How I learned physics
Evangelizing, appreciating, and reminiscing
There are a few reasons why I’m writing this post. One reason is because physics has been the most foundational and influential part of my education. I think that physics is an important part of any technical education and an extremely useful lens to look at the world through. Anyone serious about understanding how the world works should know some physics, especially if they want to understand topics like energy, economics, and engineering.
Another reason is that the quality of physics education varies widely, and is often poor. Many of the issues people cite when complaining about physics are comments on how the subject was taught rather than on physics itself. I have learned physics at multiple places and have some opinions on what works well and what doesn’t.
I also want to write this post because physics has been the most impactful subject I’ve learned and informs every part of my world view. I hope to continue learning physics for the rest of my life, but I want to preserve an account of my first few years with the subject while they’re fresh in my mind.
The first physics class I took was my junior year of high school. Before this, my only experience with the subject was through popular science books about cosmology and other topics. I loved books like The Quantum Age, Astrophysics for People in a Hurry, and Pale Blue Dot. These books didn’t discuss technical details, but they filled me with wonder and interest. I badly wanted to understand the topics of these books more deeply.
Despite my interest, there were many kids in my high school who took physics before me. It was common for advanced students at my school to take AP Physics 1 their sophomore year. At this point I don’t remember why I didn’t do this, but I did end up in this class during my junior year. AP Physics 1 is a pretty standard mechanics class that covers projectile motion, forces, rotational motion, and some basic fluids topics. It is an algebra-based course so there was no calculus involved. I did well in this class, and had a wonderful teacher who was the ideal mix of serious and fun. One of my favorite parts of this class was each Friday when we would watch an episode of How It’s Made and discuss the physics going on in whatever manufacturing processes were involved.
After this, I knew that I definitely had not had enough of physics, so I enrolled in two physics courses during my senior year. These were AP Physics 2 and AP Physics C. AP Physics 2 covers circuits, basic E&M, some thermodynamics, and basic ray optics. AP Physics C covers mechanics and E&M, but now with calculus. It is supposed to be the equivalent of a year-long introductory college physics course. At this point I was taking calculus, so this class was perfectly timed. I remember physics really helping with my mathematical intuition and vice versa. It was the first time that I saw the explicit correspondence between ideas from math and the real world. Before this math was completely divorced from other subjects in my mind.
The end of my senior year of high school was in the spring of 2020, so it was substantially altered due to the pandemic. Starting in mid-March we stopped going to school and all my classes were “online”. This mostly meant that my teachers recorded lessons for us to watch afterwards. During this time we covered optics in AP Physics 2, and I remember being bewildered. I couldn’t get an intuitive feel for optics like I could for mechanics or E&M. I swore off the subject, hoping to never come across it again.
When I applied to college I thought that I wanted to be an aerospace engineer. I think that this was mostly due to the awe I felt from reading popular science books. I perceived “rocket science” as a good mix of intellectual challenge and practical applications. That being said, I didn’t know anything about what an engineer actually did. Once it came time to go to college, I rethought my plans and realized that aerospace engineering is quite specific. I decided instead that I wanted to study something more general and chose to take mostly math and physics courses my first year until I figured out what I wanted to do.
At Stanford, freshmen have the option of taking a physics sequence called the “60 series”. These courses are for students who took physics and calculus in high school and are considering majoring in physics. The first course is mechanics and special relativity, the second is E&M, and the third is quantum and thermal physics. Due to covid these classes were online my freshman year, but they nevertheless taught me the physicist’s mode of thinking and surrounded me with peers who had as much interest in physics as me.
Approximations, limiting cases, and dimensional analysis
My freshman year was the first time I got a hint of how professional physicists approach problems. In high school courses, physics is often presented as a series of formulas to be memorized that give the right results. As such, many conceptualize physics as the process of figuring out which formula to apply to get the right answer. In the 60 series, however, the emphasis was much more squarely on building experience with a set of mathematical tools that could be applied to arbitrary problems. Calculus figured often, but the most important new skills I learned were approximations, limiting cases, and dimensional analysis. If I could insert any physics knowledge in the general population’s brain it would not be a specific formula or topic. It would be these three skills.
For all but the simplest systems, physicists have to make approximations. The vast majority of the time, reality is simply too complex to be modeled exactly. For example, water is often modeled as an incompressible fluid. An incompressible fluid is one whose density does not change, no matter how much pressure is applied to it. In reality, water is not incompressible1, but in most reasonable scenarios, it compresses so little that it is negligible.
Therefore, much of the art of physics is knowing when to make certain approximations and when not to. This requires understanding the physical processes involved, but also when terms can be considered negligible and when they can’t. All of this must be backed by the mathematical skill to simplify equations correctly. As another example, the Earth is approximately a sphere, but not exactly. If you wanted to calculate the orbit of the Moon around Earth you could approximate the Earth as a sphere, but if you’re operating a GPS satellite, you probably don’t want to make this approximation.
The second skill is limiting cases, which might be the most widely applicable of the three. This technique allows you to check your answer by considering a limiting scenario. For example, special relativity deals with objects moving close to the speed of light. If you calculate something in special relativity and then take the limiting case of v << c, it should match the result you would get if you used standard classical mechanics. If not, you’ve done something wrong. Or, you might check how the speed of a falling object changes when its mass doubles. In this case, there should be no change, so if there is, you know you’ve gone wrong. The concept of limiting cases can be applied to any model, be it economic, financial, social, or psychological. While the technique is not as clean in these disciplines, substantial intuition can be gained just by considering what a model implies in extreme cases and whether this matches what you already know.
Dimensional analysis is the most esoteric of the three but probably the most useful for understanding new physics concepts in the wild. Dimensional analysis relies on the fact that all physical quantities have dimensions. For example, mass has dimensions of…mass, and acceleration has dimensions of length / time2. If you multiply mass by an acceleration you should get a force, which has dimensions of mass * length / time2. In other words, dimensions must follow the same relationships as the quantities themselves. This is useful because oftentimes you will calculate something like a force in a complicated way involving multiple steps and many different quantities and constants. There might be charges, masses, lengths, etc. all mixed together. At the end, you can tally up the dimensions of your answer and if it doesn’t come out to mass * length / time2, then you know you made a mistake. If you get good enough with dimensional analysis you can even guess the form of certain equations without doing any math!
By my sophomore year I decided that I wanted to study electrical engineering, but I stayed in the most physics-adjacent part of EE. I took graduate quantum mechanics in the EE department. It was taught by a physicist, and this was my first experience with the centrality of linear algebra in physics. Some people say that quantum mechanics is just applied linear algebra, and these courses highlighted this point. This year I also took a mathematical methods class in the physics department. This was largely mechanical practice of solving differential equations and proving things about matrices, but the reps paid off in later classes.
By the beginning of my junior year I had an existential crisis about whether I wanted to be a physicist or an engineer. I really loved physics and was able to do some of this in EE, but I didn’t feel like this was enough, so I added a second major in physics. This meant I had to take the core physics classes my junior year: advanced mechanics, intermediate E&M and quantum mechanics. I also took some electives like statistical methods and semiconductor physics.
Active learning
By this time covid had passed, so I had the privilege of experiencing the core physics courses in their intended form. In a traditional physics course, the professor lectures a few times a week, and the students solve problem sets outside of class. The students work on the problems with their classmates and the TAs, and can go to office hours with the professor. The structure of problem sets and office hours was the same at Stanford, but the lecture component was significantly different.
Instead of the professor lecturing the whole time, each lecture period was organized around a worksheet that the students would complete. The professor would lecture for 10-15 minute chunks and then have the students work a section of the worksheet. We would discuss amongst ourselves and ask for help if necessary. The class would then reconvene and the professor would review the problems and move onto the next topic.
Once I took a physics course this way, every other way seemed foolish. Most professors know that courses run better when the students engage during class. However, in many lecture classes attendance is low because the students don’t enjoy the lectures and don’t feel they learn more from them than reading the textbook or notes. To fix this, many professors resort to asking questions during the lecture or even cold-calling students. These strategies are bad because they make the students feel uncomfortable (especially if they can be cold-called) and don’t add a new dimension to the lecture. It’s hard for the professor to pose a one-off question that is the right level of difficulty, understandable, and useful. Trying to achieve this in real time and reacting to the students’ expressions is nearly impossible.
The advantage of a worksheet is that the problems can be thought out and paced beforehand, challenging students just the right amount. Students also feel much more comfortable being confused and asking questions with a small group of their peers as opposed to with the professor in front of the whole class. It engages everyone, not just the students willing to speak up. Even better, students can review the worksheet after the fact, giving a much more actionable set of notes than simply writing down whatever the professor writes on the board.
This teaching technique helped me immensely and allowed the courses to be much more cohesive and fast paced than they otherwise would have been. Every class I had with this format was consistently well attended. Even when attendance wasn’t part of the grade, students came and engaged because they saw the benefits2.
I understand that not every physics course can be formatted this way. Most courses will have too many students or lack the correct classroom space to accommodate small groups of students working together. It also requires substantial effort from the professors in the form of writing well paced worksheets and structuring their lecture around them. I had a few professors who were not good at this, but their courses were still better than the traditional lecture style. This was the most effective way I learned physics by far, so it is worth being considered more widely.
The end of my college physics career allowed me to tie everything together and see how all the topics I learned were connected. I took statistical mechanics, which underlies all of physics and builds on all the content in a typical physics degree. I also took another math methods course that gave me a taste of how modern theoretical physicists work and the tools they use. I took a computational physics course and even got to take an advanced quantum course with Leonard Susskind! On top of this, I was able to take many of these courses with the other physics majors–a group that was much more tight knit than other majors. Studying physics was an immense privilege and I truly couldn’t imagine a better place to do it.
I was initially attracted to physics because of the aesthetic beauty of the subject and the generalizability it promises. Everything must obey physics, which lends itself to endless curiosity and wonderment. But eventually, every person who studies physics must ask themselves: what’s the point? Physics doesn’t have as much of a standard career path as other majors (most physics majors don’t do a PhD), so people answer this question in many different ways. For some, the answer is simply that they liked physics enough to study it, but don’t need to pursue it further. They might become a software engineer or quant or start a company. For others, the point of studying physics is to understand the world. Taken to its logical conclusion, this means doing physics for the sake of the subject.
For a long time I was very undecided on what I wanted out of my education. I used to lose sleep over whether I wanted to be a physicist or an engineer, but I realize now that I can basically be both. I study optics, which is ironically the only part of physics I ever had a visceral negative reaction to. In high school I found it confusing, hard to visualize, and convoluted. If anything this fact has been a lesson in leaning into the parts of a subject you find the most confusing, because this is where the most opportunity for growth is.
Why you should learn physics
I would be negligent if I didn’t spend at least a little time explaining why you should learn physics. Of course I can’t tailor this to your specific goals and ambitions, but here are the best reasons I can come up with.
Physics is a monumental human achievement on the level of our best movies, books, cities, symphonies, and cathedrals. The beauty of architecture and art can be admired from afar, but it takes a little more work to get through some math and appreciate physics. Physicists work at the level of abstractions and data, not emotion, so speaking their language takes more deliberate effort, but it is well worthwhile. It doesn’t take too long to get to a conversational level and be able to marvel at some of your fellow humans’ greatest accomplishments. Leonard Susskind’s The Theoretical Minimum is highly regarded for taking someone from complete beginner to competent, although I haven’t read it myself.
Another reason is that studying physics gives you a unique ability to cut through bullshit. As the world grows more complex the opportunities to be fooled or confused grow, a good bullshit detector is becoming more important. Physics is unforgiving, so studying it exercises the parts of your brain committed to detecting anomalies and figuring out the truth. Domains like history and economics don’t follow neat formulas like those in physics, but the skill of reasoning about complex systems from a small set of axioms transfers well.
Another reason to study physics is that it’s hard. This means that you not only will be challenging yourself, but you’ll also have the chance to ponder the open questions that occupy physicists. In most situations physics works great, but there is so much we don’t know. The more human brain power we can commit to these mysteries, the better–so come join me!
Even at the bottom of the ocean floor water is only compressed by about 10%!