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