Mathematics necessary for relativity, quantum mechanics, etc?

[lesdavies123]

Hi,

First of all, I am an undergraduate Chemistry student, but who is also very interested in various areas of science (most actually :P), but like many other people I am especially curious about Einstein's theory of relativity, quantum mechanics and I especially would like to be able to read the papers of the famous physicists like Heisenberg, Dirac and Einstein and understand the mathematics of their theories in depth. As you may know, studies in Chemistry are disappointingly spare in mathematics courses (pretty basic calculus and linear algebra), so as of next term I'm taking extra math classes for the heck of it, but I want to start as early in my math studies as possible (anxious to start) so can you guys tell me what branches of mathematics I'm missing, that I should study. Book suggestions for theses subjects would also be very appreciated. Thank you for your answers!

 

[Fredrik]

Unless you're specifically interested in the history of physics, I would strongly recommend that you read modern presentations instead of the original papers. Actually, even if's the history you're interested in, I would still recommend that you read the modern presentations first.

At the very least you need a standard introductory text on calculus, like the one by Serge Lang, and a book on linear algebra like Axler or Friedberg, Insel and Spence. (Axler is mainly intended for people who are already familiar with matrices, but I wouldn't say that you have to have studied that before). I don't recall if Lang's book covers functions "of multiple variables" at all. (These are really just functions defined on ℝⁿ instead of ℝ). You need to study the basics about those too. It's especially important that you understand partial derivatives.

People always mention differential equations in these threads, but my opinion is that if you have a solid understanding of derivatives, you will have no problem understanding the concept of differential equations, and no problem understanding the solutions of differential equations that you will find in physics books. In other words, it's a good thing if you study them too, but it's not essential.

My favorite presentation of SR is the one in the early chapters of Schutz. You should definitely read a presentation that emphasizes spacetime diagrams (sometimes called Minkowski diagrams) instead of algebraic manipulations. Other authors whose books are often recommended as easy introductions are Taylor & Wheeler, and Moore.

(This paragraph was added in an edit the same minute as Chestermiller's reply). If you're also going to study GR, you will need differential geometry. The books by Lee are the best. You will however find them pretty difficult if you haven't studied some analysis and topology.

The only introductory book on QM that I've studied is one I didn't like, so it's hard to recommend an introductory book. But Griffiths looks good to me. It doesn't cover as wide a range of topics as e.g. Zettili (which certainly looks like a good alternative), but the idea isn't to learn how to calculate as much as possible, but to quickly get to the point where you understand the basics. For that I think Griffiths looks good. I would supplement it with Isham. If you want to know more, you can move on to Ballentine after that.

Unfortunately, an in depth understanding of the mathematics of QM requires that you study a lot of calculus and analysis, linear algebra, topology, integration theory, some abstract algebra, and a ridiculous amount of functional analysis. Most physicist don't do this. A reasonable compromise is to study linear algebra in detail, so that you have a thorough understanding of finite-dimensional vector spaces, and let the mathematicians worry about what actually makes sense in the infinite-dimensional vector spaces of QM.

You should do a search for similar threads. There are lots of them.

 

[Chestermiller]

You also need to learn differential geometry, vector and tensor analysis, and partial differential equations.