Re: Bras, kets etc.


"Timo Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote in message
news:Pine.LNX.4.50.0609041624060.14484-100000@xxxxxxxxxxxx
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 I can see that is totally ad hoc.


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.

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?


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?

If we had deterministic predictions.

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?

The answer to that question is Nobel worthy. The debate on that
has raged for a century. It vexed Einstein. Bell thought
that there are hidden properties. Aspect's experiments
indicate that either there are no hidden properties or if there are
then they are "non-local".


Measurement tells us what properties the system has,

No. "Measurement" tells us what we can measure.
And Heisenberg said we can't measure everything.

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.

That doesn't sound like a reason to me.
You've implied that measurement is a transformation
of the original system basis to a measurement 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,

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

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.

It would if you knew the source location
a priori.


--
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


.

Relevant Pages

  • Re: Bras, kets etc.
    ... How does a measurement translate to a basis? ... basis for the description of the possible states of the system. ... But I didn't see orthogonal basis in that. ... both kinds of integrals turn up. ...
    (sci.physics)
  • Re: Bras, kets etc.
    ... If we want to consider possible outcomes of a measurement, ... S above is a superposition of basis vectors. ... The photon had some wavefunction giving the probability of detection ... Of course, with enough photons, you can approach the information ...
    (sci.physics)
  • Re: Layman Q: wave funtion and measurement
    ... >> being a linear combination of eigenstates. ... choose a basis for it doesn't mean that such basis cannot itself ... > under an appropriate measurement. ... you say that we need the addition of the collapse postulate to ...
    (sci.physics)
  • Re: Bras, kets etc.
    ... every measurement establishes a valid underlying ... This last part is why the basis is orthogonal, and why the measured states are mutually exclusive. ... Again, in classical mechanics, you have deterministic predictions, but what does that tell you about the thermodynamic properties of the objects we are deterministically predicting the mechanics of? ... The photon had some wavefunction giving the probability of detection at a particular position in the position basis, and a wavefunction giving the probability of detection in some particular spherical wave mode/multipole mode in the spherical wave basis. ...
    (sci.physics)
  • Re: The true crackpots
    ... A basis is orthonormal if the ... derive an orthonormal basis from any collection of N vectors, ... for other coordinate systems I'd ... measurement of the wavelength is not g, ...
    (sci.physics.relativity)
Loading