For My Nephew

 My nephew just finished his frosh year in college, and wants to study...astrophysics?  Not entirely sure yet, but the college where he is has only a basic physics degree anyway. He seems a little lost on what to study, so this posting is a personal one, for him, but feel free to read on if you're in the same situation.

Okay kiddo,

It sounds like you have done the basics of mechanics and geometric optics, but I'm not actually sure what else in physics. I know you haven't done Maxwell's equations and quantum mechanics, but what about special relativity and introductory thermodynamics (statistical physics)? And in math, I know you've done "calculus", by which I gather you mean functions, limits, and integration and differentiation of a single variable. Good start, but we've got a long ways to go.

Despite our conversations, it seems like you don't have a clear picture yet of the curriculum, or even the broad structure of knowledge, in math and physics.  (And since I'm a computer engineer, all of this is from that perspective, of course.) So the first thing to do is to get oriented on what you really need.

Here are a few things to help you understand the structure of the body of knowledge, which should help you figure out what classes to take (or what to study on your own).

• The Map of Mathematics. I'm sure I've recommended this before, but watch it again.
  
• The Map of Physics. Same folks, same recommendation.
  
• Map of Computer Science. (Getting the idea yet?)
  
• A #QuantumNative Engineer's Bookshelf. For you, focusing on the math and physics parts will be helpful.
  

Mathematics

For a basic physics degree, you are going to need the following math:

• Linear Algebra. Multiplying vectors and matrices, solving systems of equations via Gaussian elimination, linear and affine transformations, and eigenvalues and eigenvectors, at least. You'll also be exponentiating matrices ($e^A$, where $A$ is a matrix) in quantum mechanics, and that's easiest if you can diagonalize a matrix. You can start with my linear algebra videos, but there are lots of resources on the web. I recommend the Georgia Tech Interactive Linear Algebra online, interactive textbook. That's pretty deep, you won't need it all right away, but it's there for a reason. There is also an entire 20-hour lecture course posted as two videos in YouTube, by Prof. Jim Hefferon; I haven't watched it, so I don't know how good it is, but the comments and likes are very positive. Khan Academy has an LA course. 3 Blue 1 Brown is one of the best things on the web, and they have Essence of Linear Algebra available (full course here). In short, there are many excellent, free resources available.
• Probability, both discrete and continuous. Probability distributions, conditional probability, Bayes' Theorem, moments of distributions. For continuous, you'll need integration, so continuous comes after basic calculus.
• Statistics. I think most physics majors get away with just what they learn in a probability class unless they specialize in statistical mechanics and thermodynamics.
• Ordinary differential equations (ODEs). You said you've seen that $f'(x) = f(x)$ is solved only by the function $f(x) = e^x$, so you've seen the start of a very deep field.
• Partial differential equations (PDEs). Next step: derivatives in multiple dimensions. These are equations involving symbols like $\frac{\partial x}{\partial t}$. I first hit this when doing Maxwell's equations; I think it's pretty common for that to happen, but it means you're dealing with both a new math tool and important ideas in physics at the same time, so studying basics of PDEs first is a good idea.
• Transforms and signal processing are a big deal; you might run into Fourier, Laplace, Z and other transforms. (This is different from the linear and affine transforms above.) Often, these are tools for solving ODEs or PDEs, and might show up in an Applied Math class in your college.
• Later, you might get into more specialized topics like number theory, group theory, and graph theory. Group theory, for example, shows up in particle physics. The basics of graph theory you can learn very early, actually, and they are critical in computer science but maybe not as much in physics.

I'm sure you can find as much stuff on the later topics as I found for LA.

Physics

Surprisingly, I'm a little less comfortable talking about what basic physics you should study. Let's talk first about optics, since it's one of my favorite parts of physics, demonstrates broadly useful concepts, and happens to be what we are working on for our quantum communications research.

From among the courses we have created already, if you want the basic physics, and to minimize the quantum communications portions, these steps would be very good follow-on to your work on geometric optics. They are a bit idiosyncratic relative to an ordinary course on optics, naturally focusing on what we need for quantum communications, but they will still carry you a long way.

• OQC, Lesson 5: Coherent Light and Single Photons, leading up to a quick quantitative intro to lasers.
  
• OQC, Lesson 6.1-6.2: Interference, group and phase velocity quick introduction to constructive and destructive interference and the notions of group and phase velocity, a distinction that is crucial to understand.
  
• OQC, Lesson 7: Waveguides discusses the most important means of guiding light, of which the most famous type is, of course, optical fibers. Total up to here is about two hours worth of video, you can do this in the time it would take you to watch one soccer match.
  
• And then most of our entire module From Classical to Quantum Light, which is about 10 hours of material covering wave equations, Fourier analysis, Maxwell's equations governing how electromagnetic waves work both in a vacuum and in materials, and then into more on single photons and the like, including how detectors work. You might want to taper off when you get to the single photon stuff and defer that for after you have had basic quantum mechanics, so let's say you should do the first ten lessons of that, 45 minutes each, so about 7.5 hours.
• Among the remaining important topics to learn about are how holograms work, more on polarization, and a lot about antenna design. A good optics course will cover these things.

From the above video on the map of physics, you should have some idea of the basic list of topics:

• mechanics
• waves
• optics
• special relativity
• electricity & magnetism
• introductory quantum mechanics
• thermodynamics

That much you should take as an engineering or physics student of any sort. If you major in physics, you'll add advanced QM, particle physics, general relativity and some other topics to that list.

Given your interest in soccer, you might consider studying biomechanics, which is a bit of a specialized area and interdisciplinary, so a little biology will help, too. You should check out Professor Ohgi's Sports Dynamics and Informatics Lab as a possible destination for grad school. He works with people up to the level of Olympic athletes to understand the forces and dynamics of athletic performance. Studying some computer science before joining his lab would also be helpful.

Enough for now. Go watch those introductory things to get oriented, then study some linear algebra and probability while you watch our online courses on optics and quantum computing and communications.