Re: [Phys-L] heat content

[John Denker <jsd@av8n.com>]

> > The fact that a hydrogen atom has nonzero size is most easily
> > explained in terms of the zero-point motion of the electron.

On 02/17/2014 08:34 PM, David Bowman replied:

> I think a fairly compelling argument can be made for the contrary
> answer.

Let's just say I find the counterargument to be not compelling.
It looks to me like a bunch of word games, not physics.

> But I believe which answer is actully correct tends to hinge on the
> technical details of just what one considers to be the definition of
> the notion of 'motion'.

Well, sure, but the actual correct answer is known, the technical 
details are known, and the definition of 'motion' is known.  These
things are not up for grabs.

There are about ten lines of reasoning that all lead to the same
answer.  Since yesterday's explanation evidently didn't suffice,
let me come at it from another angle.

Consider the a bunch of non-interacting particles in the gas 
phase.  The concept of pressure is pretty well understood.
It has to do with the motion of the particles.  You can put
scare quotes around 'motion' but it doesn't scare me;  motion
is still motion.  The thermal fluctuations are still random 
fluctuations.

Suppose the particles are fermions.  At high temperatures this
doesn't make any difference.

Now extrapolate to zero temperature.  If you do it wrong, you
get zero pressure.  If you do it the right way, you get nonzero
pressure at zero temperature.  This has to do with the motion
of the particles at zero temperature.  If there were no motion,
there would be no pressure.  If there were no kinetic energy,
there would be no pressure.  Indeed, if there were no kinetic
energy, there would be no energy at all, since the PE for this 
system -- ideal gas in a box -- is zero under all conditions.

The pressure at zero temperature is sometimes called 'degeneracy'
pressure, but really it's not any different from any other kind
of pressure.  It's just pressure.  The quantum fluctuations are
random fluctuations, and they are not in any fundamental way
different from thermal fluctuations.  Thermal and quantum are
different asymptotes on the *same* fundamental curve
  http://www.av8n.com/physics/degeneracy.htm#fig-qho

The zero-temperature pressure is well known in neutron stars, 
in nuclei, and in metals.  It is not a matter of opinion.

Anybody who doesn't believe me is invited to do the calculation.
It's not a tricky calculation.  Please let us know if your answer
is different from mine.
   http://www.av8n.com/physics/degeneracy.htm

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

Here's yet another line of reasoning that leads to the same answer.

Consider an atom.  Better yet, consider a harmonic oscillator
consisting of an electron attached to a proton by a spring.
This is hard to build, but it's conceptually simple.

QM gives us a position operator for the electron. Also a velocity
operator.  Also an acceleration operator.  Also acceleration
squared.  The expectation value of acceleration squared in the
atom (aka oscillator) is nonzero.  Therefore in accordance with
the Maxwell equations, the thing will radiate.  Even in the
zero-temperature ground state, the thing *must* radiate.

Now things are getting interesting, because we know that on
average there must be no net radiation coming out of the
ground state.  Note the contrast:
 -- There must be radiation coming out.
 -- There must be no average net radiation coming out.

This is not a contradiction.  It is not a paradox.  It is easy
to understand as soon as you realize that the atom is in 
equilibrium with the EM field ... and the field has zero-point
fluctuations of its own.  This works just like thermal equilibrium
at any other temperature:  some thermal radiation goes out, and
on average the same amount of thermal radiation goes back in.

This is super-easy to see if you analyze the system using 
something resembling position operators and voltage operators.

In contrast, you will never see it using the photon-number basis,
because those are monochromatic.  They have zero spread in 
frequency and correspondingly infinite spread in time, so they
average out all the fluctuations.  Insisting on analyzing the
zero-point motion in the photon-number basis would be like starving
to death at a banquet because they have run out of your favorite
type of pickle.  I concede that one particular dish is empty,
but everywhere else you look there is plenty of food, plenty of
evidence for quantum fluctuations.