Russell Standish wrote:
On Thu, Feb 26, 2015 at 04:44:25PM +1100, Bruce Kellett wrote:
Russell Standish wrote:
On Thu, Feb 26, 2015 at 09:31:53AM +1100, Bruce Kellett wrote:
An eigenfunction in one basis is a superposition (potentially an
infinite superposition) in any other basis. Why do we not see
superpositions of positions?
Bruce
But we do! Whenever the two slit experiment is performed, for example,
or perform a momentum measurement of a photon.
You seem to have missed the point I was making. The measurement in
the two slit experiment is the spot on the screen where the particle
lands. The slits themselves are not measurement devices. So the
eigenfunctions of the position measurement are (ignoring HUP
limitations for the moment) delta functions along the position
direction. But this is only one possible basis of the Hilbert space
for the position operator. We could take any arbitrary rotation in
this Hilbert space to form another, equally good, basis. In that new
basis our observed delta function would be a superposition. If that
new basis were preferred, our observed outcome would be a
superposition. But we observe position outcomes only as eigenvalues
and eigenfunctions in one particular basis. Why is that? What
selects that basis in the position Hilbert space?
Bruce
Your basis is not the position operator of the particle (which slit it
went through in the 2slit experiment, or of the photon), but the basis
of the measuring apparatus, which as classically conceived, is
probably some sort of position basis (eg the needle of an analogue meter).
I think that is the point I made in answer to your earlier misconception.
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
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