Does Theoretical Math Really Help with Physics? | My Experience at Berkeley

[th5418]

So, I'm currently a 3rd year in the Physics program at Berkeley, and this is my first semester transitioning into upper division mathematics. I never knew it would be so theoretical, does any of this theoretical math actually help me do physics problems? Also, I plan on applying for theoretical physics for grad school, if that makes any difference.

 

[Gear300]

Yup, it does...I ran into the same question while starting off this analysis. It was hard to see the connections at first, but they are there. You'll bump into them when you advance yourself in physics...and very much so in theoretical physics. Keep in mind, though, that there are distinct differences between the work of theoretical physicists and mathematicians, despite what many people might say (theoretical physicists have the ability to analyze the properties and behaviors of mathematical structures, but that is not their scope as theoretical physicists).

 

[th5418]

I see, what classes are suggested to take? I'm currently taking Linear Algebra, plan on taking Analysis and Abstract Algebra and later on Complex Algebra to get a minor.

 

[rubrix]

Linear Algebra is VERY important. It is often required course for Physics Major. Another recommended course would be Complex Analysis. Take "Applied" version of the courses if it is offered.

You could also try a follow-up sequence to Calculus III (multivariate), sometimes called "Advanced Calculus".

P.S. do you like more of 'pure' math? I'm just asking since you are taking Analysis and Abstract Algebra. I would swamp them with some applied mathematics course if possible.

 

[th5418]

I hate pure math. But there are a limited number of applied mathematics courses here.

 

[rubrix]

can you perhaps list what courses they offer?

another thing you could do is take CS courses. They'll come handy!

you can check this thread if you like

https://www.physicsforums.com/showthread.php?t=324760

 

[snipez90]

I'm not a physicist, but I think the whole "hate pure math" but want to go into theoretical physics deal doesn't bode well. I'm fairly confident that if you want to study certain topics in theoretical physics deeply, you'll either need a background in certain math courses that are considered pure or you'll be dealing with a level of abstraction that is more or less the same you encounter in a pure math course. For instance, you probably won't get very far in general relativity without an understanding of linear algebra (I'm sure the more abstract aspects play a larger role as well). If you don't have a basic aptitude for linear algebra or analysis, then you probably won't make it very far in functional analysis which is used to study quantum mechanics. I mean a lot of the sophistication you find in mathematical analysis today is probably due to physical considerations in the first place. So yeah, the point is math helps.

 

[th5418]

http://sis.berkeley.edu/catalog/gcc_list_crse_req?p_dept_name=Mathematics&p_dept_cd=MATH

There's the link to all the math major courses. 100-199 are upper division courses. The thing is I don't hate it. I'm just struggling because my background in mathematical proofs is pretty weak.

I'm currently taking Ordinary Differentials (123) and Linear Algebra (110). Plans to take Abstract Algebra (113), Analysis (104), and Complex Analysis (185) are there. I might take Introduction to Partial Differentials if given time.

Also, does physics really use theoretical math? That's what I'm confused about. This theoretical math, I can see, is giving me a more in depth knowledge of what a vector space is and what dimensions are etc. But do I really need to know the depth of background knowledge of what those are when trying to do problems in like.. Quantum Mechanics?

 

[Monocles]

You should definitely take Intro to Partial Diff Eq. before abstract algebra or analysis. It is far more useful - more useful than complex analysis too, though complex analysis is also very useful.

I know you were just using it as an example, but I think having a strong idea of what vector spaces is VERY helpful in physics. For example, when you switch between different eigenstate representations on a wavefunction, you're really just switching your basis in hilbert space. If you haven't taken a course on linear algebra, then this idea won't seem nearly as profound as it actually is.

Things like abstract algebra and analysis being useful is rather field-dependent. Most high level physics uses idea in algebra to an extent, but learning the idea from a physics standpoint is usually much more useful than a math standpoint. I don't know as much about analysis, other than the ideas are useful in topology and algebraic topology, which crops up in a lot of places in physics (from condensed matter to nonlinear dynamics to high energy), but rarely find uses in other fields (though I don't feel knowledgeable enough to say which).

You should also consider a course in probability and statistics and numerical analysis. Both would be more useful than abstract algebra or analysis.

 

[Fredrik]

Linear algebra is extremely important. You can probably skip the abstract algebra. You will eventually need to learn some of the definitions from that course (e.g. what a "group" is), but you can learn those things on your own when you need them.

 

[th5418]

I'm interested in doing high particle theoretical research by the way, if that is any help. I'd like to take a numerical analysis class too. You think I should just skip abstract algebra then Fredrik?

 

[rubrix]

if abstract algebra and analysis is not required for your minor i would suggest swamping. But if you are considering, a double major in mathematics and physics (or might consider at some point) then you should take introductory course on them asap.

121A. Mathematical Tools for the Physical Sciences. (4) Three hours of lecture per week. Prerequisites: 53 and 54. Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Rapid review of series and partial differentiation, complex variables and analytic functions, integral transforms, calculus of variations.

189. Mathematical Methods in Classical and Quantum Mechanics. (4) Course may be repeated for credit. Three hours of lecture per week. Prerequisites: 104, 110, 2 semesters lower division Physics. Topics in mechanics presented from a mathematical viewpoint: e.g., hamiltonian mechanics and symplectic geometry, differential equations for fluids, spectral theory in quantum mechanics, probability theory and statistical mechanics. See department bulletins for specific topics each semester course is offered. (SP)

118. Fourier Analysis, Wavelets, and Signal Processing. (4) Three hours of lecture per week. Prerequisites: 53 and 54. Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images. (F,SP)

126. Introduction to Partial Differential Equations. (4) Three hours of lecture per week. Prerequisites: 53 and 54. Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform. (SP)

128A. Numerical Analysis. (4) Three hours of lecture and one hour of discussion per week. At the discretion of instructor, an additional hour of discussion/computer laboratory per week. Prerequisites: 53 and 54.Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer. (F,SP) you already fulfil requirement for above courses, and i would recommend taking them (highly recommended from bottom of the list to the top). You can scrap the first one if you take the rest.

P.S. i pasted them so someone else can confirm/suggest as well.