As the saying goes, learning proceeds from the known to the
unknown. This thread assigned us the goal of explaining to
a "sophisticated layperson" whether degeneracy pressure is a
fundamental force or not. It's not at all clear that a
direct approach will work. I gather we're starting with
somebody who doesn't know what degeneracy is, doesn't have a
particularly clear idea what pressure is, and doesn't know
what a fundamental force is, or why it's called fundamental.
It's hard to get from there to the goal in one step.
Here are some stepping-stones that may be of some use:
1) Suppose we have some ideal classical gas (g) in a
isolated container (c) with a piston (p) being loaded
down by a weight (w) in equilibrium.
c c
c c
c W c
cpppppc
cpppppc
c c
c g c
c g c
c c
c g c
ccccccc
Question: Why doesn't the weight descend all the way to the
bottom? It would have lower gravitational potential energy
there.
Answer: If the weight went down, the kinetic energy of the
gas would go up. We have a name for this change in gas KE
as a function of volume: It's called pressure.
2a) How big is a proton, a hydrogen nucleus?
Answer: about a fermi, 10^-15 meters.
2b) How big is an electron?
Answer: zero. For purposes of atomic physics, an
electron is a point particle. No structure.
2c) How big is a hydrogen atom?
Answer: about an anstrom, 10^-10 meters.
2d) What's the binding energy of an electron that close to a
proton?
Answer: about a Rydberg, 13.6 eV.
3) Why? Why isn't the hydrogen diameter one fermi instead
of one angstrom? Why doesn't the electron just spiral in
and sit on top of the proton? It would have 10^5 times
more binding energy there.
Answer: If you try to confine an electron to such a
small volume, it's kinetic energy would go up.
Conceptually (and even semi-quantitatively) you can
think of this as a "box" of a certain size. The
electrostatic attraction is like the weight in question
1, trying to make the box smaller, but it is opposed by
the pressure of the electron-gas in the box.
4) In question 1, if we lower the temperature of the gas,
it shrinks. At zero temperature the weight goes all the
way to the bottom. Does question 3 imply that the size
of the hydrogen atom should be proportional to
temperature, too?
Answer: No. The kinetic energy of an electron is
proportional to temperature at high temperature. As the
temperature decreases, the KE decreases but not all the
way to zero; it levels off at a nonzero value that
depends on the size of the box. This has got nothing to
do with the electron's charge; neutral particles such
as neutrons and even photons behave the same way.
5) How can I understand this in terms of high-school
Newtonian mechanics?
Answer: you can't. We don't live in a Newtonian
universe. We live in a universe governed by the laws of
quantum mechanics. Get used to it.
6) But how come you used one set of rules for question 1 and
another set of rules for question 4?
Answer: I didn't. Quantum mechanics gives the right
answer for question 1 _and_ question 4. No cheating
required. The compression of the gas in question 1 can
very accurately be described as shortening the
wavelength of the wavefunction of the gas particles in
proportion to the shortening of the region they're in.
This is particularly easy to demonstrate with a gas of
microwave photons. You can start out with one liter of
microwaves at 1 GHz and compress them down to half a
liter at 2 GHz. You can demonstrate the key idea using
a violin: Start with your finger near the nut, say A# on
the A string. Pluck the string. Then _slide_ your
finger up to shorten the active part of the string by a
factor of two. You'll hear the frequency go up an
octave. The total energy of the vibrations will be
increased, because you did work on them as you
compressed them into the smaller region (although the
energy levels will not be super-obvious in this demo).
Remember: The quantum description is always right. The
classical description is an approximation that is
sometimes valid at high temperatures and low densities
(and not necessarily even then). If you take any
approximate law and extrapolate it beyond its range of
validity, you'll get fooled for sure.
7a) How come you never told me this before?
Answer: You didn't ask. If you do experiments in the
regime where the classical approximation is valid, it's
hard to notice non-classical effects.
7b) Why can't you give me a simple classical explanation of
degeneracy pressure?
Answer: You're specifically asking about an explicitly
non-classical phenomenon. So don't be surprised if you
get a non-classical explanation. If you're allergic to
bread and cheese and tomatoes, don't order a pizza.
8) Why is a sodium atom markedly larger than a neon atom,
and why is its first ionization energy markedly lower?
Answer: Electrons are fermions. That makes their
wavefunction slightly different from waves on a violin
string. On a violin, you can excite the fundamental
mode as much or as little as you want, over a very wide
range. For electrons, you can't do that. You can put
at most two in the fundamental mode (one spin up, one
spin down) and at most two in the first partial, and at
most two in the second partial, and so on. If you want
to put a lot of electrons in a small box, you will have
to occupy some pretty high-numbered modes. These have
lots of momentum, so the zero-temperature pressure is
higher than it would be for non-fermions. The valence
electron in a sodium atom has so much kinetic energy
that the nucleus can barely hold on to it.
You won't notice this at high temperatures and low
densities, because then you're trying to put, say,
hundreds of fermions into millions of modes, so you can
always find an available unfilled mode for whatever
you're trying to do.
9) What's this got to do with neutron stars?
Answer: The analogy is rather tight. In one case, the
central force is gravitational; in the other case it's
electrostatic. In one case the fermions are neutrons,
in the other case they're electrons. But the essential
part of the story is the same: It's a box full of
fermions. The pressure is due to the momentum and
kinetic energy, which has to do with wavelength.
10) What does this tell us about the fundamental forces
(gravitational, electroweak, and strong)?
Answer: Nothing. Particles have momentum and kinetic
energy quite independent of which fundamental force(s)
they interact with. And boson/fermion character is
independent of the fundamental forces. If you're trying
put too many fermions into a mode, you can't "force"
them in there using a stronger "force". It's just not
going to happen. They will either go into another mode
or not go in at all.