How Can We Solve this Second Order ODE for Electron Behavior?

[CSteiner]

Homework Statement

I'm taking an online introductory chem course, and while explaing the failure of classical mechanics to describe electron behavior, the teacher brought up the following ode which is based on Newton's second law and coulombs law:

-e^2/4(pi)(epsilon-nuaght)r^2=m(d^2r/dt^2)
she said that given the initial condition r=10 angstrom, the radius would be 0 at 1 nanosecond

2. Homework Equations
How did she solve this? I thought for separation of variables, you needed both an initial x and f(x). Here there is only the f(x), the initial radius.

The Attempt at a Solution

The only thing i can think of is that inital t is 0. even so I am still rusty at doing double integrals an eliminating the integration constants. Any help would be appreciated!

 

[vela]

Your instructor's point was probably that if the electron was initially in a 10-angstrom circular orbit, according to classical electrodynamics, the orbit would decay in an extremely short time, so atoms would not be stable. To do the actual calculation, you'd have to know how the electron radiates when it's accelerated.

 

[UVCatastrophe]

Adding to the post above...

That's not an easy differential equation to solve, but fortunately the solutions are well-known. Actually, the problem has been around since before Newton invented calculus, and it's called the Kepler problem. Note that the spatial dependence of the Coulomb potential is identical to the gravitational potential (they both behave like 1/r). If you're interested in solving this, you'll need a centrifugal term and I refer you here: http://en.wikipedia.org/wiki/Kepler_problem

However, this two-body problem is a red herring. Don't actually use your precious time solving the two-body problem.

Your instructor's point was probably that if the electron was initially in a 10-angstrom circular orbit, according to classical electrodynamics, the orbit would decay in an extremely short time, so atoms would not be stable. To do the actual calculation, you'd have to know how the electron radiates when it's accelerated.

The whole reason your professor mentioned this is because of a specific problem for atoms that motivates the development of quantum theory (which is highly relevant for chemistry). Rutherford et al discovered through empirical observation that atoms are made up of positive charges surrounded by negative ones in "orbit." Great, so people wrote down those equations, but they found a huge problem. The classical solutions to the two-body problem, which work so well for celestial mechanics, fail here, due to the fact that any orbiting particle must be accelerating, and charged particles are known to radiate when they accelerate. So the particle radiates, losing energy, and spirals into the nucleus in less than a nanosecond...

Except that doesn't happen, because we know matter is stable. This conundrum is a big motivation for quantum mechanics. You'll see in your class that electrons have truly bizarre, non-classical behavior, and long story short-- they don't radiate nonstop and matter is stable in quantum mechanics.

The problem you should do is found in Griffiths Introduction to Electrodynamics, Problem 11.14. Use the Larmor formula for the emitted power, and the fact that $P = \frac{d E}{dt} = \frac{d E}{dr} \frac{dr}{dt}$, plug initial conditions and calculate the lifetime. Remember, this is a classical calculation applied to a situation where classical mechanics is known to fail.

 

[CSteiner]

Thanks, guys! To be clear, I knew that that was my teacher's point, and that the solution of the ODE doesn't even actually apply to the electron, I was just interested in the theoretical process of solving such an ODE. I was assuming it was a relatively simple ODE that could be solved using regular separation of variables or techniques of solving homogeneous linear ODES. I see now that's not the case, which explains why my attempts were failing. Thanks for clearing it up for me!