Russell Standish wrote:
On Thu, Feb 26, 2015 at 09:55:26PM +1100, Bruce Kellett wrote:
Does a measuring apparatus always have to be eigenvalue of some
position operator, though?
If you are doing quantum mechanics, yes. The result of any
measurement is an eigenvalue of the corresponding operator, and the
system is left in the corresponding eigenstate.
What about variants of the experiment that
record the results of the measurement as bits in a computer
memory. Surely that would be in a basis that is eigenvalue of the
charge of the memory cell transistor, not a position operator at all?
>>
That is not a measurement unless you can specify the relevant
quantum operator. It is usually the case that most measurements, of
whatever quantity, boil down to pointer positions. That can be
recorded digitally if you like, but the basic measurement is still a
position measurement and you need a basis in the corresponding
Hilbert space in order to specify what are the
eigenvalues/eigenvectors of the possible results.
Bruce
I changed the title of this subthread, as I think it is an interesting
point worth exploring further.
I have heard this claim made vaguely before, though I don't remember
whom - do you have any references where someone has advanced this
argument?
That most measurements are measurements of position was not my primary
concern in this, but it is an interesting question, I agree. The idea is
quite old -- my fading memory traces it back to at least Eddington, and
it may not have been original with him, although I do not have exact
references to hand.
I still think the claim unlikely - the measurement of an interference
pattern of coherent light doesn't seem to involve any position basis
that I can see, for example.
That is a good example. How do you measure an interference pattern --
from double slits or any other source of interfering coherent waves? An
interference pattern is a series of light and dark fringes on a screen.
Ultimately, a variation in the probability of detection of a particle or
photon with position. I do not see how you could remove the fundamental
requirement of a variation of something with position from this. And
variation with position is measured by something equivalent to observing
particular impacts on a screen. Each impact is, within UP limits, a
point along the longitudinal axis of the screen -- a position measurement.
I realise this seems a bit like whack-a-mole, but you are defending a
strong thesis, and in the absence of a well-articulated reasoning for
it, to see potential counter-examples deconstructed in front of my
eyes is educational. :).
I can multiply examples of this for such things as spin measurements and
so on. Momentum is most often measured by time of flight between two
points, or by the curvature of tracks in a strong magnetic fields.
Measurements of point positions are essential in all these measurements.
There may well be measurements that are not fundamentally position
measurements, but that is not really of relevance for my main point,
which concerns the nature of each and every position measurement.
I said: "...the basic measurement is still a position measurement and
you need a basis in the corresponding Hilbert space in order to specify
what are the eigenvalues/eigenvectors of the possible results."
Measurement results are eigenvalues of quantum operators. Eigenvalues
and eigenvectors are relative to some basis, so you have to specify the
basis to specify the measurement. The default basis for position
measurements is each real number along the line, represented by delta
function eigenvectors, and the corresponding single value on that real
line. But why? This is only one possible basis in this Hilbert space. We
could take any arbitrary rotation in the space; then the eigenvectors
and eigenvalues would be superpositions of those we currently use.
The point is that "position" is not a basis. Position is an operator in
a Hilbert space, and we can choose an arbitrary basis to represent that
Hilbert space. The position operator would have different
representations in different bases, but why do we invariably choose the
"classical" basis for this Hilbert space? I think it probably has
something to do with Bohr's intuition that we need to anchor our quantum
experience in a classical reality, but this causes problems for the
quantum program of trying to derive the our classical experience from
purely quantum models.
Bruce
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