Mathematical Physics for Graduate Studies: Overview

[AllRelative]

I am currently an undergrad in pure Math. Until now, the courses I found the most fun/interesting were Probability 1&2, Geometry and Group Theory. I still have 3 semesters to go.

Prior to math I did some university courses in physics which were Classical Mecanics1, Optics and Intro to modern physics(half semester on special relativity and half on intro to quantum mecanics)

I am thinking of going for graduate studies and I am thinking of going in Applied Math. Mathematical physics looks like an awesome field since I love both disciplines.

I wanted to learn a little more about the amount of physics needed to get into mathematical physics graduate studies.

Thanks!

 

[undefined314]

I'm an undergraduate myself, so I'm not comfortable telling you what a mathematical physics grad program would be looking for. However, I can offer you a basic outline of where your math studies may align well with your undergraduate physics coursework, so you can think about if you want to focus your mathematics studies to compliment what you'll do in physics.

Often, the first three semester constitute a basic intro sequence: Newtonian mechanics, E&M, optics, and quantum mechanics, with a bit of statistical mechanics/thermo and special relativity thrown in somewhere along the line (universities vary quite a bit in when those topics are covered).

If you look at upper-division physics courses, you'll see a much more advanced "redo" of the previous topics, and some "special electives," like general relativity (surprisingly often, this is not a required course), solid-state physics, more work with optics, etc.

In general: An undergraduate-level understanding of Real & Complex Analysis, Ordinary and Partial Differential Equations, and Linear Algebra will help you.

Upper-div classical mechanics typically covers formulations with the Lagrangian and Hamiltonian, and offers a more rigorous treatment of rigid-body rotations. Your first semester of physics, in theory, gives you all the physics concepts needed to solve problems involving a spinning top, but it certainly wouldn't be fun attacking that sort of problem using only that background.

Some related topics include Calculus of Variations and Lie Algebra.

For QM, note that a lower-division QM course generally only covers cases where the Schrodinger equation can be solved analytically, and while those cases are very important for the learner, they are not completely generalizable. The upper-division ones usually dig into Heisenberg's formulation with matrix mechanics, which in many cases, is easier to apply approximation methods. (They should also cover topics in perturbation theory, depending on how in-depth the courses are.)

If you take the upper division quantum mechanics courses, you can expect to see more of the applications of intense linear algebra and group theory.

Thermo/stat mech is all about dealing with large systems, think of something on the order of N=N+1, where there is no observable distinction between when the system had some number of particles, vs. if it had one more or less. Imagine a macroscopic sample of a gas that you might've worked with in a chemistry class. Nobody every asks you to determine the path an individual particle takes, or the velocity of an individual particle. (Food for thought, imagine what kind of memory (or storage) would be needed just to store the positions (in 3-space) of 1 mole of monatomic gas molecules.) Instead, you're mostly concerned with the macroscopic state of the system. Understanding probability theory and the use of statistics is important here.

With the upper-division E&M course(s), you'll be dealing with the formalism of vector calculus. In lower-division courses, you might often work primarily with the magnitude of vectors, and leave the direction as an afterthought, or give more heuristic justifications of where your final vector points. This is not so in a good upper-division course in electricity and magnetism. Expect LOTS of work with the vector calculus. (And solving many PDEs along the way, as-typical.)

If you end up doing general relativity, expect to work with nonlinear systems of PDE's. Unfortunately, if you try to work out the gravitational field due to two black holes in general relativity, you cannot simply superimpose the fields due to each black hole as if each were isolated in space. You must solve for the system.

 

[mpresic3]

In my experience with at least 4 universities, I found The Mathematical Physics graduate course, (none of the grad schools had an undergrad course in Math physics), varied from school to school. Your best bet is to ask your advisors, or at very least, examine the curricula as in the school course catalog.

For example, at one school, the first semester was a complex analysis course with a few weeks of hypergeometric functions. The second semester (which I did not take), concentrated on algebraic methods, like Lie algebras. Most other schools made the courses treat partial differential equations and special functions encountered in classical electrodynamics by Jackson, or Quantum Mechanics which were generally taken concurrently. Another school taught differential forms because that is what the professor wanted to teach. Your school may be teaching very specific and idiosyncratic.

These subjects are "core" courses used for your graduate physics courses. I doubt that taking these courses will give you at taste of what mathematical physicists currently do. You also do not see experimental physicists roll balls down inclined planes like you may see in freshman/ sophomore labs.

In talking to mathematical physicists, one that I remember said, that this is a very demanding field. He said, you have to understand physics like a theoretical physicist and understand math like a mathematician. Luckily, you may not need to select this direction as a career until much later.

I'm glad you liked the first physics courses you completed. Before you choose this direction, you will need many more physics course. One of two events are likely. You may love the advanced physics courses, and continue with mathematics, or you may learn to hate the advanced physics courses, and continue in math and applied math.

My own thoughts are applied math is more diverse, and broader. I know some applied mathematicians who work on modeling the lens of the eye and work with biophysicists. I know some who work in operations research. Some may work along mathematical physicists as well, (I presume). These fields have little to do with each other and need different skill sets.

I do not know as many mathematical physicists, but I think their field is less broad, and less diverse.