I’d run into these learning roadblocks before. I remember struggling almost helplessly with Abstract Algebra. I’ve written before how I started using the local library as an oasis of peace and calm in an attempt to get to the heart of the topic and gain understanding and ... frankly ... just get the homework done.
Another time I got way behind my class in a physics course and spent a long weekend in the mountains camping with Linda and Mike. They fished while I did every problem in my four inch thick “Physics Problem Solver” book.
Then there was the time I was first learning electromagnetic theory and struggling with div, grad, and all that jazz. In this last case several meetings with an excellent professor finally showed me the light. If you didn’t understand the math, you would never figure out the physics.
So this is not a new feeling; that “I don’t understand any of this stuff” combined with the feeling in your gut like you're on a roller coaster ride and the bottom just fell out. Been there...done that. So I certainly recognize the current circumstances. One difference is that, this time, since I’m taking the course online, I can repeat the lecture. In fact, I’ve repeated some of them three or four times. But if you can’t understand what the guy is talking about, listening several times doesn’t help that much.
So I’m trying to diagnose my current problem and seeking a solution. I’m thinking about the other times in my life when the problems zoomed over my head like one of those government drones, and I just could not grasp what is going on like waking from a dream and trying to figure out what the dream was as all recollection slips away. Even when I do make progress, it is at a much slower pace than I expect, and it is quickly ruining my plans and schedule.
I certainly know when the trouble started. I’m currently in the middle of an about two-year review of physics in preparation for taking my PhD Qualification Exams. The Classical Physics went well, even if I had to remember a lot of math that I’d happily forgotten in the last twenty or more years. I’ve gone over a giant list of common integrals and especially sweated out the Trigonometric Functions and their common Integrals in an attempt to sharpen wits that have dulled with age and time. But since everything is “open book,” it hasn’t been a failure of memory or even process. Sure the math is hard and I’ve forgotten more than I have remembered, but that is not what is inhibiting my success now. After all, I’m well prepared for that with a Master’s Degree in Mathematics, even if it has been a lot of years since I practiced much of this stuff.
No, that isn’t the problem. One reason I’m writing this note is to try to diagnose the problem. This is sort of a conversation with myself, and you all are welcome to listen in if you like. I’m not sure what is giving me the difficulties, but the great clue to the solution is the fact that I didn’t struggle like this in the Classical Physics subjects. It was only when I started this current Quantum Mechanics course that I seemed to have lost my way.
The mathematics of standard Quantum Mechanics is no more difficult than the math in classical physics topics and is very similar to electromagnetic and field theory that I’m pretty good at ... albeit that that was a topic I struggled with initially years ago.
I believe the problem may be in how I understand things and how I learn new subject matter. New ideas and concepts are always added to and built upon what you already know. New knowledge is added to old knowledge like building a multiple story building. I used to tell my students to take each new fact and idea and connect it to the web of information they already possess. I imagine a spider web like structure of interconnected facts and data which, taken as a whole, gives knowledge and understanding. That’s how I’ve always learned.
In this case I do have the fundamental ideas. And, quite frankly, Quantum Mechanics isn’t that different or strange to me. I grew up in a world of science that quantum theory was a familiar topic. It is the starting point that I may be struggling with. After all, Quantum Mechanics is not that different than most field theory and QM uses the Hamiltonian just like Classical Mechanics and wave equations similar to standard field theory. I think it is the conceptual understanding that is giving me the most problem.
Of course, I’m not the first to struggle with the fundamental concepts. The history of the development of QM is filled with brilliant scientists that struggled with the concepts. Even the genius of Albert Einstein, a great man of visualization and "gedanken" experiments, could not agree with all the Copenhagen Convention and he supposed hidden variables to understand and eschewed probability based explanations, no matter how well they seemed to describe and predict what was happening.
Even the use of the statistical Probability Density Function, which should be quite familiar to me, is a struggle because it is not a probability that an electron is located at a specific location. Rather, in some mystical and almost non-scientific way, the electron is located simultaneously at all the locations. It is not a “cloud OF electrons,” but a "CLOUD of electrons." They seem to be “smeared” out across the atom and they get “shared” and can even “tunnel” through impossible barriers...at least statistically, and I know that works from my introduction in the 1960’s to tunnel diodes that so brilliantly implemented the underlying theory in a practical application.
In classical mechanics it is quite possible to have concrete mental visualization of a situation that you then describe mathematically and solve for results. After all, Newton is said to have equated an apple falling from a tree to the moon "falling" around the earth in his discovery of the rules of gravity.
I just can’t seem to visualize the underlying concepts of QM. The topic uses familiar terms such as particle, wave, spin, energy, position, momentum, etc., but they are all in such a new and mysterious way that I don’t grasp them. Sure an electron has spin like a planet or moon, but what in the world is “integral spin” and a “spin of 1/2”? It is boggling my mind.
Sometimes I just wish the early pioneer scientists had chosen brand new terms like “Quark” rather than use familiar concepts like “color” to describe something that has nothing to do with light, or at least rainbows.
I’ve studied QM before and I’m well aware of the mystical interpretation from books such as the “Tao of Physics” or the “Dancing Wu Li Masters.” But these were books written for laymen and you didn’t have to take the understanding and apply it to difficult matrix mathematics and time-dependent Schrödinger equations. I suppose one could divorce the mind from understanding and just learn the math, but my mind doesn’t work that way.
I want to “know.” That’s why I’m taking these courses and beating my head against the books. I want to know and understand, not just “do.” So, assuming that is the problem, what is the solution?
Is it possible that I’m just too old and my mind has lost the flexibility to grasp these new and amazing principles? Could be, but that’s a big cop-out. Maybe I lack the fundamental knowledge that would be the basis for the new facts and understanding. Maybe my spider web lacks the tie points to connect the new information. Especially if the missing element is mathematics. I am still very, very rusty on almost all my higher math knowledge. Perhaps I should take some math classes before attempting QM. I had considered that when I made my initial plans for my doctoral pursuit, but rejected it as too time consuming and I didn’t think it would be necessary. Maybe I was wrong in that regard.
I know that scientists such as Richard Feynman understood the physics at the root level. Of course, Feynman was a brilliant mathematician and so the math didn’t get in the way of a more fundamental understanding. But Richard could explain the subject in everyday terms ... well, nearly everyday. He is my mentor and model, although I don’t expect to equal his brilliance, I do try to mimic his technique and style and ... most importantly ... his joy in learning and discovery. Sadly these current problems have sucked the joy out of this and, earlier this morning, I actually considered quitting this entire enterprise.
I’ve recovered from those defeatist thoughts, and I have an idea. For my current problem, I’ve written my self a prescription: Feynman’s Lectures on Physics. I will once again sit at the feet of my mentor and try to catch a glimmer of his wisdom in his written words. If he can’t explain it to me, then no one can.
Plus I’m going to stop the QM while I further sharpen my math tools. That was what helped most when I struggled with E-M and Maxwell’s equations. Make Div, Grad, and Curl as familiar as your fingers and then you’ll grasp the field theory.
Eigenvectors and Ehrenfest’s theorem, Lie groups and complex vector spaces must become as as my fingers and toes ... well charted territory and capable of reaching out and grasping. That and a careful reading of the three volumes of Feynman’s Lectures on Physics along with several of his books that are in my library, but my initial reading was shallow. I am going to dig deep.
So, for now, I’m putting my class on Quantum Mechanics on hold ... another advantage of online classes. If this was traditional college, I’d have to drop the class and register for another class next semester.
So now, after writing this little essay explaining my problem ... mostly just to myself ... and deciding on a course of action that is—most importantly—a change, I have a plan of action. It feels so good when you quit hitting your head against the wall. As of this morning, I’m taking a new track. I have a new plan. I expect to spend a month on this new plan and then get back to the original course. That shouldn’t upset my schedule that much, assuming this works. If it takes longer, and that is a very real possibility, then it will just take longer. I have the time...I think!
Wish me luck.
Here’s a picture of Richard Feynman and his bongo drums. You have just got to love a physicist who plays drums as a hobby. He sought the joy in everyday experience. I’m trying to recover that joy from my current state of disappointment. Maybe I should get a pair of drums.
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