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Regarding JD's response:
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.
So requiring that something actually *move* in order for motion to exist
is not compelling to you. To each their own. I agree the disagreement
is solely about words. I'm certain we both agree on the physics.
But I believe which answer is actually 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.
True. However, an accepted definition of motion is "the action or
process of moving or being moved". Zero point motion has nothing
actually moving. Why is it then called motion? In a quantum ground
state (for a time-independent Hamiltonian) nothing changes its position,
its orientation, its shape, or any other property as a function of time.
Nothing is moving.
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;
I didn't mean to scare anyone. I was merely identifying or pointing out
the word whose definition is not satisfied by the phrase 'zero point
motion'.
motion is still motion.
Very true. And zero point motion seems to not have any.
The thermal fluctuations are still random fluctuations.
True. A system with a unique ground state has no thermal fluctuations
when it is in the zero temperature zero entropy quantum ground state.
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.
Not necessarily. In general it has to do with just energy, not
specifically kinetic energy. But in the special case of a system which
is in a box with infinitely hard impenetrable walls rather than the more
general and realistic case having partially soft partially penetrable
walls, the energy does indeed happen to be entirely kinetic. But the
(kinetic) energy and every other observable is time-*in*dependent when
the system is left undisturbed. The pressure is a rate of potential,
i.e. possible, virtual work that would be done on the system per unit
wall area if the location of the walls were to be virtually
infinitesimally displaced. For such a virtual displacement the particles
themselves in the quantum ground state remain in their respective
occupied single particle states adiabatically as the walls are virtually
displaced. The virtual displacement virtually changes the energy
spectrum of the single particle states in the system. The total energy
of the system is th
e sum of
the changed energies of the occupied single particle states. As the
individual energies change the total sum value changes under the virtual
displacement. The rate this total changes per unit virtual wall normal
displacement distance, per unit area of the walls is the zero point
pressure. But nothing in the system is doing any actually moving when
the system is left undisturbed. And when it *is* disturbed by having its
walls displaced, the only thing about the system that is changing is the
individual particles just have their energies adiabatically and trivially
track, in real time, that imposed by the wall displacement.
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 zero mean PE is only *if* the walls are required to be perfectly
impenetrable rather than have any elastic give. Otherwise both kinetic
and potential energies contribute to relevant energy, which is just the
total energy of the states involved.
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 curve is a graph of total *average* equilibrium energy for a SHO mode
in a fixed temperature heat bath. It is not a graph of any fluctuations
per se, at all, let alone both thermal and quantum ones. At the zero
temperature limit of the graph there are neither thermal nor quantum
fluctuations in the energy. The entropy is zero there, after all, and
the ground state energy is unique with a well-defined *non*-fluctuating
nonzero value.
The zero-temperature pressure is well known in neutron stars,
in nuclei, and in metals. It is not a matter of opinion.
Of course. But that doesn't mean a system in its ground state has
anything about it that is actually moving in some way when the system is
left to itself. Pressure is simply not the same as motion.
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
I'm sure you know how to properly calculate degeneracy pressure. But no
matter what its correct value happens to be that doesn't mean there is
any motion in the system possessing that degeneracy pressure. Pressure
is not the same as motion.
======================
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.
1) Note this is not an isolated system. It is a charged oscillator
interacting with the external Maxwell field.
2) The velocity operator for the non-interacting system is just the
momentum operator divided by the mass. And the acceleration operator is
just the position operator multiplied by the negative spring constant
over mass ratio. The squares of these operators are proportional to the
kinetic and potential energies respectively. These squared operators
have nonzero expectations because there are nonzero probabilities for
multiple different outcomes for a position measurement and for a momentum
measurement. I agree the distributions for potential position
measurements and for potential momentum measurements have nonzero
variances.
3) That doesn't mean the system is radiating in its ground state,
Maxwell's equations notwithstanding.
Having uncertain values in a quantity is not the same as having
time-dependent values. For me (at least) motion requires having
time-dependent values, not just uncertain values.
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.
That's one way to look at it. One could also say there just isn't any
radiation going in or out when the system is in its ground state,
Maxwell's equations and quantum fluctuations in the value of quantum
variables notwithstanding. There is certainly no measureable radiation
going in or out.
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,
The oscillator system is also monochromatic, i.e. one frequency mode,
when it is considered by itself apart from the Maxwell field.
so they
average out all the fluctuations.
But measuring the radiation supposedly going in or out would require
measuring at least one photon's worth of radiation in transit. That is
not going to happen.
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.
I don't think the other frequency dishes are relevant. If the background
EM field is a vacuum it has no photons at any and all
frequencies--including the frequency of the oscillator. If there *are*
occupied by photon modes at different frequencies the external field is
actually driving the oscillator off-resonance and the system is not being
left to itself in its ground state. I believe tickling the system from
the outside with different frequencies is cheating when it comes to
looking for motion in the system in its ground state.
In short, my point is a point about words, not physics. Physical
arguments won't affect that. My point is that zero point 'motion' ought
to be called that if something is actually moving, i.e. changing in some
way over time. It ought not be called that if nothing is actually moving
in any measureable way. I'm perfectly happy with zero point energy and
degeneracy pressure. But they don't demonstrate any actual motion of
anything.
BTW, JD, what did you think of my point about phase space areas
corresponding to pure quantum states?
David Bowman
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