Brainfarting reading Griffiths QM (small step in solving Hydrogen atom

[ThereIam]

I can tell this is simple, but I'm just not seeing it: (pages 146-147)

Radial equation = d$^{2}$u/dp$^{2}$ = [1 - p$_{0}$/p + l(l+1)/p$^{2}$]u

Later... (having stripped off the asymptotic p$^{l}$e$^{-p}$ parts)

d$^{2}$u/dp$^{2}$ = p$^{l}$e$^{-p}${[-2l-2+p+l(l+1)/p]v + 2(l+1-p)dv/dp + p*d$^{2}$v/dp$^{2}$}

And he says, "In terms of v(p), then, the radial equation [as I put it above] reads

p*d$^{2}$v/dp$^{2}$ +2(l+1-p)dv/dp + [p$_{0}$-2(l+1)]v=0.

Wot?

On a loosely related note, should I bother to memorize these sorts of derivations? And at what point in my physics career ought I be proficient at busting out the power series method to solve differential equations?

 

[WannabeNewton]

What exactly about the derivation isn't clear? It looks straightforward to me. Just plug the expression above (4.61) into the LHS of (4.56) and use (4.60) in the RHS of (4.56). Then simplify using a bit of algebra.

You should understand the technique used here as it is a very important one and you must become proficient in it. It is called Frobenius' method. Of course there's no point in memorizing as that will not make you proficient. You should do a lot of problems which involve the use of Frobenius' method. Griffiths has a couple of them.

 

[ThereIam]

Thanks, that's what I was doing. I must have just been screwing up the algebra.

Yeah, I was asking two different questions: 1) Should I memorize these derivations and 2) by what point in my physics career should I be proficient with the Frobenius method (not "Is it important to know how to use the Frobenius method?"). You didn't answer either of those, haha, but thanks for the first part. I do assume that I need to learn Frobenius (and I actually do feel like I understand it, but as you pointed out, practice is in order).

Thanks!

 

[ChrisVer]

In what point in your career do you need to know a method is absolutely not an answerable question, because it makes no sense. It helps you solve some problem, so it's useful to know. In what point of your career do you need to know how to find some limits?
Also memorizing a method is not appropriate. In one month or year, it will be forgotten.

 

[strangerep]

[...] by what point in my physics career should I be proficient with the Frobenius method (not "Is it important to know how to use the Frobenius method?").

Hah!

You should have heard the loud collective mega-brainfart in the lecture hall when I was first exposed to solving the Schrödinger eqn for the Hydrogen atom. The lecturer just threw around terms and methods (e.g., separation of variables, Legendre functions, Laguerre polynomials, etc, etc), which no one in that class had yet studied in their math courses.

Unfortunately, the answer to your question is: if you're not already familiar with a technique, which then slams you out of nowhere in a (sloppily-presented) physics class, then that's the time to learn the technique. At least you have PF as a resource, where nice people will advise about helpful textbooks, etc. I had no such help available.