Re: [Phys-L] what's quantized and what's not

[John Denker <jsd@av8n.com>]

On 04/24/2016 09:06 AM, Moses Fayngold wrote:
> Here I can only try to formulate again what has been proved right by 
> an overwhelming scientific evidence, starting from 1905. 
> Mathematically, quantization is the reduction of the set of allowed 
> values of some characteristic from continuous range down to a 
> discrete set. In QM, this is manifest in representation of 
> observables by their respective operators whose eigenvalues may form 
> a discrete set. Particularly, the Hamiltonian operator representing 
> energy E of a system may have a discrete set of eigenvalues. In this 
> sense, the system's energy is quantized. Actually, quantization is 
> known already in CM, e.g., frequency quantization of a finite string 
> or elastic rod.    Physically, energy quantization in QM is observed 
> in interactions and energy exchange between different systems. The 
> system's evolution may be continuous and described by the 
> corresponding wave equation or unitary transformation. But the 
> observed interaction outcomes may be discontinuous with the instant 
> finite energy changes. In this sense, energy is quantized, while 
> still satisfying conservation laws (when  averaged over vacuum 
> fluctuations). That such changes may fall below available 
> experimental sensitivity, thus appearing continuous (e.g., low 
> frequency limit of EM radiation), falls short of "conflicting 
> evidence". Otherwise, one could pronounce seemingly continuous flows 
> of fluid an abundant evidence against the existence of atoms.  By 
> contrast, the frequency threshold and practically instant 
> photo-emission in PEE, discrete spectra in atomic and molecular 
> Optics, etc., are the solid evidence of quantization of light.   We 
> should be very vigilant to avoid the interpreting classical limits
> of QM as an evidence against Quantum.

I disagree with most of that.  I disagree with the overall drift
and with many of the details.

> We should be very vigilant to avoid the interpreting classical limits
> of QM as an evidence against Quantum.

Quantum mechanics is right, so far as we know.  However, a great
deal of what is /said/ about quantum mechanics is wrong.  Sometimes
the classical limit tells us something about why the right things
are right, and why the wrong things are wrong.

> Actually, quantization is known already in CM, e.g., frequency
> quantization of a finite string or elastic rod.

That seems kinda inconsistent with the previous warning about
arguments based on classical mechanics.

In fact if you look at the oscillations of a string, rod, organ pipe,
et cetera, you find that the resonances are not particularly sharp.
There is a nontrivial linewidth.  In accordance with Floquet's theorem,
if you drive the thing at frequency f, it will respond at frequency f,
whether or not that's one of the resonant frequencies.

The same is true of atoms (whether or not you consider this analogous
to classical resonances).  I've never seen an atom emit a photon
"instantly".  I've never seen an atomic transition with zero linewidth.
There is not "overwhelming" evidence of this;  indeed there is no
evidence at all.

An experiment optimized to look for "instant" or near-"instant" transitions
would be incompatible with assigning a definite energy to the photons.
Here "incompatible" has a precise technical meaning, as explained by
Heisenberg.

One of the most fundamental principles of quantum mechanics says that
it doesn't suffice to /talk/ about the energy.  If you want to know
the energy, you have to measure it.  The same goes for other observables
such as the spin components Sx, Sy, and Sz.
 *IF* you measure Sx, the result is quantized.
 *IF* you measure Sy, the result is quantized.
 *IF* you measure Sz, the result is quantized.

However, you can't measure all three of them at the same time.  You can't
measure any one of them without greatly disturbing the others.  If you
measure Sx, there are some things you can say about Sy and Sz, but quite
a few things that you cannot say.  You ought no even imagine that they
are quantized.  This is the essential message of the Bell inequalities.

The "*IF*" is an important part of the foregoing statements.  The
converse is perhaps even more important:  If didn't measure it, you 
have no idea whether it is quantized or not.

The same goes for energy.
 *IF* you measure E, the result is quantized.
 HOWEVER if you measure something else, you get a different result,
 and in all likelihood it is inconsistent with the idea that the
 system is in a definite-energy state.

We see this All The Time in atomic physics.  An atomic electron has
a perfectly well defined position.  You can measure the position if
you want.  However such a measurement is incompatible with measuring
the energy, incompatible with the spectroscopic N,l,m quantum numbers.
If you measure position you forfeit the option of measuring the energy
and vice versa.

I don't know how to say it more clearly.  If you imagine that the
electron has a definite energy /and/ a definite position, that's just
wrong.  It's inconsistent with the well-known laws of physics.  The
simpler assertion that it always has a quantized assertion is essentially
just as bad, because it is tantamount to asserting that none of the
incompatible variables can ever be measured.

Energy basis states are not the only states.
The energy-state basis is not even the only basis.

Many of the things we deal with on a daily basis are not even remotely
energy eigenstates.  Examples include
 -- AC voltages
 -- planetary orbits
 -- pendulum clocks
 -- musical instruments
 -- other sounds
 -- waves on the surface of water
 -- NMR π/2 tipping pulses
 -- etc.


> Can we wiggle a point charge and thereby produce radiation field?"
> And my answer to this would be (b) Yes. The generated radiation field
> would be a special case of what we call "Dipole radiation".

That's true, but it totally misses the point.  We all know what the
physics is.  The question is whether the physics can be explained
by extrapolating Coulomb's law, and the answer to that is No.

Typical textbooks are infested with diagrams that purport to do this,
but it's nonsense.