Re: To AC -- on electrons

Zoe wrote:
On 11 Jun 2006 01:55:22 GMT, AC <mightymartianca@xxxxxxxxx> wrote:

On Sat, 10 Jun 2006 01:11:36 GMT,
Zoe <muze10@xxxxxxx> wrote:

[snip]

Maybe you can answer some questions I have about the wave-particle
duality of photons.

First, if a wave needs a medium in which to travel, how do photons
travel through a vacuum in which there is no medium for a wave to
travel?


Not all waves require a medium in order to be able to
propagate. Various kinds of waves appear to be able to
propagate through vacuum.

Electromagnetic waves, the classical waves of which light
was said to be ultimately made before quantum mechanics came
along are one case of this.

The probability amplitude waves of quantum mechanics, which
describe (probabilistically) the motion of the individual
photons that make up classical light waves are another such
case.

Still: for quite a while, it _was_ thought that a real
medium was required in order for classical light waves
to be able to propagate. There was no idea of quantum
mechanics yet when the Maxwell equations were written
down of course, so no one worried about probability
waves then.

But the fact is that the equations describing classical
light waves actually do not require or imply the existence
of any medium. They are simply mathematical equations
after all, which allow for the electric and magnetic
fields in the equations themselves, whose motion they
describe, to create waves which propagate all on their own,
by the following rough mechanism: a time varying electric
field creates a time varying magnetic field in a region
nearby, which in turn creates a time varying electric field a bit
further away ... and this cyclic process can continue ad
infinitum, producing an _electro-magnetic wave_ that moves
through empty space, even in the absence of any medium of
which that wave could be said to be a disturbance.

Still, even after the equations describing this process were
written down, almost everyone imagined that, nevertheless,
there _must_ be a medium through which light
travelled. Maxwell in fact had derived the final form of the
equations by assuming such a medium existed, and by making
up a detailed mechanical model for it.

So your question is very natural.

But when numerous experiments failed to detect any effects
of this supposed medium -- it was called luminiferous aether, or
aether for short -- and when it became clear that relativity
theory explained the failure of all such experiments as well
as many other observed phenomena relating to light and other
questions, the idea that light waves required any medium
in order to propagate was given up.


Do they change and act as a particle when passing through a
vacuum?


No.

Also, in the double-slit experiment, it seems that when photons are
fed through a double slit, one at a time, with a detector to keep
track of which slit each goes through, there is no interference
pattern on the other side. Yet, without the detector, there is an
interference pattern.


Yes, assuming such a detector could be built. But a photon
can only be detected once, since photons are destroyed in
the process of detection, so if a photon is detected going
through slit 1, it could never reach the back wall at all.

But: a similar experiment could be done for electrons or
neutrons, say, and in that case it _might_ in principle be
possible to really build a detector which could say what
path the neutron or electron had followed. In that case, the
prediction of quantum mechanics is that what you say above
would happen.

Double slit experiments described in many places you will
read about them are often, to a very large extent, gedanken
experiments. That is they are descriptions of what one
thinks would happen in the given situation.

Not everything that you read about has actually been done.

Some real experiments of the general type _have_ been done,
though, and all of those that have been done do agree with
the predictions of quantum mechanics.

The Copenhagen interpretation says that until the particle is detected
at any location along a probability wave, it actually exists at every
point, therefore it will pass through both slits, resulting in an
interference pattern.

No. This is not quite right.

What the Copenhagen interpretation says is that before the
experiment is done and a particle is detected, together
possibly, with what path that particle actually took through
the experimental apparatus, the only information that can be
given by theory about what might happen in the experiment is
to specify the form of the probability amplitude.

This probability amplitude evolves in time according to the
Schroedinger equation.

Now the evolution is a well-defined notion, so there's
no ambiguity about what it means to say what I've said
here. It's a purely mathematical statement.

But then, the Copenhagen interpretation says, there occurs a
measurement process, by which the experimenters eventually
become aware of what has actually happened in the
experiment.

Now clearly, this measurement process is not very well
specified so far, so there is ambiguity there.

There is also some ambiguity about what is meant by the
probability amplitude or wavefunction, and people in fact are
divided in what they say about it.

Some people hold that it is a real physical thing, and that
leads to the kind of statement that you made above
about the particle actually existing at every point.

For my part, I hold that the probability amplitudes
themselves are not strictly speaking physical quantities, so
just the fact that a probability amplitude can be said to
`exist' everywhere along the possible paths that a photon
might travel, does not mean that the photon _actually_
exists at every point of those paths.

You see?

For me, the wavefunction is a theoretical object, and
discussions about what it does, though very common
and even useful, are meta-theoretical discussions.
Those are not quite as bad as meta-physical discussions
in my view, but they can become almost as bad, if
you are not very careful in what you say.

I know that this may seem like a very subtle difference, but
I think it is also a very important one.

The theoretical statement that is made under this view is
*not* that the photon actually passes through both
slits. The statement is that it _might_ pass through either
one.

That's why the basic objects that quantum mechanics deals
with are waves of probability amplitudes in fact.

Here's a bit more detail on how the theory goes
in the double slit experiments. I've reverted to
talking about photons, since you started out that
way. But all of the caveats I mentioned apply.

Let's call the source of the photons behind the two slits
s. Let's call the point on the other side of the two slits,
1, and 2, at which the photon is to be detected: x. For the
moment, forget about the possibility that there are photon
detectors d1 and d2 that could tell us through which slit
the photon actually went.

The detectors can be included in the analysis and we can
discuss how it is that they destroy the interference
pattern. But this is more complicated. I'm not going
to do it for you.

If there are no detectors, we can write the total
probability amplitude for the photon to be detected as:

PA(x,s) = < x | s >

Now since the motion is completely unknown, before
it actually happens and the photon is detected or not,
it's necessary for us to include all possible contributions
to the probability amplitude in our calculation of PA(x,s).
That includes the possibility that the photon passes
through slit 1, after leaving the source s, as well
as the possibility that the photon passes instead
through slit 2. Thus, we write:

PA(x,s) = < x | 1 > < 1 | s > + < x | 2 > < 2 | s>

= pa1 + pa2

In other words, to find the total probability amplitude that
the photon starts at s, and finishes at the point x, we add
up the probability amplitudes for all of the possible ways
in which the photon could go from s to x, and also, to find
the probability pa1 that the photon goes first from s to
slit 1, and then from slit 1 to point x, we multiply the
probability amplitudes for the successive steps along this
path.

Now of course, x could have been any point at all behind the
wall in which the two slits were cut. So if we can further
specify the forms of the basic probability amplitudes for
going from point a to point b: < b | a >, we will have
actually written down a quite general result, in fact. I
won't do that explicitly, but it isn't in fact too hard to
do.

Now in fact, there were a lot of hidden assumptions
in that discussion.

It was assumed that the slits are very narrow in comparison
to the wavelength of the light source producing the photons
involved in the experiment, and it was also assumed that
there was no possibility that the photon simply doesn't make
it through one or the other of the slits, or that the photon
possibly hits the wall of the slit and is reflected back
towards the side of the wall on which the source s, is
sitting, when it reaches a slit.

But one could handle all of these possibilities too, in an
exactly similar fashion.

The Copenhagen interpretation then says further that in
order to find out the actual probability that a photon makes
it from s to x, one should `square' the probability
amplitude. Actually, since the amplitudes are generally
complex numbers, `square' is shorthand for multiplying
the amplitude by its complex conjugate.

So after doing all of the above, and putting in the
explicit mathematical amplitudes for a photon
to propagate from point a to point b as required,
we then calculate:

P(s,x) = | PA(s,x) |^2 = | pa1 + pa2 |^2

Working out the square of the sum of pa1 and pa2 in this
expression for the probability of detecting a photon at x
will result in an expression involving the sum of the
squares of pa1, pa2, as well as an interference term
involving the product of the two.

The expression for the probability will yield the prediction
of the double slit interference pattern.

If we were to do a similar analysis using electrons instead
of photons, and with detectors in place that could allow us
to see which slit the electrons actually went through, then
the expression for P(s,x) would be significantly changed.

It would emerge that whenever we could say which slit an
electron went through, no interference pattern would result
for those electrons, but whenever we could not say for sure
which slit an electron went through, an interference pattern
would result.


Tell me, please, is it possible for an
indivisible particle to divide and pass through both slits at the same
time?


No.

This possibility would lead to serious problems: half a photon
would presumably have half of the energy, and if the slits
were finite sized, this could be expected to make a
detectable difference in the pattern observed. And the
suggestion would not resolve the basic problem in any case,
which is that light is observed to have _both_ particle-like
and wave-like aspects to its behaviour in these experiments.

Photons can be said to be detected as discrete events on the
other side of the wall containing the slits in the right
circumstances, say, if the flux of light is made very low
and if one had a CCD camera on the back end with a fast
cycle time: one could then see every once in a while, a
discrete amount of charge being ejected from a specific cell
on the camera, which would correspond to a single photon
hitting that spot.

Similarly one could use ordinary film, and do successively
longer exposures. For short exposure times, one would see
basically random patterns of exposed spots on the film,
corresponding mostly to individual photons hitting the film
if the light intensity were very low. But over time, an
interference pattern would build up, either for the CCD or
for the ordinary film, showing that there was a wavelike
behaviour.

However, if the experimental apparatus doesn't allow one to
detect which slit the photon went through, then we simply
cannot say where it has gone.

Neither can we say that it has passed through both slits,
thought it's very clear that it is the fact that there
exists a possibility for a photon to pass through either
slit that allows for the interference pattern to be built
up: just block one of the slits, and the interference
pattern will go away.

This is the basic puzzle in quantum mechanics, and my
tendency is to resolve it in this case by saying that the
wavefunctions, which `exist' in both of the slits, are not
real physical objects. They are theoretical objects which we
manipulate according to definite rules to obtain the
prediction of what will happen in the experiment.

Thus I claim that it's not possible to say that there is any
dividing of the single photon in these experiments, or any
passing of pieces of it through both slits. The experiment
cannot detect what particular possible path the single
photon has followed, at the same time as also detecting an
interference pattern characteristic of a wave that passes
through both slits.

Nevertheless, it may be possible to detect individual photons,
or electrons in the detectors at the back of the experiment.

Having come up against that oddity, I find myself speculating. Bear
with me.

What if the interference pattern, suggestive of a wave, is really due
to an optical illusion, something that the human brain gets tricked
into seeing?

No. The brain of the experimenter is irrelevant to what
happens. The interference pattern is quite real: it can
be preserved on film, or registered in a position sensitive
electronic detector and the results then recorded in a
computer, and we can then look and see what the result
was, play it back at our leisure and at any speed which
we desire. It isn't a matter of the limited nature of
human vision, or the assumptions built into
our visual perception systems.

The interference patterns are not optical illusions, and
they are not produced by the influence of the brains
of the experimenters. The experiments can be run
with no experimenters anywhere near to the apparatus,
and the same thing will happen.

Since keeping track of single photons keeps the
interference pattern from developing, then maybe the wave-like
interference pattern is a result of the brain's inability to keep up
with the high-speed movements of the photons?

It's not keeping track of single photons that matters, per
se, but trying to keep track of how and by what path each
individual photon has passed through the experimental
apparatus. As I said, one can't do that for single photons,
since photons are destroyed as soon as they are detected.

But setting up a double slit experiment where it was
possible to do that for single electrons would destroy the
interference pattern.

I mean, if you don't
keep track of the photons, they move so fast and pile up so fast that
they leave the illusion of light and dark fringes? Which would mean
that photons or electrons are really only particles, never waves? The
wave-like pattern is only in the eye of the beholder?


Electrons or photons, whenever they are capable of being
detected as singles, are always only detected as particles.
But under the right circumstances both can be made to
display wave-like behaviour.

Both aspects are real enough in both cases. The basic
peculiarity is that certain kinds of experiments which one
might expect are possible to do if photons and electrons
are either particles or waves are in fact _in principle_
impossible to do ... namely an experiment which would
allow one to say what slit an electron passed through
in a double slit interference experiment, and still detect
a double slit interference pattern. The wavelike behaviour
of the electrons would be destroyed if the experimental
apparatus were such as to constrain the electrons to
behave like particles to the extent that it were possible
for experimenters to find out where the electrons actually
went.

[snip]

David

.

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