Re: [Phys-L] zero point motion and E-M emission

[John Denker <jsd@av8n.com>]

On 06/09/2013 10:25 PM, Bernard Cleyet wrote:

> At zero there is energy available to emit radiation (E-M). 

Hah!  That's a good question.  I remember asking myself 
that question once upon a time.  It took me a year to 
figure out the answer.

> Zero point motion is a fiction?

That's not the answer.

Consider a hydrogen atom in its ground state.  The size 
of the atom *is* the zero-point motion of the electron.
If there were no zero point motion, there would be no
atoms.

So, do you want the conventional answer or my answer?

 *) The conventional answer is that the electron in
  the ground state cannot radiate.  If you do the math 
  using the usual energy-eigenstate photon-number
  basis, you can more-or-less convince yourself that
  this is true.  Note that photon number is second
  order in the ladder operators "a" and "a†".

 *) However, we can also work things out in terms of 
  the /position/ of the electron and the /voltage/
  of the electromagnetic field.  This is a whole 
  different kettle of fish.  Note that the voltage
  is first order in the ladder operators "a" and "a†".

  If you do the math this way, the electron does
  radiate.  It's an electron, and it's moving, so
  how could it not radiate?

  However, what's sauce for the goose is sauce for
  the gander.  There is also some zero-point excitation
  in _every mode_ of the EM field ... and there are a
  lot of modes.  This electromagnetic radiation shines
  on the electron and imparts energy to it.

  If you do the math correctly, you find that the
  energy going from the electron to the EM field
  is exactly equal to the energy going the other
  way, on average.


=========
  This is true at any temperature, if the atom is 
  at the same temperature as the photon gas.  Zero 
  temperature is just a special case ... and not
  even very special.  That is to say, the thermal 
  fluctuations do /not/ go to zero at T=0.  Based
  on extrapolation of the high-temperature behavior
  you might have expected them to, but they don't.
  The whole picture is diagrammed and explained here:
    http://www.av8n.com/physics/oscillator.htm
=========

Hint:  If you want to do this calculation yourself,
start with the one-dimensional field, i.e. the field
in a coaxial cable.  The three-dimensional quantum
field theory is more work, but not significantly 
more edificational.

Note that in the course of doing the calculation,
as a by-product, you will rederive the fluctuation-
dissipation theorem.  Also, all of this is intimately
related to the second law of thermodynamics and to
the Heisenberg uncertainty principle.