoidance only applies to electrons of the same spin. I haven’t attempted to represent these complications pictorially, because I think it would make things more confusing.

This exercise reminded me about the limits of models and analogies. They help to a certain extent… until they don’t. Mathematics allows for a level of abstraction beyond any picture we can draw or ‘see’ with our mind’s eye. I tell my students that math is their friend in this regard, and that’s why it’s worth learning. Every time I teach Quantum, one or more students will tell me: “I understand the concepts, I just can’t handle the math.” My response is that: “If you don’t understand the math, you don’t actually have a good grasp of the concepts.” That’s the nature of quantum mechanics. Feynman was right – no one really understands it. But the math helps. A picture of Orbital Azkaban only gets you so far. The mathematical framework gets you further and beyond.

P.S. If you’re wondering who the dolphin-whale is on the right side of the white board, it’s a student rendition of one of my wild ideas earlier this semester.