It was the day of the Lord 12 February 1989, such as they count in the Old World. I was in the Old World that day, which I know precisely, because I found a trace of it in my keepsake box. Two days before Valentine’s Day of that year which brought about the Fall of the Iron Curtain, I was sitting in a giant lecture hall at the Rheinish Westfaelische Technische Hochschule Aachen (short, RWTH) for the midterms in Theoretical Quantum Mechanics. In German, that’s called a Klausur, which is the local spelling of a Latin word that means, “sequestration.”
It was the most important midterm of my career: I knew I had no business in Experimental Physics, and getting into Theoretical Physics was competitive. The first course in Theoretical Physics, Mechanics, hadn’t shown a whole lot of separating powers. But the second one would coincide with the Bachelor’s Degree, which meant professors were looking at the outcome to pick the students they would mentor.
It happened to be the year that Professor K. taught the class. K. was the dean of the College of Physics and an incredibly respected name. His father had been a famous exponent of the Copenhagen School of Theoretical Physics and the son – not as distinguished, but still a powerful force – had been groomed since birth to become the leader of the Physics movement. I am not kidding: his name was Hans-Albert, which dad had borrowed from his colleague, who had named his own son Hans-Albert… Einstein!
Impressing Hans-Albert (I feel free to call him that, two and a half decades later) was thought to be of fundamental importance. It also was my only chance of getting into the Theoretical Physics circus made famous by Sheldon Cooper on Big Bang Theory. You see, when I watch that show, I recall my own college years: Sheldon is the Theoretical Physicist that gets to look down on Experimental Physicists and to pee on the shoes of the Engineers. That was us, in the day.
By the time of the Klausur, half the students had already abandoned the course. You needed it to get your B.S., of course, but you might want to take a professor with a less fundamentalist approach. Hans-Albert was many things, but easy to understand was not one of them. Didn’t help that his general attitude was “you should already know that” whenever someone asked a question in class.
Still, the lecture hall was full enough that the TAs had the hardest time seating us sufficiently far apart. Ideally, they wanted an empty row and two empty seats between any two students. That time, though, we were far too many to do that. Hans-Albert mused about splitting the Klausur in two, until the TAs (mindful of their own free time, no doubt) pointed out you can’t just force half the class to show up again. For the same exam.
Concessions were made. People willing to sit in the front row would get to sit with only one empty seat separating them from a random other student. Watched like hawks, of course. People in the middle and back rows would be spaced farther apart, with no chance of interaction at all.
When I got into the hall, I was eyeing a seat in the middle. Don’t stand out! was my motto. Hans-Albert saw me. Hans-Albert didn’t like me. I had been seen in the Fachschaft, the student association. That automatically marked me as an incompetent troublemaker. That’s because the student association was a constant thorn in the administration’s workings, since the constitution of the university guaranteed student participation in all processes. When Hans-Albert wanted to hand out a spot on the tenure list to one of his cronies, he didn’t need buy-in from the students – we had only two votes to the combined 50 of the professors – but he needed to tell us what he was doing.
Of course, Hans-Albert didn’t think about it that way. He thought of the students that were a thorn in his eye simply as the kind of people that sat on gremiums they didn’t belong in and that said things that sounded stupid and naive. I’ll grant you, much of the time that was exactly what we did. But I wished Hans-Albert hadn’t actually thought of me that way, in that moment.
That’s because Hans-Albert decided to move me. From the middle of the rows to the very last row, in the far corner of the far-side aisle. The move was very visible and entirely unexplained (to the other students). What it meant was that I had only one other student in my general vicinity, the guy to my down and left (since I was on the top right corner). I wasn’t going to be able to copy anything, Hans-Albert seemed to suggest. And if I did, it could be only from one person. Easy trace.
The other students had gotten the message: I was singled out because I was a notorious copycat cheater. I wasn’t going to stand a chance, and I was going to fail this exam. After all, why would Hans-Albert do something as visible as that forced move?
I didn’t panic. I knew why he did it, and I knew it was no reflection of my level of preparation. Also, I love proving people (particularly including myself) wrong. Finally. I absolutely loved quantum mechanics. It was a beautiful theory that surprised everyone by actually working for no reason. It made wild statements about nature that made no sense, and it still worked better and more accurately than anything we had ever known prior to it. It was the triumph of creativity over reality – exactly what I stood for in my own mind.
You see, another thing that was gnawing at me was that I was not what other people perceived me to be. In particular, I didn’t look like a theoretical physicist. To a German, I didn’t even look like a physicist. Or a college student. Or a high-school graduate. To a German, I was (and am) an Auslaender, a foreigner, which has similar connotations to the stereotype of a “Mexican” in the United States. Only that culture is more important to Germans than to Americans, and hence being stereotyped as uncultured and ignorant was a much worse stigma.
In any case, being in the Fachschaft and looking like I didn’t belong in the first place must have made the nerves of good Hans-Albert tickle. After all, “Belonging” was his whole life – why should someone that clearly didn’t belong get a chance? It’s not that I think he was mean, it’s that I think he would be the one person that would have the hardest time understanding that looks can be deceiving.
I settled down and waited to get my three white sheets with the questions. The TAs handed them out separately, one sheet per TA, starting at the bottom left (you guessed it). I was the last person to hold the three sheets in my hand.
The questions, mind you, couldn’t have been much shorter. Here the translation:
Let `E_1` and `E_2`, `E_1 ne E_2` be eigenvalues of the stationary Schroedinger equation and `u_1` and `u_2` their associated normalized eigenfunctions. At the time `t=0` the system is in the state given by:
`Psi (vec x, t=0) = cos alpha u_1(vec x) + sin alpha u_2(vec x), 0<alpha<pi/2`
a) What is the probability that after measuring the energy at `t = 0`, the system is found to be in the state `u_1`?
b) Determine if there is a value for `a` and a time `tau > 0` for which the state `Psi(x, t= tau)` is orthogonal to `Psi(x, t=0)`. If there is, determine the minimum such value.
c) What is the probability of finding the system in the state `u_1` at the given time `t>0`?
Prove that the inequality
`E_n geq 1/2 ℏ omega`
for the eigenvalues of the energy of the harmonic oscillator
`H = 1/(2m) P^2 + m/2 omega^2 Q^2`
can be derived from the uncertainty relationship
`Delta Q Delta P geq 1/2 ℏ`
and the properties
`(u_n, Qu_n) = 0, (u_n, P u_n) = 0`
Prove that the equalities above are consequences of the invariance of H to mirroring (`Q rarr -Q` and `P rarr -P`).
What are the eigenvalues and eigenstates if the Hamilton operator of the harmonic oscillator, `H = ℏ omega (a^+ a + 1/2)` is replaced with
`H = ℏ omega (a^+ a + lambda a^+ lambda^(**) a + 1/2), lambda` complex
I read the questions and I knew what to do. Furthermore, to my utter astonishment, all three questions were very straightforward. Too straightforward, I thought. I agonized for a long minute: was I missing something?
I gave up. The agonizing part. I just started thinking. Didn’t write down anything yet. For a good ten minutes, I was just letting my mind wander to the solutions. I added another ten minutes to make sure I had thought correctly. I didn’t know the solutions at that point, but I knew what I had to do and where I had to go.
Of course, by then Hans-Albert had already seen a student that didn’t belong sit there for twenty minutes solid without as much as putting the pen to the paper. To Hans-Albert, I was vindicating his prejudice.
Then I started putting pen to paper. It was a fountain pen, in blue ink. That matters because you can see where I made changes: I used an ink eraser and wrote on top, which happened exactly twice in the entire set of solutions.
Despite it consisting of three parts, my solution to the first question was dispatched on one sheet of paper, with only three lines on the second page. That’s even more impressive if you count that every sheet had to be labeled with name and student ID number.
It looked good. I had even found that in part c), the probability of finding the system in the state u1, was constant. That was quite unexpected, since you would generally assume it also depends on the time. It was the kind of thing that told me Hans-Albert had been playful in a very Sheldon Cooper-ey way. Bazinga!
The second question, another long one on the assignment, was dispatched of just as quickly. I made one mistake (putting down Ψ instead of un, since Ψ is usually used for the function) that I simply struck through and another (a mistake in an integral) that I wiped. Otherwise it reads like a Mozart version of a Theoretical Physics exam. Dictated by the angels to an unworthy mind.
When I got to the third question, I was stumped at how short it was. It was obvious what the changes were to the harmonic oscillator, but I couldn’t race through to the solution in my own mind. I didn’t know intuitively what the change would bring about. I needed to calculate.
So I did. It went on forever. Well, really, it was done with four inches to spare on the sheet of paper. I wrote it in one go, without thinking, because at that point all I wanted was to see what would happen with that transformation. My former self’s version of a nailbiter.
Turns out all it did was shift the eigenvalue down by the absolute value of λ. Gee, I should have known that. Another nifty trick out of Hans-Albert’s playbook. But by now I was in the mood to do some theoretical physics. I noticed that the equation had an analytic solution at the lowest eigenvalue. Nobody asked about it, but I just decided to calculate it. Then I realized that my solution was not normalized, so I did that extra. Nobody asked, again.
I was done with everything in under an hour of the three hours we had. I checked everything, tried to help the guy to my down and left, who was hopelessly lost (which was probably interpreted as going the other way around). Then I checked again and again and couldn’t find a mistake.
I saw everybody else spending time and paper. There were sheets full of notes stacked an inch high on most desks, desperate pleas for help to the TAs, attempts at sneaking a peek at notes. People were frantic, scribbling on and on. It was obvious this was a difficult exam. The second midterms and finals were the only way to the rescue, because there was no chance this was going anywhere..
It’s at the one hour mark that I realized I wouldn’t be able to figure where I made a mistake. That was the thing: it was too easy. But I didn’t know what I had gotten wrong, and all my results made perfect sense in a very deeply physical way. Even the Easter eggs I found planted by Hans-Albert, like the fact that the probability in time didn’t actually depend on the time, or that the changes in the harmonic oscillator simply shifted the eigenvalues, were just the kind of thing that tells you you are on the right track.
One hour was gone. Two more hours to go. I was terrified, and I was bored. I didn’t know what else to do. So I stood up.
The walk down, it became clear to me, was a walk of shame. Not showing up to the exam required a note from a doctor or a plane ticket, so we had to hand in our question sheets even if we didn’t have a solution, to prove we had been there. So the entire class saw me walk down after an hour and hand in a stack of paper so thin, it could only be a complete admission of defeat. I had handed in my resignation. Order had been restored.
Hans-Albert was gloating smugly. I put my stack on the desk, first one to do so. He didn’t even look at it. Admittedly, he couldn’t tell I had written anything, it really looked like I had given up and walked away.
I was out of the lecture hall and went for a coffee and a treat. I felt queasy, the kind of queasy you feel when a ton is at stake and your presumed success sounds too good to be true.
None of my colleagues made it out of the hall before the time was up. I was surprised and slightly panicked. How was it possible it had taken them so much longer to finish the questions?
The days after the Klausur were even worse. The word had spread that I had totally flunked the exam to the point of giving up after a single hour. Sure, I claimed that I had finished everything, but nobody believed that. I wasn’t particularly remarkable, and even our numerous Sheldons didn’t finish in time. Plus, everybody had seen me handing in nothing!
Hans-Albert was a cruel man. I guess he thought of it as some kind of public reward/humiliation ceremony, but instead of having the TAs hand out the corrected exams in homework group, he did so in class. (If I thought that was a particularly wasteful use of time, I would have been presuming that the rest of the semester would have been more informative. Alas, it was entirely about scattering experiments, one of the most soundly boring applications of quantum mechanics you could imagine.)
After the first few beaming students were called to the front, it was obvious that H-A was handing out the results in descending order of grade. Our Sheldons were richly rewarded, as all eyes were transfixed on them walking to the front while the rest of us were trying to anticipate who was going to be next, or were surprised when someone did better than expected.
I wasn’t anywhere near the top. Well, I told myself, it was clear you misunderstood something. You probably got one of the questions wrong. I was wondering what it was, and wondering why we hadn’t actually talked about the questions in study class. It is true that the professors recapped the exam only when they handed out the grades, but it would have made the wait much easier to handle if we had already known what was right, and hence what would be graded as wrong.
Each question granted 4 points. A total of 12. By the time Hans-Albert handed out the 8s, I was starting to wonder what I could have gotten so wrong. I understood that I might have completely misunderstood one of the questions, but anything below meant I had misunderstood two. At that point, anything might have been possible. I might have gotten all of them wrong.
Hans-Albert got to the 6s. My queasy feeling started turning into a riot in my intestines. I would have run to the bathrooms, had it not been for the fact that I was absolutely certain I was going to be called next. Which never happened. By then, my study group (all of them Sheldons) had turned their bemused smirch (haha! you are not as smart as you think!) into a look of horrified puzzlement. I am sure they thought themselves potentially contaminated by the ignorance that might be spreading from me. I was already kissing my study group sessions good-bye – one of the few social outlets of the day for me.
Fact is, the way they set up the system, you really didn’t have to have a minimum grade at all. All you had to do to pass the class was to get 12 points total on the two midterms. Even if you had only 1/2 a point in the first midterm, you could theoretically pull off the semester even with just an 11.5 in the second. Heck, even with a zero you could pull it off, if you got a perfect score on the second.
So, Hans-Albert went on. His initial smile turning into a disapproving frown the farther down he went. I felt humiliated, betrayed by my own confidence. I hated every second I had slaved over the textbooks, over the lecture notes: how could they all lead me so astray?
Hans-Albert was down to the 4s. The horrified look on my friends’ faces, who all had their grade in hand and hence had time to waste, turned into pity. They knew what this debacle meant for me: I, who had been aspiring to become a Theoretical Physicist, was likely to abandon not just theory, but the science altogether. I wasn’t cut for it, especially after I had adamantly claimed I had actually turned in the exam completed.
Zero. There were quite a few that had zero points. Hans-Albert showed no mercy, letting them come to pick up their exams in a display of contemptuous consistency. Admittedly, at the zeros people didn’t quite seem to care. They probably already knew things hadn’t gone well, and most of them probably didn’t hand in anything to be graded, as everyone assumed I had done, too..
There were good reasons why we had so many zeros that semester. Hans-Albert was not very congenial, and his lectures were incredibly theoretical. I loved them for it, Statistical Physics even more so than Quantum Mechanics. He seemed to pull reality out of random quirks of math. Like the fact that the quanta of certain systems were determined by the fact that solutions had to be real numbers and not complex ones.
But that wasn’t what most students were used to. After all, most of us would end up in Experimental Physics, where theory was just used to determine what experiments to try. So the fact there were all these things in his lectures that were entirely unmoored in actual reality was disturbing and hard to learn.
Zero. Hans-Albert had handed out the last of the zeros. The stack in front of him had vanished, leaving the teal desk exposed.
At that point, things got weird. I mean, OK, I got a zero – but why wouldn’t you actually hand it to me? It couldn’t be mercy, since there had been none shown to anyone. Maybe my exam had gotten lost. Maybe it had been soaked in coffee? Maybe it had been blown away by the frigid West winds blowing from the Atlantic?
One of the Sheldons (real name Joerg) made a gesture towards me. It seemed to say, “Do something!” So I raised my hand. Hans-Albert didn’t care. One of the TAs walked over to him and pointed me out when it became clear I wouldn’t give up. The remainder of the class just wanted to get the professor’s official explanation of the exam question, so they looked at my interruption with open hostility. What good question could possibly come from someone that hadn’t even gotten the exam results?
I asked, as plainly as I could muster, “What happened to my exam?” Hans-Albert did his trademark blinking, followed by his equally trademarked puzzled look. Then he turned to the TAs, as if to ask who I was and who was responsible for my presence in his lecture. Then he turned around and asked for my name.
I told him my name. Instantly, I realized that by virtue of doing so, every single student in the lecture hall knew my name. All of them knowing that I was the guy that had given up after an hour, that I was the guy that had not gotten his exam back. I wished I hadn’t said anything and used back channels to figure out what happened.
When Hans-Albert heard my name, his look switched from surprise to revelation.He called me to the front. The humiliation would be complete. He had me stand right by him, next to the projector (one of those slide projectors of pre-modern days), facing the students. I was about to cry.
Silence befell the lecture hall. We were quiet even on a rowdy day, but that moment was so still, you could hear the fan of the projector all the way back to the last row. All eyes were fixed on Hans-Albert and me, and my colleagues were probably thinking the same thing as I, “What is Professor K. going to do to Marco?”
When Hans-Albert was sure he had everyone’s attention, he said, “Students, look at your colleague. You must remember this face, because one day he is going to be a great Theoretical Physicist. You will all be able to say, I studied with – what’s his name – Marco Gazzetta in Professor K.’s Quantum Mechanics class.”
He turned to me and said, “Here is your solution on slides. You did perfectly. You have a perfect score. Please tell all other students what you did.”
I froze up. I couldn’t speak. Hans-Albert quipped, “Not all scientists are great explainers!” The audience laughed, mostly because Hans-Albert himself had that reputation, which he might have known.
He sent me back up and used my slides to explain. I was so shellshocked, I couldn’t even turn to see what other people’s faces looked like, but they were probably as surprised at the sudden turn of events as I was.
When we got out, everybody surrounded me to congratulate. People I had never spoken with cheered me like I was some weird sort of magic wizard who mastered soccer playing and was being sent to the World Cup. I was a superstar. My 15 minutes of fame had arrived.
Things didn’t turn out as magic as they seemed back then. I was so bored with the second half of the semester that I totally bombed that exam (4 points). I did well in the degree program, earning top grade. But I realized there was no future in Physics for me. The days of quanta had long gone, and math had nothing new to offer to the theoretical physicist. I switched to computers, that were and are much more exciting.
But I held on to that exam paper for 25 years. A quarter century later, I still see myself sweating in Professor K.’s class, thinking I’d be forced to admit that physics, which I so much love, was just out of my reach.
[Note: This anecdote is actually the combination of three separate events, of which only one involved Professor K. Fiction, my dear friends, is much more interesting than truth.]