Physics 2000 Science Trek Quantum Atom

Vibrating Charges Revisited

I'm not sure I understand what you mean by "vibrating." How does the Bohr model for an atom relate to the ideas we were talking about earlier? You compared "vibrating charges" to oscillating springs; how does that fit in with Bohr's idea of jumps between orbits?

Let's step back from the Bohr model for a moment and look at an earlier model for the atom-- don't worry, you've seen this before. Remember when you were playing around with orbits, and I said you'd created a simplified version of an atom?

That was just an electron orbiting around a proton.

That's right; in this model (which is sometimes called the "Rutherford model"), an atom is made up of electrons orbiting a nucleus in the same way that planets orbit around the sun. The electrons are held in their orbits by the electric force, just as the planets are kept in theirs by gravity, and the entire atom resembles a miniature solar system. This is known as a "classical" model, meaning that it doesn't use any quantum ideas, just Newtonian mechanics applied to the electric force.

So does the Rutherford model relate to the "charge on a spring" idea?

It certainly does. An electron orbiting a nucleus moves periodically, just like the charge on a spring you saw earlier. Both are undergoing simple harmonic motion; if you project the orbiting electron's motion into one dimension, it will look just like a mass oscillating on a spring.

So if an electron is orbiting, you could say it's "vibrating" at a certain frequency, which depends on the radius of its orbit. That makes sense. But when we discussed the "charge on a spring," you said that if an electron vibrates at a particular frequency, it must produce electromagnetic radiation at that same frequency. So if you had, say, a hydrogen atom with an electron orbiting at some fixed radius, it would always be giving off radiation--that can't be right.

Yes, Rutherford's picture of the atom has a couple of fundamental problems. First of all, there's no apparent reason why an electron's orbit couldn't have just any old radius, and thus any old frequency. That flatly contradicts the experimental evidence of spectral lines.

Well, isn't that why Bohr came up with the idea of electrons' being restricted to certain special orbits?

Yes, that's a partial solution. Let's assume that an electron's orbit, for some mysterious reason, can only have certain discrete radii. Now suppose that an electron is happily cruising along in one of these legal orbits. As you've said, the electron is "vibrating," so it ought to be producing radiation. So, let's say our electron emits an electromagnetic wave of the proper frequency. That's all well and good until you start to think about the energy contained in that wave...

Waves have energy?

Take another look at that vibrating charge. Notice how when the wave reaches the negative charge on the left, that charge starts bouncing up and down? The original + charge never touched that negative charge; only the wave did.

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If the wave made that second charge move, then it must have carried energy from the + charge to the - charge.

So the moving wave contains energy...and energy is conserved. Then in order to emit the wave, the electron has to give up some of its own energy.

Exactly; the electron would have to slow down, which would decrease its kinetic energy. But if it did that, it wouldn't be able to remain at a fixed radius; it would be pulled in closer to the nucleus. (Note: this actually does happen for macroscopic objects in orbit; stars in a binary system, for example, spiral slowly in toward one another, because as they orbit they give off energy in the form of gravitational waves.)

And meanwhile the electron would still be wiggling, so it would give off another wave (at a different frequency), which would make its orbit decay even more...when would this ever end?

It wouldn't. The electron would keep right on spiraling inward until it crashed into the nucleus, at which point there would be no more atom. So if this classical picture were correct, atoms would be highly unstable, and nothing made of atoms could possibly exist for more than a fraction of a second. You and I couldn't be having this conversation if it weren't for quantum mechanics.

Okay, so it sounds like you're saying that the Bohr model and the Rutherford model actually have nothing to do with each other. Rutherford's classical picture was just completely wrong, so Bohr had to come up with something entirely new.

Well...yes and no. This classical model is inaccurate, but there are cases in which the correspondence principle applies. Just as Newtonian mechanics is a good approximation to relativity at low velocities, Rutherford's model is a good approximation to the Bohr model for closely packed energy levels.