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Math Background Problem in Physics (Grad School)

  1. Oct 6, 2008 #1
    Hello Everyone,

    I just started grad school this semester, and I realized that I have a background problem compared to my classmates. The thing is when I talk to them about that, they also say that they feel they have a background problem compared to others.

    Then I thought about it, I thought that if I would go to one of my professors and ask him:

    1. Professor, would you how calculus of variations work again.
    2. Professor, would you solve this differential equation for me.
    3. Professor, would you tell me about the saddle point method.
    4. Professor, would you derive me laplacian operator in an n dimensional general coordinate system
    5. Professor, would you explain me the virial theorem.
    6. Professor, ... etc.

    I am sure that the Professor will not be stuck (maybe a little, maybe s/he will look through a book a while).

    So as you see I am creating a scenario (maybe I am going mad). Then I realized that it is because they have a lot of experience, they did a lot of practice. And most importantly their knowledge is NOT FRAGMENTED. So they can go through what they know to the point.

    So how did they get this experience, how did they defragment their knowledge. Now it is obvious to me that you cannot learn physics from coursework alone or using a single textbook for a subject. On top of it there are rules of the game, which is called "math", that if you only know half the rules you cannot play the game. So for example you know about the schroedinger equation right, and the discrete eigenvalues and eigenfunctions their completeness and closure, cool, now show that this holds!!!

    And do the books of type "Mathematical Methods in Physics..." help. No way. They are full of methods, techniques and so and so. One of them talks about matrices in its linear algebra section but has nothing about infinite dimensional vector spaces, you say "OK I will just pick this book up which has that section". To your amazement the notation is nothing like the one you used before and for example it does not tell you about hermitian operators in infinite dimensional vector spaces which is what you need at the time, "great here is another one I will just use this and it seems mathematically more rigorous". Then what, a completely new notation again, you say "anyway I have to learn from this" linear vector spaces is on page 300 and you cannot read 300 pages in one week and it would be ridiculous to do so because what you need is on page 300. When you go to that section to your amazement.

    "If f(z) is a holomorphic function in a simply connected region where in the neighborhood of z you can find any z' which is an utterly megatrone complex number which has a prime number in its complex part and the difference of the limit of f(z) and hermite polynomials of the form dadada as zadfdsfadf goes to lajfljdfkaldf ..............................."

    So you realize that any physicist must at least know mathematics as a math major.

    I hope you can see what I mean, I am sick of this, I have a terribly fragmented knowledge, If I will be stuck with a problem I want to get stuck in a part about physics, not about mathematics, on top of it if you are stuck with math then you also get behind in terms of your physical knowledge too.

    I see here that people recommend books to eachother, read shankar QM, no read sakurai, read landau lifgarbagez, read math methods by this, no no read a rigorous real analysis book
    read linear algebra by serge lang etc. etc.

    If you try to read all of them (including doing the exercises) and do your other duties, the time it will take will not be a reasonable time even for a genius of geniuses.

    So would somebody who has been in this situation I described above, or who really understands my condition please recommend me what to do to have a strong background in mathematics. I know that it is impossible to be the master of all that is mathematics, but I also know that there are people who do not suffer that much from it.

    And I am not in a hurry, I can read 3-4 extra books, very slowly, doing the exercises, I can also read pure math books as long as they are "self contained" and I would love to. Please anything that comes to your mind, just save me from this cursed endeavor of filling in the gaps and not connecting the dots.
  2. jcsd
  3. Oct 6, 2008 #2
    How much more do you need to learn exactly and how deeply? I'm still an undergrad, but whenever I come across something (say in undergrad research) that I haven't heard of or don't understand or don't remember, I Google it, or better yet Wikipedia. Then, if I don't understand their definition, I Google or Wikipedia the terms I don't understand.

    This can take a few hours for me, but in the end I know what I needed to know and I didn't need to buy a book.

    But if it's something more in-depth or you want to know an entire subject... then yeah, you'd need a book... I can't help you there, I'm afraid.
  4. Oct 6, 2008 #3

    George Jones

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  5. Oct 6, 2008 #4
    Honestly I think your best bet is to concentrate on learning quantum mechanics, which combines all of the math skills you want to practice with the most important physical framework for most fields of research. You can also learn the calculus of complex functions in the context of electromagnetism, which would be my second recommendation.

    Why am I suggesting physics books instead of math books? I think that without taking time off from graduate school you couldn't learn math like a math major, as you said there is simply not enough time to learn all that on the side. The upside is that you do not need to learn math like a math major to do most kinds physics research, let alone physics coursework. I am suggesting physics books because they will be written in a style that is accessible to you with your current skill set. I think you are not at a great disadvantage, 90% of the mathematical skills I have studied cannot be applied to physics coursework.

    I learned this by reading the proofs many times in several different QM books. Math books are so full of other stuff that these proofs barely appear within them, maybe as a single line. For example, what is referred to as "bra" space in physics (dual space) is actually the set of all linear functionals over the original "ket" space, but even in graduate school you are thought to think of "kets as column vectors, and bras as row vectors." Incidentally, I find the terms "bra" and "ket" to be sexist and vulgar, IMO they should have no place in physics!
    Last edited: Oct 6, 2008
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