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As a 12th grade student , I'm currently acquainted with single variable calculus, algebra, and geometry, obviously on a high school level. I tried taking a Quantum Physics course on coursera.com, but I failed miserably, as I was not able to understand the mathematical operators (one being the Laplace operator).

I would really like your advice as to what precise part of mathematics I should teach myself, and some suggestions regarding proper textbooks, if possible.

Thanks a lot! :D

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Multivariable calculus, differential equations, and linear algebra are definitely prereqs. –  RghtHndSd Apr 17 '14 at 23:27
Several courses in set theory, measure theory and topology (both algebraic and set theoretic topology). If only to ensure that you don't abuse terms and ideas from these fields, and cause mathematicians grind teeth and twitch when you spout something about these topics. –  Asaf Karagila Apr 18 '14 at 0:14
@AsafKaragila are you trying to tell me that some spaces... aren't a Hilbert space? –  Omnomnomnom Apr 18 '14 at 1:21
Try to maintain your drive my friend. It will be wondrous. –  Arturo Don Juan Oct 30 '14 at 16:36
You'll learn everything you need along the way. Later on, you can refine everything as you like. I guess ${\tt @RghtHndSd}$ comment is a correct advice. Usually, before the QM first course students take a 'Wave' course ( like the Berkeley one ) where we see a tremendous analogy with vector spaces ( Hilbert, etc... ). –  Felix Marin Nov 7 '14 at 18:34

6 Answers 6

up vote 9 down vote accepted

I am going to plug my undergraduate professor's book again, but it is honestly the best book I know to prepare oneself for the math involved in QM. (I should know, as I experienced his course as a Math/Physics double major.) The book is Applied Analysis by the Hilbert Space Method by Samuel S. Holland. It is now available in paperback and relatively inexpensive. This book is custom tailored for the math and physics student on the cusp of taking a first course in QM. There's even a chapter on the Schroedinger equation, with the solution to the hydrogen atom worked out in detail. I cannot recommend highly enough.

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I've browsed to the contents of this book and it looks absolutely incredible !!! Thanks a lot , mister Gordon , I'll buy it today ! –  Victor Apr 18 '14 at 11:18
@Victor: My pleasure! I assure you as someone who worked from the original notes and has worked every problem in that book, it will give you an incredible preparation. It is no coincidence that the math that I am most expert in is this sort of math; Prof. Holland was an incredible teacher. –  Ron Gordon Apr 18 '14 at 11:35
Let me praise you for your wonderful suggestion , Mr. @Ron Gordon , I got the book a few days ago and it's extraordinary.Every problem is solved in such detail that even a highschool senior like me cand understand the matter in hand. Thank you so much, best of luck to you ! –  Victor Jun 18 '14 at 9:55

It depends on what type of QM course you want to take. Courses in QM for engineers, undergraduate physics majors, graduate students in physics, and graduate students in mathematics are all pretty different. I will assume you're seeking an "undergraduate physics major" understanding of QM.

I have two recommendations:

  • In my opinion, the best Quantum Mechanics book for self-study is Shankar. My main reason is that the first third or so of the book is a survey of the mathematics you'll need in the other two thirds, i.e., it answers precisely the question you've asked. You'll need to know calculus first (including vector calculus), but from there Shankar will give you what you need. I also like this book because it includes tons of fully worked out examples, and pages upon pages of "what does this mean" type exposition. Usually, you'll see this book being used in graduate or advanced undergraduate quantum mech courses, but that is just because it is a very long book about a very involved subject, it doesn't mean it isn't accessible to the beginner.

  • When I took quantum mech I read Lang's Linear Algebra concurrently, and it made everything so, so much easier. You don't necessarily have to finish it, basically just get real comfortable with inner products and dual spaces, up through the spectral theorem.

Some additional remarks:

  1. I think it's an exaggeration to say a course in quantum mechanics requires functional analysis or operator theory, even though that is essentially what you're doing. The mathematically rigorous forms of those disciplines are pretty advanced, but you don't need to understand them at that level to do quantum mechanics. You just need to be able to use them. Shankar will teach you how to do that. (Of course, I wouldn't discourage you from learning them eventually, but they aren't strictly necessary for the purposes of QM.) He should catch you up on the basics of probability theory, too.

  2. I would not say that about abstract linear algebra, however. You will need to understand vector spaces, dual bases, inner products, eigenvalues, etc. on a rigorous level.

  3. Group theory and representation theory are definitely not necessary. Although they are important to quantum mechanics, you will not see them until very advanced levels.

So, long story short: you need to know calculus up through vector calculus and linear algebra up through abstract linear algebra.

It's worth mentioning that, although what I've mentioned above would be sufficient for the mathematics side of things, it would be real good if you had seen mechanics, electricity and magnetism, and thermodynamics beyond an introductory level beforehand. It's possible to do it without previous physics courses, but you'll find yourself saying "so what?" a lot, as the weirdness of quantum mech will not seem as jarring to you if you have not seen how things work at larger scales.

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I've studied and understood all Newtonian mechanics, same goes for electricity and magnetism. As for thermodynamics, I've got 2 years of learning behind me and I, thereby, understand all there is to understand about the laws of ideal gases... –  Victor Apr 18 '14 at 11:23
@Victor "I understand all there is to understand about x" is possibly one of the boldest statements a human can utter. Your list of math you've learned does not include multivariate calculus, meaning you are missing an entire world of all the physics you listed. For example, there's no way in heck you've done significant work with Maxwell's equations without multi. –  jpmc26 Apr 18 '14 at 19:35
@Victor I think you will be pleasantly surprised at how much there is left to learn, even in the most elementary physical topics, once you study vector calculus, calculus of variations, and linear algebra. You're never finished with mechanics, E&M, and thermo - even in grad school! –  Alexander Gruber Apr 18 '14 at 19:37
You're both completely right, my answer was arrogant and, evidently, unjustified. My sincere apologies. I was a bit irritated at the moment by a math problem. –  Victor Apr 18 '14 at 20:05
What Alexander Gruber is saying here reminds me of Heisenberg, when asked about symmetric and self-adjoint operators, "what's the difference?". –  user135041 Apr 25 '14 at 21:24

I think that Higher Maths for Beginners – Zeldovich, Yaglom and Elements of Applied Mathematics books have the math you need for QM. They're written by Zeldovich, a co-father of Soviet nuclear bomb project.

The latter book has a related volume called Elements of math physics. Noninteracting particles, unfortunately it's not translated to English. The Russian version used to be my favorite text for math physics.

One of my favorite math physics text was Methods of Theoretical Physics by Morse and Feshbach. It's huge, and is written like a handbook, no need to read the whole thing.

If you read German, then there's this crazy handbook The mathematical tools of a physicist by Erwin Madelung, I'm not sure if it's available in English, but I saw it in Russian.

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Since you seem to be russophone, do you know any nice websites where I can download pdf's of freely available texts on math and/or physics in Russian? –  Raskolnikov Apr 18 '14 at 9:52
@Raskolnikov, I studied in Russian, which doesn't make me a russophone. There are many sites with illegal PDFs, I won't advertise them here. The legal math content resource is eqworld.ipmnet.ru –  Aksakal Apr 18 '14 at 12:41

I have taken two courses in QM. There were a range of different math courses that are extremely useful. I would recommend studying multi-variable calculus, linear algebra, partial differential equations and probability theory.

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Of course, it also depends on what point of mathematical perspective of QM you are looking for. Me - I study QFT in terms of categories. –  Sanath Devalapurkar Apr 17 '14 at 23:31
Well, I would say that a theoretical approach would fit me best , what do you suggest in this case? –  Victor Apr 17 '14 at 23:45
Yea in that case you are definitely going to want to look into real, complex and functional analysis along with the other subjects I suggested. –  RDizzl3 Apr 17 '14 at 23:47

It depends a lot on the point of view you want to take on QM. For an experimental physicist's point of view, you will need real/complex analysis, linear algebra and probability theory. If you want the theoretical physicist's/mathematician's point of view, then add functional analysis (maybe with a focus on $C^*$-algebras/algebras of operators) and representation theory.

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I hate to be that guy, but in order to properly understand Quantum Mechanics, you need to have a solid understanding of Newtonian Physics. Since you're already on Coursera, try and slow down a bit and take one or two introductory physics courses before moving on to quantum. For the first few weeks (or likely the entire first course), you won't be learning any new math since it's all basically single variable calculus, but as you progress, you'll see that physics courses generally bundle in the required math with them, though not necessarily to the same level of depth (for example, the Laplace operator is covered pretty extensively when discussing electricity, that way people who come to learn quantum mechanics are "already prepared").

Instead of diving in headfirst with some hard math, start by learning Newtonian physics until you are ready to progress into quantum mechanics - you'll have the proper skills to advance onwards to quantum once you're done, though it certainly won't hurt (it'd be quite fun really) to do a few math courses as you go along.

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I know all Newtonian Phyisics by heart , same goes for the first two principles of Thermodynamics and most of Electricty and Magnetism.Regarding Quantum Physics , I am perfectly acquainted with Einstein's Photoelectric Effect, Special Relativity , Wave-Particle Duality ,Compton Scattering , DeBroglie Hypothesis , Bohr's Model for the hydrogen atom.I am no begginer,trust me.Good day. –  Victor Apr 18 '14 at 10:08
I had assumed you were relatively new to physics as you hadn't mentioned it in the OP. My apologies –  Noamyoungerm Apr 18 '14 at 19:52
@Victor, if you know Newtonian physics in Hamilton formalism, then you're good to go. Otherwise, I found this little book most helpful : Landau, Lifshitz, Mechanics –  Aksakal May 12 '14 at 2:03

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