Re: [Phys-L] electron location & wave function

[John Denker <jsd@av8n.com>]

On 04/10/2013 10:44 PM, I wrote:

> At the risk of slightly overstating things, one could reasonably say
> that it is the /analyzers/ that cause things to be quantized ... and
> in between analyzers, there is no reason to imagine that anything is
> quantized.                [1]

As advertised, that is somewhat of an overstatement, so we should fill
in the rest of the story.

This harks back to the photoelectric effect.  In the earliest days of 
quantum mechanics, Einstein posited that a photon of frequency ω had an 
energy of ℏω, and ...
   the energy to eject a single photoelectron must necessarily come
   from a single photon          [2]

This of course satisfies conservation of energy.  Photoemission requires
some energy, so that the electron can overcome the potential known as the
work function.  The photon supplies the required energy.

This differs from classical notions as follows:  Suppose a blue photon
has just barely enough energy.  Classically it would be possible to use 
two red photons instead of a single blue photon to provide the required 
energy.

So, in the situation Einstein considered, the photon energy is effectively 
quantized, whether you want it to be or not ... which seems like an 
exception to statement [1] above.

One should not over-react to this.  There are lots of situations where you
*can* use the energy of two photons.  For example, a typical green laser
pointer uses a doubler.  The doubler takes in two IR photons and puts out
one green photon.  As a more direct example, there is such a thing as
multiphoton photoemission.  I get 50,000 hits from
  http://www.google.com/search?q=multiphoton+photoemission

To summarize, statement [1] is an overstatement in one direction, while
statement [2] is an overstatement in the other direction.

It becomes mostly an engineering question, rather than just a fundamental
physics question:  If you want to see simple Einstein-style quantization,
you need to /design/ the experiment to be predominantly sensitive to 
single-photon photoemission rather than multi-photon.  There are several
factors that must be considered.  Two that come to mind immediately are
 -- second-harmonic effects, depending on nonlinearity, and
 -- two-step processes, depending on the density of intermediate states

Here's another way to understand the meaning of statement [1].   Returning
to the photon polarization example discussed yesterday, it is easy to see
how a photon can have 45° polarization, but becomes "quantized" along the
X and Y axes when it hits an analyzer.  So, why do we not ordinarily see
an electron that is "halfway" between being photoemitted and not emitted,
analogous to 45° polarization that his "halfway" between X and Y?

One key requirement is that the medium treat X and Y polarization more-
or-less the same.  More specifically, the medium must not randomize the
phase of either the X-component or the Y-component.

In contrast, the equation of motion for the photoemitted electron is
wildly different from the non-emitted electron inside the metal.  More
specifically, both of them are subject to all sorts of weird interactions
with the environment, so any attempt to set up a coherent superposition
of emitted and non-emitted would fail after a very short time.

Sometimes the /physics of decoherence/ favors one basis-set over another.
In the photoelectric effect, the energy eigenstate with N=0 photons is
treated very differently from the eigenstate with N=1 photons, and this
is very significant.

Interestingly enough, there are plenty of other examples where the
/physics of decoherence/ does not favor the energy eigenstates.  A
famous example involves molecules with stereo-isomers.  As discussed 
in Feynman's chapter on the ammonia maser, we must consider two possible
basis sets:
  a) "left" and "right" i.e. the states of definite dipole moment
  b) "gerade" and "ungerade" i.e. the states of definite parity.

It turns out that (b) are the energy eigenstates ... whereas with the
exception of ammonia and a couple of other extremely small molecules,
what we observe in nature is virtually always (a).  The details are
complicated, but the basic idea is that the /dipole moment/ is what
couples to the environment -- via the EM field and via collisions with
neighboring molecules -- so decoherence tends to collapse things onto
basis (a), the states of definite dipole moment.

This is another nail in the coffin of the idea that energy is always
quantized.

Energy states are not the only states.  They are not even the only 
basis states!

On 04/11/2013 02:15 PM, Derek McKenzie wrote:
> > If a quantum system is in an energy eigenstate, and later found to be
> > in a different energy eigenstate, then whatever happened in between
> > cannot be described within quantum mechanics.

Nonsense!
 -- Pick up any book on intermediate quantum mechanics and read about
  "perturbation theory".
 -- Or pick up any book on pulsed NMR.
 -- Et cetera.

There are some hard-to-understand aspects of quantum mechanics, but
this is not one of them.