Re: Bras, kets etc.

On Sat, 2 Sep 2006, Ron Baker, Pluralitas! wrote:

"Timo A. Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote:
On Fri, 1 Sep 2006, Ron Baker, Pluralitas! wrote:
"Timo Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote:
On Thu, 31 Aug 2006, Ron Baker, Pluralitas! wrote:
"Timo A. Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote:

If the basis
vectors correspond to measurements you might make, the probability of
outcomes of the measurements.

How does a measurement translate to a basis?

If the measurements correspond to a mutually exclusive complete set of
properties, then the possible measurement results _are_ an orthogonal
basis for the description of the possible states of the system.

I get the concept but I'm not ready to buy in yet.
Seems to me we might be getting to the crux of the
matter.
I'm thinking of Bell's inequality and the Aspect experiment.
I've read several descriptions of Bell's inequality. They
all describe a logical tautology based on something
that seems equivalent to your "orthogonal 'measurement'
basis". Then they say that QM predicts something different
and totally gloss over the QM prediction.

"Measurement" is the crux.

I've listened to the Feynman lectures. What he describes
is totally (forgive my California accent) classical waves
except for "measurement".

I am definitely not ready to accept that one can
simply posit an orthogonal basis and say it is
a 'measurement'.

"Classical waves except for measurement" is a pretty good description of
QM. "Classical waves + annihilation/creation operators" for QED.

The relationship between basis and measurement is more the other way
around:

Even so.

a "good" measurement - with complete and mutually exclusive results -
tells you a useful basis.

Yes, even so. When I typed in the previous reply
I was also thinking:
"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> + ...

as far as our measurement is concerned, it's enough, _if_ the measurement
results are complete and mutually exlusive. So, yes, not every
measurement, only special measurements - orthogonal (ie no overlap between
them) measurements that also provide all possible outcomes.

And you're spot-on where you later wrote:

So a measurement of one property cannot be a
complete description of a system.

A measurement might have a basis but the measurement
basis is not equivalent to the system basis.
The measurement basis is at best be a lower dimensional projection
of the original system basis, isn't it?

Yes, but that's enough for practical use. Descibe an object in classical
mechanics in terms of its position r(t), and one can do useful physics,
without even bothering about orientation, let alone deformation,
temperature, chemical composition.

To head towards philosophy, how could we ever know we have described _all_
the properties of a system? The best we could possibly know is that we've
described the properties we can measure or at least detect (a crude
measurement). How can we know if there are unknown properties?

Measurement tells us what properties the system has, and what the possible
values of the properties are. What more do we need? What more can we have?

[cut most of rest since I think we have the crucial points above]

But I don't yet see a solid link between a particular
PMT click and the spherical basis to the HE.

There isn't because the PMT click is a measurement in a position basis or
a time basis. 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, but this is technological
problem rather than a mathematical or logical problem. I'll it to you to
decide if it's a physical problem :)

Conversely, getting a click in the detector circuit attached to a
single-photon multipola antenna wouldn't tell you the direction the photon
came from.

--
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
.

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