Re: Whence Degeneracy Pressure?

["John S. Denker" <jsd@MONMOUTH.COM>]

[This is a continuation of my previous note.]
[I hit the send button by mistake.  Sorry]

John Mallinckrodt wrote:
> ... Degeneracy pressure, however, arises from properties of the
> wavefunctions themselves.

Certainly degeneracy arises from properties of the
wavefunctions themselves.  Equivalently we can say
that degeneracy arises from properties of the creation
operators themselves.

Be carefule about the "pressure" part.  We can't measure
the pressure unless the particles interact with the
container somehow.  But this is mostly a technicality,
not a major conceptual point, so I won't pick apart the
term "degeneracy pressure" unless somebody else thinks
it's necessary.

===========================

Here's a parable:  Back in the olden days, people believed
that objects at rest would stay at rest, and objects in
motion would come to rest.  Friction was considered natural
and ubiquitous, not requiring explanation.  Continued motion
required explanation.
Nowadays, people say that objects at rest stay at rest,
and objects in motion continue in motion.  Inertia is
considered natural and ubiquitous, not requiring explanation.
Friction results from physical processes that are
subject to explanation.

It seems to me that high-school physics gives people the
impression that everything is linear and commutative.
But it's just not true.  Putting on your shoes does not
commute with putting on your socks.  The energy required
to accelerate your car from 0 to 5mph is _not_ the same
as the energy required to accelerate from 60 to 65mph.
The pressure of an ideal gas is !!not!! a linear function
of the density.  It's just not.  It's linear when the
density is low (the non-degenerate regime) but it's not
linear at higher densities.

Read my lips:  Life is nonlinear.

If you expect everything to be linear, you're going to
be mystified over and over again.

Maybe some people think that for simplicity introductory
physics should focus mainly on the linear first-order
approximation to everything.  But if you make a list of
100 things, 98 of which are linear, and somebody asks
about the other two, well, c'mon, those two are nonlinear
because you _selected_ the nonlinear ones.  What did
you expect?  The degenerate regime is nonlinear because
that's what degenerate means!

Asking for a Newtonian theory of degeneracy is like asking
about dehydrated water, or nondissipative friction, or
perfectly circular triangles.

Degeneracy pressure is just pressure, in the regime where
there is degeneracy, i.e. in the regime where pressure is
not a linear function of density.

This nonlinearity comes from the creation operators for
the particles, and is quite independent of their interactions
such as the electromagnetic interaction.  You can see
this very clearly by observing the behavior of, say, a
tritium atom in a fairly dense gas of hydrogen atoms.  It
will have (to an excellent approximation) the same
electromagnetic interactions as a plain hydrogen atom would.
But a plain H atom would be subject to identical-particle
effects, whereas the lone T atom would not.  To see this, one
could measure the diffusion constant, observe spin waves,
et cetera.  You could describe _some_ of the identical-
particle effects by pretending all the H atoms were classical
but subject to the "force" from a "molecular field" ... but
keep in mind that that's just a figure of speech and/or a
fairly lame approximation.

Let's not make it more complicated than necessary.  Pressure
comes from the momentum and kinetic energy of the particles.
If you try to put a lot of fermions into low-energy states
in a box, the creation operators won't let you.  The only
way to make a high density is to start putting them into
states of higher and higher energy and momentum.  This results
in more pressure than a 19th-century physicist would have
predicted.  But it's just pressure.  There's nothing very
special about it.

As the saying goes, learning proceeds from the known to
the unknown.  It's hard to explain degeneracy pressure to
people who don't know what degeneracy is and don't know
what pressure is.  So don't start there.  Start with the
two-slit diffraction pattern, or with the waves radiated
by two bobbing ducks on a pond, and show that there are
places where two waves can add to zero.

Classically, one and one makes two.  In wave mechanics,
 -- one and one makes zero, sometimes
 -- one and one makes four, sometimes
 -- one and one makes two, on the average.

So the classical case is just some washed-out average
of the real (quantum) story. For fermions, two states
that differ by the exchange of identical fermions always
differ by a minus sign, so superposing them is destructive,
and that is the origin of the exclusion principle.  But
it's just wave mechanics.  It's not magic.  See Feynman
volume III chapter 4.