Understanding Math: Acquiring Mathematical Skills for Problem Solving

[CPL.Luke]

so I'm studying physics right now, and I'm encountering a number of situations that require relatively obscure concepts that I would have no idea where to find if I wasn't reading about the problem in the book.

for instance right now I'm reading quantum mechanics by boas, and I see that bessel's functions are crucial in the solution of the schroedinger equation for the hydrogen atom, and it occurs to me that had this nifty little trick not been in the book I wouldn't have been able to see that that equation was soluable, and I also now lack any real knowledge of bessel's functions outside of the trick used in the book, so its not like I would see the solution in another problem that was similar as I have not grokked it yet.

I also saw in the library that there are over 2 dozen large books on the theory of bessel functions, so where would I be able to get the grit of the theory without having to read a massive book on the subject?

I'm planning on reading Boas over the summer however I also see the problem that when some concepts are introduced they seem very insignificant (the introduction to bessel's functions in my diff eq book being an example) that my brain doesn't see the need to spend time and grock that concept. So I have to ask when it was that other people here acquired the mathematical skill to study these problems and have a general idea of where the answer to them lies (whether or not they are entirely familiar with the subject or not).

 

[JasonRox]

I know nothing about what you're curious about, but I have one thing to tell you.

Sometimes it's not necessary to read the whole book but maybe only 2-3 chapters to get all that you need out of it.

For example, if a student reads the first 3 chapters of Munkres Topology, he/she will have a relatively good understanding of what Topology is in it's basic sense. Which is a hell of a lot more than what any other regular math major might know.

Note: The textbook has like 12 chapters.

 

[Crosson]

Bessel functions, and other special families of functions that arise as a solution to diff eqs, are only interesting in so much as they apply to particular physical problems. In fact, the reason why the sine and cosine are so important is that they are the solution to the simple harmonic oscillator: I am sure you are not overwhelmed by the fact that there are hundreds of books on trigonometry.

All the special functions of classical physics are solutions to second order linear equations, and the properties of such solutions are studied in general in Sturm-Liouiville theory. I think that if you study Sturm-Liouville theory (sometimes it is in the last chapter of a PDE book) you will better understand the role of special functions in classical and non-relativistic quantum physics.

I know nothing about what you're curious about

http://en.wikipedia.org/wiki/Bessel_function