Specialized mathematics courses a physicist should take?

[bjnartowt]

Hi everybody, I'd like to ask: besides your "calculus", "ordinary/partial differential equations", "complex numbers", "probability", and "linear algebra" that a physicist should be familiar with, what else should I study? My goal to studying this math is to understand my physics-graduate-course-core-material from as many perspectives as possible. For instance:

- it seems in quantum mechanics, the wavefunction is part of the set of all functions that are square-integrable to 1, and that it is complex-valued. Would my understanding of QM therefore be furthered by...I think...a "real analysis" course?
- Group theory also seems important. Therefore...should I study "abstract algebra"?
- So-called "differential geometry" for general relativity, I think?
- Are Lie groups/algebra/chickens important in physics?
- ?

There are other examples I could give. Or, maybe I could study math on my own? Advantages to self-study:

1) My grad advisor doesn't want me to load up on courses, and I'm about 60% in agreement with him, knowing that I tend to bite off more than I can chew
2) I've discovered a few tricks and techniques to self-study that has made it easier (which I will be more than happy to share with askers)
3) in self-study, I won't be stuck considering "pathological" examples that are very far-removed from physical reality...like this Wierstrauss (sp?) function I just skimmed a Wikipedia article about 5 minutes ago that is supposedly everywhere continuous but nowhere differentiable.

So yeah...my original question (to recapitulate): what specialized math is good for best-understanding the core-classes of physics that I will be crunching through in 1 month's time? For instance: I guess Arnold's "Mathematical Methods of Classical Mechanics" would be good for...classical mechanics.

I think I just answered my own question (get math-books with corresponding titles!), but please feel free to make recommendations.

 

[ZapperZ]

Do a search on here on Mary Boas' "Mathematical Methods in the Physical Sciences" text. There's been a lot of discussion, and recommendation, of this text.

Zz.

 

[DukeofDuke]

Please share those tricks and tips on self study! I intended to do that this summer and failed miserably...

 

[bjnartowt]

Okay, DukeofDuke! I have some "dos" and "do nots":

1) DO read multiple books on one topic. To me, it's pleasing (not to mention encouraging) to see the same principles postulated in different places in different forms, and surrounded by different descriptions and proses of different authors.

2) DO teach yourself PHYSICS rather than ALGEBRA. Meaning: I used to do out every algebraic step in derivations, and that got me wondering about things like "does 1/1 - x = (1 + x)(1-x^2)"? That's stuff you have to learn in an algebra class, not in a quantum mechanics class. Your time is valuable: spend it studying what it means for a particle to have a wavefunction of "such and such" in an infinite potential well, and (I think) an expected momentum-value of zero, not whether you integrated (cosine(a*x))^2 properly.

3) Sometimes, it's better to have an end in mind. I found myself "failing miserably" when I was cramming, chapter by chapter, the context of even the best textbooks into my brain because "Brad-you're-supposed-to-know-this". Then, I was asked by a professor (whom I'm doing research for) to "calculate the quantum state of a photon in the presence of an electron/positron pair", and he said "To do that, you need to know special relativity". Then, you should have seen me gobble up special relativity textbooks! I had an *end* in mind, rather than "You-should-know-this-and-gosh-damn-you-if-you-don't-you-miserably-lazy student".

4) I find I work best when I don't have an indefinite amount of time set aside for studying. Like the dialogue: "Ok, Brad, you're awake and it's 7 AM or so." "Great!" "Time to study!" "How long?" "Why do you ask? Start studying." (Translation: until like 11 PM that night). The result of that bit of self-lambasting: facebook.com, gmail.com, ooh, just one more YouTube video, because this is so damned tedious...

It's nice...I just keep reminding myself that Man was not made for physics...rather, physics was made for Man. We're worth so much more than any number of days and nights in which heavily-mathematical textbooks have been successfully cracked. ::gets down off of soapbox:: I have some other ideas...but they don't immediately come to mind. If you have more troubles, please ask me how to resolve them. I might have a semi-helpful reply...

 

[DukeofDuke]

Okay, DukeofDuke! I have some "dos" and "do nots":

1) DO read multiple books on one topic. To me, it's pleasing (not to mention encouraging) to see the same principles postulated in different places in different forms, and surrounded by different descriptions and proses of different authors.

2) DO teach yourself PHYSICS rather than ALGEBRA. Meaning: I used to do out every algebraic step in derivations, and that got me wondering about things like "does 1/1 - x = (1 + x)(1-x^2)"? That's stuff you have to learn in an algebra class, not in a quantum mechanics class. Your time is valuable: spend it studying what it means for a particle to have a wavefunction of "such and such" in an infinite potential well, and (I think) an expected momentum-value of zero, not whether you integrated (cosine(a*x))^2 properly.

3) Sometimes, it's better to have an end in mind. I found myself "failing miserably" when I was cramming, chapter by chapter, the context of even the best textbooks into my brain because "Brad-you're-supposed-to-know-this". Then, I was asked by a professor (whom I'm doing research for) to "calculate the quantum state of a photon in the presence of an electron/positron pair", and he said "To do that, you need to know special relativity". Then, you should have seen me gobble up special relativity textbooks! I had an *end* in mind, rather than "You-should-know-this-and-gosh-damn-you-if-you-don't-you-miserably-lazy student".

4) I find I work best when I don't have an indefinite amount of time set aside for studying. Like the dialogue: "Ok, Brad, you're awake and it's 7 AM or so." "Great!" "Time to study!" "How long?" "Why do you ask? Start studying." (Translation: until like 11 PM that night). The result of that bit of self-lambasting: facebook.com, gmail.com, ooh, just one more YouTube video, because this is so damned tedious...

It's nice...I just keep reminding myself that Man was not made for physics...rather, physics was made for Man. We're worth so much more than any number of days and nights in which heavily-mathematical textbooks have been successfully cracked. ::gets down off of soapbox:: I have some other ideas...but they don't immediately come to mind. If you have more troubles, please ask me how to resolve them. I might have a semi-helpful reply...

Dang, if you come up with anymore post them! #3 was really good especially, I too have noticed that "ok you got to get STRONGER at physics because uh...you just do..." doesn't create long term motivation no matter how hard you scream it. And #4 is kind of a problem too, facebook gmail and fml I rotate those three all day

 

[bjnartowt]

Also DukeOfDuke, when doing physics examples to gauge your ability to "do" the physics you're self-studying, it's important to understand the spectrum of errors that may occur. The errors you should be LEAST concerned with are algebra-mistakes, and dropped signs. The errors you should MOST be concerned about are inabilities to set up force-balances and stuff (for instance).

Do you know what Lagrangian mechanics is, and how you get the same equations of motion as Newtonian force-balances? Sometimes, you're given a complicated-looking physical system, and asked to solve for x(t). However, the important step, really, is getting the Lagrangian. Once you have that, it's just symbol-crunching and ODE-solving techniques. The "physics" lies in getting that Lagrangian. Perhaps you are overwhelmed at solving the whole problem? Maybe you should just consider the "physics" part of the problem. I stopped in the middle of a huge problem when I saw I'd gotten the right Lagrangian. Then, I paid attention to limiting cases in the solution that verified that I understood the physical implications of the solution.

 

[bjnartowt]

Now, if I'm getting too long-winded (which I tend to do), I don't oblige you to politely nod your way through all this un-asked for advice I'm heaping on your plate. So yeah...I'm just going to enjoy flapping my yap about my experience studying physics...see what flapping helps your cause : )

 

[mal4mac]

Okay, DukeofDuke! I have some "dos" and "do nots":

1) DO read multiple books on one topic. To me, it's pleasing (not to mention encouraging) to see the same principles postulated in different places in different forms, and surrounded by different descriptions and proses of different authors.

I don't see this as a necessary. If you understand Newton's laws from your set textbook, and can do all the questions set, why bother reading another book on the topic? Why not go and read Dickens instead?

2) DO teach yourself PHYSICS rather than ALGEBRA. Meaning: I used to do out every algebraic step in derivations, and that got me wondering about things like "does 1/1 - x = (1 + x)(1-x^2)"? That's stuff you have to learn in an algebra class, not in a quantum mechanics class.

If you don't know something as basic as that then you better learn (re-learn) how to do general algebraic simplification now and quick! Otherwise how will you cope if you need to simplify a similar expression in an exam? At this stage, you should know this and there is no algebra class coming along to fix it (that was last year), so you better fix it!

3) Sometimes, it's better to have an end in mind. I found myself "failing miserably" when I was cramming, chapter by chapter, the context of even the best textbooks into my brain because "Brad-you're-supposed-to-know-this". Then, I was asked by a professor (whom I'm doing research for) to "calculate the quantum state of a photon in the presence of an electron/positron pair", and he said "To do that, you need to know special relativity". Then, you should have seen me gobble up special relativity textbooks! I had an *end* in mind, rather than "You-should-know-this-and-gosh-damn-you-if-you-don't-you-miserably-lazy student".

We don't all have a professor providing a prompt like that. I'd recommend reading the textbook prefaces carefully. That should tell you the end you are aiming for.