What to study after Quantum Mechanics?

[mgal95]

Hallo everyone!
I am studying Physics at University level. This Fall I will enter the third year of my studies. I find the curriculum inadequate and thus try to learn stuff on my own.
I have already taken the basic courses in Calculus (single and multivariable), Complex Analysis (analytic functions, contour integration, Cauchy Theorem, residues), Linear Algebra, Ordinary and Partial Differential Equations and some elements concerning linear operators (what is needed for an introductory quantum mechanics course. Nothing too rigorous). In addition to those I have studied on my own some Real Analysis (basic topology on metric spaces, completeness, compactness).
As long as my physics background is concerned, I have already taken courses in classical machanics (lagrangian and hamiltonian formulation, phase space, liouville theorem- all near goldstein's book level), basic electromagnetism (first 7 chapters from griffiths and this Fall The course will cover material from Jackson's book), special relativity (four-vectors, electromagnetic field Tensor, but without any mathematical rigor- not even the notion of the dual basis was introduced) and some other courses concerning thermodynamics, optics, waves, computational physics, programming (in C). This summer I studied on my own Quantum Mechanics from Liboff's Introductory Quantum Mechanics -and a bit from Shankar- (wavefunctions, basic one-dimensional problems, harmonic oscillator, orbital angular momentum and spin, hydrogen atom, elements of matrix mechanics, Heisenberg picture and time-independent perturbation theory plus the WKB approximation method). I skipped everything that had to do with applications in atomic and molecular physics as well as the scattering in three dimensions).

What I want is someone to guide me on what to study next. I am mainly interested in the mathematical foundations of physics and I am planning to do a masters (and possibly a PhD) on theoretical physics. I am extremely intrigued by Classical and Quantum Field theory. I am planning on studying Arnold' s Mathematical methods of classical mechanics, some general topology (from Munkres), algebra (maybe Lang?) and measure theory in order to have a strong mathematics background. I will take a course on Differential Geometry this Fall and on General Relativity next year (unless I study GR on my own).

Which books would you recommend for the above? What should I study next? Maybe Relativistic Quantum Theory? And after that? What should I learn before moving to Classical Field Theory (including GR) and QFT? I do not want applications (I will do those in university courses). I really want also to understand the mathematics behind all these. From where should I study Tensor Algebra (and Analysis probably), Algebra (Lang is a really big book!) and measure theory? What other mathematical background would you suggest is needed for the above? Can I, for instance, proceed directly now on Lie groups? Which books do you recommend?

Thanks and please excuse my english

 

[Dr. Courtney]

Have you taken the PGRE yet?

 

[mgal95]

Have you taken the PGRE yet?

I' m sorry, but how is that relevant?

 

[artfullounger]

Condensed matter might be an interesting theory oriented area to look into after QM. You might need to do some stat mech first depending on how rigorous your thermo course was however :x

 

[Dishsoap]

Why not study the applications to AMO physics? You said you skipped that, but why?

 

[mpresic]

You did mention you found your curriculum inadequate, but if you look at the admissions part of all those charts of graduate programs on graduateschool shopper, you find this is exactly the background most graduate schools seek. You may want to start examining the classes you are already enrolled in and go to more depth in those courses. This is especially warranted when you mention you skipped some sections. You do not want these skipped sections to fall between the cracks when you encounter problems on a future qualifying exam or the physics GRE.
If you absolutely have to study something unrelated to your coursework, I am not sure I liked many treatments in differential geometry from the mathbooks I examined as it relates to general relativity. I rather suggest the book, Tullio Levi-Civita, the Absolute Differential Calculus (Calculus of Tensors). I heard Einstein himself often consulted with Levi-Civita. I also recommend Morse and Feshbach as a good reference.

Many of you young guys/ women are in a hurry to get to the modern physics and mathematical abstraction. My suggestion is you will see that soon enough. Take the time to enjoy biting into the fruit and let it dribble down your chin. Bite the fruit and devour it completely. You can find important physics in well-trodden disciplines if you look hard enough. Feynman reinforces this throughout his books for public consumption.