What sort of math classes should be taken for quantum theory/physics major?

[Seele]

I'm beginning college winter semester, and am just curious what sort of math classes I would be most appropriate for understanding the mathematics behind quantum theory. I have several years background in cosmology and astronomy, however the subject of the quantum has just been to tantalizing to ignore. Mostly, I am interested in the probabilistic nature of things at the quantum scale, but beyond that I've been wondering what sort of math courses would most help me in solving probabilistic equations.

My mathematics background as it stands now, consists of advanced algebra, some calculus courses, and I am now beginning to teach myself linear algebra. however, I am somewhat at a loss as to what courses would be most applicable to quantum mathematics.

Was just wondering if anyone had some suggestions as to what I should consider taking in the near future.

 

[G01]

Differential Equations and Linear Algebra are used a lot in Quantum Mechanics. As for the actual requirements for a Quantum Mechanics course at your school, you should probably talk to your adviser.

 

[Seele]

indeed I should talk to an adviser, but I wanted to get a little perspective on the issue before I did that, so at least I would be able to get some background on what he is recommended for me to take.

As far as linear algebra goes, is there a particular order in which I should take the classes, or are they more or less independent of one another?

 

[Nolen Ryba]

Having somewhat recently scaled this hill myself, I'd say it's most important to understand the ideas in linear algebra and how they apply to differential equations. Also, you will want to know the very basic ideas of probability. I'd list off the key concepts to focus on (in this order) as:

-the idea of projection and how it is used to decompose a vector into components
-the eigenvalue problem
-certain sets of functions (function spaces) form a vector space and the dot product generalizes to an inner product
-linear differential equations can be understood as linear operator equations on function spaces
-to put it together, in the quantum mechanical formalism the state of a particle is a function ("wave function") or in other words an element of a vector space (function space), observable quantities (position, momentum, etc.) are associated with operators on this vector space, and the values of these observable quantities that can be observed are the eigenvalues of the operator.
-the idea of expectation value and how the above ties into the above

You need to make sure you understand the above, but do not get me wrong, doing a lot of computations along the way (in particular, with the last item) help you "see" how this works out. Also, I should note that most of these ideas are far from specialized to quantum mechanics. Studying such things as PDEs or linear systems / signal processing can also gain you much insight into this. Hope this helps.

 

[leon1127]

linear algebra (Advanced Matrix Theory is a plus)
Vector Calculus
Advance Probability
Calculus of Variation
Fourier Analysis
Theory of Integration
Functional Analysis

 

[Daverz]

linear algebra (Advanced Matrix Theory is a plus)
Vector Calculus
Advance Probability
Calculus of Variation
Fourier Analysis
Theory of Integration
Functional Analysis

If time is tight (and it usually is if you aren't named Julian Schwinger), I'd say only linear algebra and Fourier analysis (usually part of a PDE sequence) are necessary.

And one usually needs to pick up the method of residues somewhere (a math methods course or a full complex analysis course; complex analysis is the most beautiful of all the undergraduate math subjects IMO). If time allows, as much of the other math as you can handle will make you a more sophisticated user of mathematics, particularly if you are aiming for theoretical physics. I'd put the analysis courses ahead of calculus of variations. Also a modern algebra course (but by then you are close to earning a BA in math.)

 

[jtbell]

Your college's Web site and printed catalog should list the prerequisite courses for each course. Note that most in most colleges (in the USA), QM first appears not in a course called "Quantum Mechanics," but in a first- or second-year course that is often called something like "Introduction to Modern Physics." This course usually assumes that you know only basic differential and integral calculus, and it introduces other mathematical topics that are needed at this level. The book that I've been using for this course introduces or reviews partial derivatives, complex numbers, and differential equations. This is enough to do simple examples like the "particle in a box."

A full-blown "Quantum Mechanics" course usually assumes that the student has already been through an introduction like this. At this point it's helpful to have been through a full differential equations course, linear algebra, Fourier analysis, etc. But for a beginning treatment as part of a "modern physics" course, usually all you need in advance is basic calculus.

It's very common for a physics courses to introduce or at least review math that is specifically needed for that course. Another example of this is electricity & magnetism. An intermediate-level E&M course usually spends a lot of time on vector calculus at the beginning.

 

[Poop-Loops]

My school has a bunch of math (Linear algebra, Diff EQ's, Vector Calculus, etc.) and a mathematical physics series (Fourier analysis, Lagrangian mechanics, more Diff EQ's, calculus, and linear algebra) before you can take QM. You take QM 3rd year of your major. And like jtbell said, the intro to QM was in Modern physics, which gave a more conceptual view of it and why it was needed. QM proper just jumped into Schrodinger's equation.

 

[eastside00_99]

Yeah, I wouldn't ignore vector calculus and a course in modern physics also. At my university it goes something like this:

vector calculus and basic physics --> diff eq modern physics --> e&m 1, mechanics 1, linear algebra --> e&m 2, mechanics 2, quantum 1 --> quantum 2.

But you have other required courses you have to take. So my advice is to develop a plan of study with your advisor and on your own for the next couple of years that includes the quantum sequence. Of course, formal eduaction processes and you could probably learn quantum on your own without taking a course as you are starting to do. But, you do want a degree and the courses give you that.

 

[mathwonk]

take some english courses so you can explain your insights to other people, and take some art appreciation and music courses so you can decorate your home and buy good dvd's for it as well. these will also help you "woo women'. if you are not yet married.

 

[Poop-Loops]

Man, English/humanities in general courses are so annoying. I had to write a 5-6 page essay on something that would have taken me 3 pages tops to explain. Ok, two pages extra, right? Too bad all my other courses (math, physics, chem, comp sci, etc) have ingrained it into my brain that being concise and elegant is the way to go, not long-winded and drawn-out.

It sucks even more because I like humanities. If it wasn't for the crappy homework I'd take a bunch more.