Classical Orbits and Quantum Mechanics

How does the correspondence principle apply to the Bohr model and the Rutherford model?

I'll start at the beginning. You know Bohr's basic theory: electrons in atoms can only be at certain energy levels, and they can give off or absorb radiation only when they jump from one level to another. If an electron falls to a lower energy level, a photon escapes; by the conservation of energy, we know that the energy of this photon is equal to the energy the electron lost--that is, the difference between the higher energy level and the lower one. But we also know that the photon's energy is equal to Planck's constant times its frequency; thus, if we know what the energy levels are, we can figure out what the frequency should be.

Yeah, but how do we know which energy levels are allowed?

Well, Bohr came up with a formula for that. He didn't have much theoretical justification for it, but it agreed quite well with experimental data. In 1885, about 30 years before Bohr's work, J.J. Balmer had studied the frequencies of hydrogen's spectral lines and had discovered a nice equation that fit these frequencies perfectly. Bohr found that his theory agreed precisely with this formula if he assumed that an electron's angular momentum was restricted to a certain set of values. (It had to be an integer multiple of the quantity Planck's constant over 2 pi, or h bar.)

 I can show you what Balmer's formula was and how Bohr derived the electron's angular momentum from it.

Given the angular momentum, Bohr could easily find the electron's speed and orbital radius, which would allow him to calculate its kinetic and potential energy. This in turn meant that the difference in energy between any two orbits could be found, so the frequency of he corresponding photon could be calculated.

Okay...now what's all this about closely packed energy levels?

As I said, the angular momentum had to be an integer multiple of h bar; the integer used was known as n. A value of n=1 corresponds to the ground state, where the electron possesses its lowest possible energy. Examining his formula, Bohr noticed that as n grows larger, the difference between consecutive energy levels becomes smaller and smaller; in fact, it approaches zero as n approaches infinity.

So the energy levels are "closely packed" when n is large. But what does this have to do with the classical model?

Remember how we can derive the electron's orbital radius and speed from Bohr's formula? These values in turn allow us to calculate its orbital frequency--i.e., the number of orbits it completes per unit time.

That's what we called the "vibrating frequency," right?

Correct. Now, here's the amazing part: As n gets larger and larger, the difference between the energy levels n and n+1 gets closer and closer to Planck's constant times the orbital frequency at level n.

Wait a second...we said the difference between the energy levels had to be equal to Planck's constant times the photon's frequency. So at very high values of n, the photon's frequency would be pretty much the same as the "vibrating frequency" of the electron--which is what Rutherford's model predicts!

That's right; this is where classical and quantum mechanics overlap. Another way to think about this is that the frequency of an emitted photon always lies between the orbital frequencies of the two energy levels involved (a fact that can be proved, with a little algebra). When the energy levels get close together, there isn't much "space" between them, so the photon's frequency is squeezed closer and closer to the orbital frequency.

 I can show you an algebraic proof of this relationship between the Rutherford and Bohr models.

This is great. Now I can see the connection between the classical way of looking at things, which seems to work well for big objects like stars and planets, and the Bohr model, which describes the way things really are on the atomic scale...

Well, you don't quite have the whole story yet. In fact, no one knows exactly what's "really" going on inside an atom, but we do know that the Bohr model isn't quite right; electrons don't go around in orbits at all. The Schrödinger model agrees more closely with the experimental evidence we have, so we presume that it comes closer to describing reality.

Oh, not another model! How does the Schrödinger model explain radiation from atoms? Does it still fit in with the idea of "vibrating charges"?

Well, that's another story...

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