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

[John Denker <jsd@av8n.com>]

On 06/11/2013 11:29 AM, Rauber, Joel wrote:

> one should not be able to arrive at different results depending on
> the basis set that one chooses for expressing vector (state).

Yes, but that's not the whole story.

 ++ Just switching from one basis to another doesn't
  change anything.
 ++ /Measuring/ something against one basis commonly
  gives you different answers from measuring the same
  thing against another basis.  The results may well
  be "incompatible" in the Heisenberg sense.

Let's be clear:  There's a lot of complexity in the
measuring apparatus.  Quantum measurement is by far
the hardest part of QM to figure out.

This is absolutely central to the original question that
was asked.  This has directly observable consequences.

In particular, let's investigate the so-called ultraviolet
catastrophe.  Does a catastrophe occur, or does it not?
That seems like a simple question ... but there are two
different, incompatible answers.

We can write the energy per mode as
  E = ℏ ω (a† a + 1/2)
    = ℏ ω (N + 1/2)

where that additive 1/2 tells us about the zero-point motion.
It really matters ... especially if you are going to sum over
an infinite number of modes. 

A) This is definitely observable.  Suppose you build a voltmeter 
 or other /linear/ measuring device -- something that is 
 linear in the field voltage (a† + a) -- and then calculate 
 the energy by squaring the voltage.  You *will* observe an
 ultraviolet catastrophe.  That is, as the bandwidth of 
 your voltmeter gets larger and larger, the observed RMS 
 voltage goes up and up.  Even though the high-frequency 
 modes have no photons in them, they still have zero-point 
 fluctuations.  Observable fluctuations.

B) In contrast, suppose you build a photon counter or other
 device that looks at the number operator (a† a).  The
 high-frequency modes have no photons in them, so you can
 sum over as many modes as you like and it won't change
 the observed signal.

If you plot the black-body spectrum as measured by the two
devices (A) and (B), you get two very different plots.  One 
goes up and up at high frequencies, while the other converges 
to zero at high frequencies.

To summarize:  Change of basis is one thing.  Change of
measuring apparatus is something else entirely.  Mastery
of the subject means being able to understand both types
of measurement, and being able to reconcile the results.