The Connection Between Classical Orbits and
Quantum Mechanics
All right, 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 the
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 "wiggling 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
"wiggling 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...
Um...I hate to disillusion you, but 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 "wiggling charges"?
All right, 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.)
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 the
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.
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 "wiggling 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
"wiggling 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.
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...
Um...I hate to disillusion you, but 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 
Oh, not another model! How does the Schrödinger model
explain radiation from atoms? Does it still fit in with the idea
of "wiggling charges"?
Well, that's another story...