Math electrons and atoms

Ignoring relativity and quantum mechanics an electron would spiral into the nucleus of a hydrogen atom in 1.3x10[sup]-11[/sup] seconds.

This result is obtained by integrating from r = r[sub]0[/sub] to r = 0 an equation derived from the combination of the Larmor power radiation equation, and the equation for the total energy of the electron.

The total energy of the electron equals the kinetic energy plus the potential energy.

E[sub]tot[/sub] = KE + PE = (1/2)(kq[sup]2[/sup] / r) - (kq[sup]2[/sup] / r)

Both of the terms of E[sub]tot[/sub] are undefined at r = 0 and the integral of these terms equals infinity - infinity = 0. How can equations that are undefined at a specific point be combined with other equations and yield correct results?

I guess I should say that I can see how the math works but I can?t figure out how you can start out with something that physically doesn?t exist combine it with something else and get something that physically does exist.

I don’t know the problem you describe, so I have to ask, exactly what integral do you have to take? E[sub]tot[/sub]dr?

Also, you can have a function which is undefined at a point, and take the integral of that function over an interval which has that point as one of its endpoints, and the integral can be well-defined. They’re called improper integrals.

Ignoring relativity and quantum mechanics doesn’t leave one much else to talk about, then.

You’re assuming that electrons are solid objects subject only to Newtonian mechanics. Since I am sitting here, typing away, and my electrons have not imploded, I would say that it’s safe to assume that they’re not. This issue has been done to death on the “sci.chem” usenet group.

There’s nothing to see here, folks…keep movin’…

Maybe that’s it. Other than the big hairy constant in front of the integral its very simple

Int r[sup]2[/sup] dr from r[sub]0[/sub] to 0

This is a problem from “Introduction to Electrodynamics” by David J. Griffiths.

I believe he is trying to show how quickly an electron would radiate away its energy if it weren’t for the QM groundstate. In any case I would be willing to bet, that since he’s a theoretical physicist with many published papers, books and texts, that he at least knows as much as you do my dear Dizzwire.

It might help to think about the assumptions that were made when the formulas you are using were derived. If you look at coulomb’s law, for example, to have the distance between the two particles mean anything, the size of the two particles has to be small relative to that distance. Otherwise, your distance isn’t even well defined. Even ignoring quantum mechanics, its still a problem to assume that the size of an electron is small with respect to all sizes. That is, as you take the limit as the distance between two particles goes to zero, at some point the size of an electron becomes significant and your formula is no longer valid. Thus, even ignoring quantum mechanics, those formulas are not designed to have any meaning for distances very near zero.

Hope this helps.


Ah, what do you know? I have that book. If you have the third edition, I believe the problem you’re looking at is 11.14. If so, I just did it and got T = 1.55 × 10[sup]-11[/sup] s, so I guess I did it right. I also concur about your integral, r[sup]2[/sup]dr. But it’s not simply a matter of integrating the energy or the power, as I’m sure you figured out. It’s more like the energy divided by the power, with the acceleration put in terms of r. Lots of r’s being thrown around and cancelled.

But even if it didn’t cancel so neatly, as I was saying before, you can also take the values of certain improper integrals that go to infinity at 0, such as r[sup]-1/2[/sup]dr.

Oh, and there seems to be some difficulty here about what exactly the question is asking. People, it’s a classic* problem to show mathematically that classical electrodynamics falls short. The whole point is that it isn’t what really happens. Read Problem 3 from some physics students’ homework (PDF) to get a better handle on it…

    • Yeah I know I said I wasn’t familiar with it. My bad.

That’s pretty impressive Achernar. I’ve been working with this text for awhile and it still took me a fair amount of time to figure out how to solve it. You just pull the book out and do it in a couple of minutes. This is very depressing.

Don’t be discouraged! I took a course on it two years ago, so I was working with this text for a while as well. Plus I got kinda lucky with this problem; the first thing I tried worked out. Trust me, it’s not always that easy.

Based on the questions you’re asking, I think you’ll do fine. You must be doing alright if you’re up to Chapter 11. My year-long course only got us to Chapter 8. :stuck_out_tongue: