What level of math is needed to truly understand 20th century physics?

[bobsmith76]

I'm a reasonably smart person. I have knowledge of 11 languages and I'm a proud studyaholic, meaning I've been studying voluntarily for about 30 hours a week since 2007, and 50 hours a week for the last 4 months. I'm tired of reading popular science physics books which report what scientists have learned since Planck. I want to do actually do the equations and understand how scientists know these things. Back in high school physics my teacher showed us how to derive e = mc^2 with relatively little math. I have no interest in making a contribution to science or becoming a scientist because my strength is in poetry, literature and philosophy, but nevertheless in order to make a contribution to philosophy you have to understand science.

Ok, how far can a non-scientist with a lot of ambition be expected to go in understanding 20th century physics, if he studies about 30 hours a week? also, I should add that I have no understanding of calculus. How much Calculus or math higher than Calc do I need?

 

[espen180]

The prerequisites required depend on the depth you want to obtain.

To even get started on Quantum Mechanics (QM) you need knowledge of partial differential equations, Fourier series, and preferable some knowledge of operators and linear algebra.

A somewhat more accessible branch of physics is Special Relativity (SR) for which you don't really need much calculus.

However, the extention of SR to include gravitational fields, General Relativity (GR) is a monster requiring a deep understanding of Riemannian geometry, differential geometry and tensor analysis.

Of course, if you are willing to teach yourself calculus and other math needed, there is no limit to the depth of your studies, but if you are not, I think progressing beyond SR would be difficult. Even classical mechanics is rich in its use of calculus. It can really be concidered a prerequisite for any physics at all.

Here is a list of relevant math in physics branches: (Brackets indicate optional topics)
Classical mechanics: Calculus, vector analysis, linear algebra, differential equations
SR: Algebra, some calculus, (differential geometry), (tensor analysis)
GR: Tensor analysis, differential geometry, riemannian manifolds, differential equations
QM: Differential equations, complex numbers, linear algebra

 

[Ted Sand]

There are elementary but important parts of quantum physics that need only a little bit of algebra, for example the photoelectric effect, wave-particle duality, electron energy levels and explanation of spectral lines.
Some amazing science and applications are based on those concepts.

 

[bobsmith76]

Espen,

I couldn't have asked for a better response. Thank you

 

[jtbell]

To even get started on Quantum Mechanics (QM) you need knowledge of partial differential equations, Fourier series, and preferable some knowledge of operators and linear algebra.

(I added the emphasis above)

I strongly disagree with this. In the US at least, physics majors usually get their first exposure to QM as part of a second-year "introductory modern physics" course that assumes only basic differential and integral calculus. Textbooks for these courses introduce / review the basics of partial derivatives, complex numbers and differential equations as needed, so students can work with the Schrödinger equation in one dimension to solve simple classic examples like the "particle in a box" (infinite square well). They learn about the probability interpretation of the wave function, how to calculate expectation values of position and momentum (introducing the momentum operator), and other basic topics.

The last time I taught such a course a few years ago, I used this book:

https://www.amazon.com/dp/013805715X/?tag=pfamazon01-20

There are other books similar to this. They don't get into matrix representations, operator algebra, momentum space, Fourier transforms, bra-ket notation, etc. Students get that in a higher-level QM course. But this gives them a start, with some of the basic concepts and examples.

 

[Fredrik]

Espen's answer is good, but I disagree about the differential equations (and the Fourier series). People always mention differential equations as a prerequisite to QM in these threads (there are many of them; you might find them useful, so I suggest you search for them). I have never understood why. You will only encounter one differential equation in QM, and the book will tell you how to solve it. Similar comments apply to all of the classical theories, with the exception of classical electrodynamics. (If you're not going to get deep into the engineering applications of electrodynamics, you don't have to worry about those differential equations either). If you ever get to general relativity, you will have to deal with an equation that's so hard to solve that you will only be looking at the two or three simplest solutions, and the books will tell you how to find those.

So my advice is: Don't worry about differential equations. Just let the physics books teach you what you need to know about them.

I think it's reasonable for someone like you to study calculus, linear algebra, non-relativistic classical mechanics, special relativity and quantum mechanics, but it's likely that you won't be able to study general relativity, quantum field theory, and the mathematics of quantum mechanics in detail any time soon. (Most physicists never study the mathematics of QM. They just study linear algebra to get some idea what they're doing).

Linear algebra is extremely useful in QM, and the basics are very useful in SR too, so don't underestimate its importance.A few book recommendations:

Calculus: Any introductory book will do. The one by Serge Lang looks really good to me.

Linear algebra: I really like Axler. Some people will say that there are easier introductions, and others will say that there are books with a more complete coverage of linear algebra, but I believe that this is the best book for someone who intends to study QM.

Classical mechanics: I don't know what to recommend here. I studied Kleppner & Kolenkow a long time ago. It wasn't bad, but I have a feeling there are better books.

Special relativity: Taylor & Wheeler gets the most recommendations. I haven't read it myself, but I don't doubt that it's good. The best intro I've actually read is the part about SR in Schutz's GR book.

Quantum mechanics: Griffiths is a nice introduction, and Isham is an excellent supplement to it. Ballentine is a good book for advanced students (but is probably not of any use to you right now). You should also read "QED: The strange theory of light and matter", by Richard Feynman (lectures about light for people who don't know mathematics).

General relativity (without mathematics): "Black holes and time warps: Einstein's outrageous legacy", by Kip Thorne.

General relativity (with mathematics): I'll just quote this guy:

There are really three tiers at which you can really learn anything substantive about general relativity.

Tier I: Primarily algebra based with light calculus. This is suitable for a first or second year university student. Prior knowledge of physics should include the basic rudiments of mechanics from the F=ma point of view. The hallmark textbook is: https://www.amazon.com/dp/020138423X/?tag=pfamazon01-20

Tier II: Calculus based with light differential geometry. This is suitable for a third or fourth year university student. Prior knowledge of physics should include an upper level E&M course and exposure to lagrangians and hamiltonians. The hallmark textbook is: https://www.amazon.com/dp/0805386629/?tag=pfamazon01-20

Tier III: Full general relativity with differential geometry. This is usually relegated to graduate classes, but can be tackled by advanced undergraduates. Prior physics knowledge is basically an entire undergraduate physics/astrophysics education. Some representative textbooks are:
https://www.amazon.com/dp/0226870332/?tag=pfamazon01-20
https://www.amazon.com/dp/0716703440/?tag=pfamazon01-20

Differential geometry: "Introduction to smooth manifolds" and "Riemannian manifolds: an introduction to curvature", both by John M. Lee.

 

[Shackleford]

We had a two-semester sequence of Modern Physics. The first class included special relativity, early quantum, and an introduction to modern quantum mechanics. The second class was straight QM. We got as far as time-independent perturbation theory. We used Gasiorowicz.

 

[DukeofDuke]

I have to disagree with Espen.

You only really need one technique to solve the differential equations (it's called separation of variables) and it's included in every quantum book.

Honestly, beginner's quantum is pretty easy. I'd say Griffiths level quantum is easier than Griffiths level E&M though you might have some difficulty if you haven't had a good classical mechanics education.

I'd recommend just picking up the griffiths QM book (it's standard) and reading through it.

Without calculus you pretty much can't do anything though. I don't think you really *need* much calculus but familiarity with it is necessary. If you have calculus AB and BC level stuff under your belt (from the AP exams) then I think you'd be fine just stay away from 3D stuff.
You'll also need a bit of linear algebra. So basically you might have a hard time understanding stuff the first time because you'll focus on the math.

 

[espen180]

When I say that I think knowledge of calculus and differential equations is required to do classical mechanics or quantum mechanics, I think I should clarify. What I mean is that in order to do pretty much anything more than copy equations from a textbook, you will at some point need calculus or linear algebra. A classical mechanics course I followed recently seemed to focus more on the math behind the theory of dynamical systems than on applying the theory on example problems.

As I said in my post, the math required depends on the desired depth. A suitable illustration, I think, is the difference between accepting energy conservation as a law of nature in of itself, or understanding it from Noether's theorem applied to the Lagrangian.

It may be comparable to the difference between understanding a theory and understanding the framework behind the theory.