Re: Bras, kets etc.


"Timo Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote in message
news:Pine.LNX.4.50.0609111602100.7407-100000@xxxxxxxxxxxx
On Wed, 6 Sep 2006, Ron Baker, Pluralitas! wrote:

"Timo A. Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote:
On Tue, 5 Sep 2006, Ron Baker, Pluralitas! wrote:
"Timo Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote:
On Sat, 2 Sep 2006, Ron Baker, Pluralitas! wrote:

"Conversely I am not ready to accept that
every measurement establishes a valid underlying
orthogonal basis."

If we want to consider possible outcomes of a measurement, and
quantumly
write the state as

S = a|1> + b|2> + ...

Previously I thought that was meant to signify the
measurement result, but that is actually the a priori
state of the thing being measured, right?

Well, the "state" is largely how likely different measurement outcomes
are.

With that view it doesn't seem to me that it is something
physical. It is just a probability calculation tool.
But we started with the HE which is relatable to the
physical. It seems to me that logical leap has been
made without explanation.

Your description of S seems oriented to population
statistics. It doesn't seem that it can be used for single
photons.


That's not the whole story, since the amplitudes a, b, etc of the
modes |1>, |2> etc are complex, and the probabilities are |a|^2 etc.

If you were to do many repeated measurements of identical systems, you'd
be able to estimate |a|, |b|, etc. Trickier to determine their relative
phases - you need to do a somewhat different measurement for that. It's
like optics: easy to measure intensities, very hard to measure phase. Ways
to measure phase in QM are analogous to ways to measure phase in optics,
basically interferometric methods.

as far as our measurement is concerned, it's enough, _if_ the
measurement
results are complete and mutually exlusive.

S above is a superposition of basis vectors.
There is nothing mutually exclusive about it.

If S is a single photon/electron/whatever state, then when you do the
measurement, you have probability |a|^2 of measuring it in state |1>,
probability |b|^2 of measuring it in state |2> etc.
|a|^2 + |b|^2 + .. = 1, and you will only measure it to be in one state
(ie you won't measure it to be in two states at once). This last part
is
why the basis is orthogonal, and why the measured states are mutually
exclusive.

Seems to me that in the course of the conversation what
has been implied is something like:

P(metaphysical "detection" event) = | <m|S |^2

Where:
S = as you said, the state of the particle, which is a linear
combinations
of its basis vectors
= a|1> + b|2> + ...

And <m| is some measurement basis. Which really
hasn't been defined in any detail.

We've gone to great lengths to describe the
quantum (with a wave differential equation and
basis vectors) but the same type of rigor has
not been applied to <m| , if it in fact exists,
or what detection is.
Why do we apply math only to part of the
problem and then shout "detection"!
Surely quantum stuff is happening in the "detector"
too, isn't it? Is there no math for that?

A good question. Yes, there will be math for that (but how exact, how
idealised, and how practically calculable?), but the details will depend
very much on the details of the measurement.

Consider a classical case of measurement (and the QM case will just be the
photon-counting version) where a diffraction grating is used to measure
wavelength. Use the grating to produce a spectrum, and use a photodiode
array or CCD to measure the power incident on each pixel. Yes, there is
the maths for that, which might make use of various approximations in
treating the grating, is likely to make a very simplistic treatment of the
effects of absorption in UV and IR in the optical system (typical would be
to just ignore those wavelengths), etc, etc. The details don't affect what
you learn: P(n), the power incident on the n-th pixel. The rather complex
details of how the measurement is actually made don't matter as far as
interpreting what the results actually mean, most of the time.

One is likely to pretend that this tells us P(w)dw, the continuous
spectrum which we'd measure if we had a non-pixelated device with infinite
spatial resolution. Convenience is sometimes paramount!

So, yes, not every
measurement, only special measurements - orthogonal (ie no overlap
between
them) measurements that also provide all possible outcomes.

Hmm. That is weak.

"Measurement" is still undefined and
who is to say measurement is not nonlinear?

What is the point of a theory in physics? Is it, to paraphrase and
perhaps
abuse the intent of, Dirac, simply to cough up predictions that can be
compared with experiment?

QM is a theory that predicts the probability of outcomes of
measurements.
What we can measure is central to what predictions we want. Why does it
need to depend on the details of the measurements?

Because it is weird and seems incomplete.
It tells us we can't know what we do know.
(Aspect: there are no hidden variables so we can't
know that the other photon has the same polarity
even though we know it does in every measurement.)

[cut]

Classically, one can measure the state. For example, Given some
arbitrary
plane-polarised state, one can use measurements of |H> and |V> - and
classically one will measure power in _both_ polarisations in general -
to
find the actual state. Quantumly, you can't do this for a single
photon,
because you only get one measurement - you'll only get a click in the
detector in the |H> path or the |V> path. Only by making measurements
for
a succession of photons in an identical state can you even _estimate_
the
state. Of course, with enough photons, you can approach the information
you get with the classical measurement, but that's just the classical
limit of the quantum case.

Suppose you have stream of unpolarized photons
incident on a polaroid filter. How would you describe
the photons before and after the filter?

Unpolarised implies incoherent, at least temporally. Diagonally polarised
would be (|H>+|V>)/sqrt(2) or (|H>-|V>)/sqrt(2), circularly polarised
would be (|H>+i|V>)/sqrt(2) or (|H>-i|V>)/sqrt(2). Unpolarised would be
something like (exp(ia(t))|H>+exp(ib(t))|V>)/sqrt(2) where a(t) and b(t)
are unknown functions of time - the relative phase between the two states
constantly changes, and the phase relationship during successive
measurements is uncorrelated.

Then let me revise my original statement
from "unpolarized" to "randomly linearly polarized".
"randomly linearly polarized" would be applicable
in the case of most natural sources, would it not?


But for a |H> aligned piece of polaroid, for all 5 cases there is a 50%
probability of the photon making it through.

Suppose the filter is horizontal and one particular
photon is a|H> + b|V> before the filter.
What is it after the filter?

|H>, times an unknown phase factor exp(i*phi). But only a |a|^2 chance of
it making it through.

So is that (ignoring phi for the moment)
S = 0 if blocked
= 1|H> else
or is it
S = a|H>

It would have to be a|H> in order to be consistent
with your description of QM, wouldn't it?
(Now that might seem to be part of a photon which
would be a problem. But if S is as I think you are
describing it then it doesn't really describe a single
photon but rather a population probability.
In which case, no problem.)

And that phi: you don't mean it to be random per
photon do you. If it were then a laser beam would
lose its coherence going through a polarizer.


Consider a beam splitter with an incident beam
of a|X> . In each of the out-going legs we
would have.... what?... (a/1.414)|X> . And if you consider population
statistics then that gives us a probability of (|a|^2)/2 of detecting
photons in either leg. That works.
But not if you consider one photon. Because if the
photom is a quantum then there
would have to be anticorrelation in detection.
I don't see that anticorrelation anywhere in the math
we've discussed so far.


Which photons are absorbed by the filter?

This is where philosophy comes into it. One school of thought would say
"1/2 them, entirely determined randomly".

That is easily demonstrated to be false using a pair
of polaroid filters.

Another school says "the
vertical photons are absorbed, and the horizontal ones make it through
(and there are equal numbers of the two)". Yet another would say "all of
the photons pass through, but only 1/2 of them in our universe, and the
others in another universe".

If S is a probabilistic description then I would be
reluctant to say that |H> and |V> are real properties
of actual photons.
Especially if trying to interpret it as physical leads
to the obviously false view that a 1|V> photon has
a 50% probability of becoming a 1|H> photon.


[cut]

I'm interested in where the classical breaks down
and figuring out why. (As was Bell.) What I've seen of QM is
basically classical waves until it comes to 'detection'.
Basically we take classical waves so far and then
say okay, no more, now we talk about probability
of detection. It is like "Here be dragons."
And there is disagreement on how far one can take
the classical view. Remember
http://jchemed.chem.wisc.edu/JCEWWW/Articles/DynaPub/DynaPub.html#ref16

This paper illustrates this issue very well. It's a very nice classical
calculation of the wavefunction. Note that the emission is dipole
radiation, and has the usual directionality of dipole radiation (ie very
little). Compare the observed "directionality" when the photon is detected
by a distant detector.

Very "here be dragons". Can I explain it? No, but it's what we observe. I
really don't like the "magical instantaneous collapse of the wavefunction"
explanation, either.

Knowledge is found not where one says "Of course"
but where one says, "That's strange."


You could do spherical basis measurements for the HE using a
succession of multipole antennas. There is a practical problem in
building
antennas suited for single photon detection,

Isn't that what a PMT is? (Or millions of such little antennas.)

It's very much larger than the wavelength. It isn't sensitive to the
multipole mode of the incoming photon,

Why not?

It simply doesn't work on the same principles as a classical antenna. The
antenna is essentially a transducer, converting an incident
free-space radiation field into a guided wave (ie the signal in the wires)
which is then sent off to some circuit to interpret. It converts an
incident (free-space) EM wave into another (guided) EM wave. The coupling
between the two modes (the incident and the guided) depends on the
geometry of the antenna, and can be used (eg with a multipole antenna) to
detect a particular incident spherical wave mode, or (eg with a phased
array) to detect (approximately) an incident plane wave mode.
The PMT, OTOH, converts an incident photon into a freed electron, which
just doesn't give the same kind of discrimination.

If the spherical is resolvable to the rectangular then
might there be filters for rectangular modes that could
be used infer the spherical?


or even the wavelength.

Does that matter?

No. This can be cured by putting a suitable filter in front of it, anyway.

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html

--
rb



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