Math Needed for Reading Jackson & Quantum Mechanics Textbooks

[Enjolras1789]

The first time I took Electromagnetics via Jackson I did terribly on account of not knowing the math needed to read the book (I despise the fact that many graduate programs do not specify what math is genuinely needed to read the textbooks they assign). At the time, I had taken 1 course each in Multivariable Calculus, Matrix Algebra, introductory analysis, ODEs, PDEs, and a Math Methods course in my undergrad physics department. I unfortunately found out that this is nowhere near enough to get through Jackson; one needs to know Green functions quite well, quite a bit about tensors, Fourier analysis, and complex analysis.

Beyond the topics I have mentioned, what topics in mathematics are necessary to read Jackson and understand it meaningfully?

Additionally, I have discovered, to my disdain at not being told this ahead of time again, that to understand Quantum Mechanics I and II, one does need to know beginner abstract algebra/algebraic structures. What areas of math are recommended to read a QM textbook meaningfully for general first year graduate quantum mechanics beyond what is already mentioned?

 

[Fredrik]

It's been more than ten years since I took the Jackson class, but as I recall, those things that you say are "nowhere near enough" should be enough. Yes, there are a few more things that you're going to need, but some of them are explained in the book, and you should be able to look up the rest as you go along. I don't remember needing to know anything about tensors beyond what's mentioned in the book, but if it turns out that you do, you might want to have a look at the SR chapters in Schutz's "A first course in general relativity". (His presentation of tensors is very easy to understand).

The problem with Jackson isn't (in my opinion) that you need a lot of prerequisites. It's that the problems in the book are very very hard. You can't really prepare for that, except by accepting the fact that you're going to spend a lot of time trying to solve problems.

You don't need anything more than what you mentioned for QM. All you really need is a thorough understanding of linear algebra and complex numbers.

 

[Enjolras1789]

I disagree completely in terms of QM. To read a book like Shankar, one needs exposure to generalizations of concepts of dot products as inner products, generalizations of cross products as outer products. The simplest argument to muster is simply this: QM is done in the Hilbert space. One needs to be familiar with mathematical spaces to understand what a Hilbert space is as opposed to other spaces. One needs to know about Lesbegue integrals to understand a Hilbert space. I am sure there is much more one needs to know, but I don't know what those are.

 

[Fredrik]

Inner products are covered in every linear algebra class and so are vector spaces. A Hilbert space is just a vector space with an inner product such that every Cauchy sequence is convergent. That last part isn't covered by linear algebra, but the teacher should be able to explain it very easily. You don't need to understand Lebesgue integrals to know what a Hilbert space is. You only need to know those if you want to understand the explicit construction of Hilbert spaces, and most QM classes don't cover that in detail. They might say a few words about it, but it's not something that students are expected to know.

Things get a lot worse when you get deep into the mathematical aspects of quantum field theory, but we're talking about quantum mechanics here.

 

[Enjolras1789]

Inner products as a generalization of the dot product for a vector space where vectors are not defined as having direction was not covered in my matrix algebra course; your first statement is not true, and the professor was no dolt. Discussion of vector spaces without vector directions was quite new.

Your second statement assumes a teacher who can teach effectively; this is not generally true. It also somewhat defeats the point of writing a book from which one can teach oneself if that's not in there. You do need to know about Lebesgue integrals, in my mind, to understand how they help motivate the creation of the Hilbert space in the first place. The book itself should discuss Cauchy sequences (a topic that I hadn't had exposure to) if it is going to talk about Hilbert spaces. People should define words if the words are new.

Your last statement defeats the point of what I wrote: "...but it's not something that students are expected to know." Why discuss something in a book if you're not supposed to know it? If it's in the book, I assume I need to know and understand it. What would be the point of telling students about the Hilbert space if you don't need to know about it? It's just an empty statement to say "We do QM in the Hilbert space" without knowing what a Hilbert space and why we do it in the Hilbert space. I always stop when I come to something I do not know when I read and look it up.

I know that I was assigned no problems concerning the Hilbert space, but that is of little consequence to me. My goal is not to pass a class, my goal is to understand everything in the book. If books include material that is superfluous, it's not a well-written book; I assume the book is always well-written by default until I see otherwise, and rather that I just need to study more and know more before having started it.

 

[Fredrik]

It's very easy to understand what a Hilbert space is. It's much more difficult to explicitly construct them. The same goes for real numbers. Do you know how to explicitly construct real numbers from the rationals? If you don't, does it make you feel that you don't understand what a derivative is. Of course it doesn't.

You absolutely have to know what a Hilbert space is, but it's not even helpful to know how to construct them explicitly. I took five(!) QM/QFT classes without ever being taught that part. Now that proves that the QM classes at my university really sucked, but it has also proved (to me) that you don't need to know how to explicitly construct Hilbert spaces to do calculations or to understand QM. I would say that you do need to understand explicit constructions to understand quantum field theories, but that's another story. I'm reading http://home.uchicago.edu/~seifert/geroch.notes/ now because I want to learn about those things.

It's funny that you say that "my goal is to understand everything in the book". I used to feel that way too. But if the book contains one contour integral, in the proof of some theorem, and there are other proofs online that don't use contour integrals, do you feel that you have to study complex analysis immediately just to understand the integral? It's better to just move on. It's the same with Lebesgue integrals. It take far too much time to learn about them, so in my opinion, they shouldn't be taught in a first (or second) QM class. They should however be mentioned. It would be irresponsible not to mention which parts we're skipping in order to be able to cover more of the interesting stuff.

I find it completely unfathomable that there are linear algebra classes that don't cover the definitions of a vector space and an inner product.

 

[Enjolras1789]

I disagree with the sentiment in your first paragraph for one reason: real numbers are a postulate for me. I was once told that you can't "understand" addition without knowing set theory, because there is a "proof" that 1+1=2 (this is all what I've heard, no claims). However, everything needs axioms from which to begin, and that real numbers exist for me is a reasonable postulate. I don't extend this argument in general, purely because I feel that postulating real numbers is a very reasonable postulate.

If someone tells me we do QM in the Hilbert space, my immediate reaction is "OK, what is it and why that one and not another one?" I agree it leads to extremely winding roads of what one must learn...but if I can't explain it to someone else, I don't really know it, do I? I also don't know what it would mean to know what a Hilbert space is without its construction.

Although I could happily do all the calculations with knowledge that I happen to be in a space which says "kick out all non-convergent integrals" (loosely), and still do problems in this class, I assume, for my greater knowledge, it must be important to know.

Is it a bad idea to say that whatever the author has written in the book is presumably necessary knowledge to understand the subject? If yes, then why put things in the book one must not know?

Thank you for your continued insights and opinions,

 

[will.c]

I disagree with the sentiment in your first paragraph for one reason: real numbers are a postulate for me. I was once told that you can't "understand" addition without knowing set theory, because there is a "proof" that 1+1=2 (this is all what I've heard, no claims). However, everything needs axioms from which to begin, and that real numbers exist for me is a reasonable postulate. I don't extend this argument in general, purely because I feel that postulating real numbers is a very reasonable postulate.

If someone tells me we do QM in the Hilbert space, my immediate reaction is "OK, what is it and why that one and not another one?" I agree it leads to extremely winding roads of what one must learn...but if I can't explain it to someone else, I don't really know it, do I? I also don't know what it would mean to know what a Hilbert space is without its construction.

Although I could happily do all the calculations with knowledge that I happen to be in a space which says "kick out all non-convergent integrals" (loosely), and still do problems in this class, I assume, for my greater knowledge, it must be important to know.

Is it a bad idea to say that whatever the author has written in the book is presumably necessary knowledge to understand the subject? If yes, then why put things in the book one must not know?

Thank you for your continued insights and opinions,

I have gladly taken the use of a Hilbert space as a postulate in my undergraduate quantum class, and so far in graduate QM, and I'm doing just fine. In fact, I'm doing excellent and the fact that I don't know how to perform a lesbegue integral doesn't bother me in the slightest. In doing that, reading Shankar was a trivial task, requiring no more prerequisite math than what he provides in the first chapter.

Jackson is an exceedingly difficult text, I agree, but unless you're actually a mathematician, I think it's a waste of time to hold off on it until you have completely mastered the entire mathematical groundwork.

It's a big mountain, and if you're interested in the top, you have to take a leap here and there.

 

[Daverz]

You didn't get Fourier analysis in the PDE course? But maybe the idea of going back and forth between time and frequency domains is not emphasized enough. Physics students could probably benefit from a course in "signals and systems", as it's probably something you'll have to pick up along the way anyway.

I'm not sure one needs an entire complex analysis course, as beautiful a subject as that is. Some lectures on contour integration should suffice.

An abstract algebra course is pretty worthless as prep for QM, where group representations is mostly what one is interested in, not Galois theory.

As for tensors, you only need to know about Cartesian tensors.

There does seem to be some stuff falling through the cracks in a typical undergrad Physics education.

 

[Ben Niehoff]

Doesn't your program offer a Mathematical Methods course? That ought to be sufficient to get through Jackson.

The tensor stuff covered in Jackson is very basic. The Green's function stuff is more advanced, but Jackson takes the time to explain it (in Chapter 3, I believe). I didn't know anything about Green's functions before, and I understand them perfectly well after Jackson.

There are some places in Chapter 2 and Chapter 7 where some complex analysis helps. But usually Math Methods is taught either concurrently or previously to taking Jackson, at which point you should have learned enough complex analysis to get by.

The last thing to remember is that Jackson is mostly a math book, not a physics book. There is hardly anything new you will learn about electrodynamics (possibly some things about EM waves in the second half, but that's it). Most of the book is teaching you various mathematical concepts, primarily Green functions and boundary value problems, in the context of EM. Greens functions will continuously come up in practically every other course.