From: *Brent Meeker* <[email protected] <mailto:[email protected]>>
On 6/12/2018 10:26 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected] <mailto:[email protected]>>
On 6/12/2018 8:25 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[email protected]
<mailto:[email protected]>>
An isolated system has energy eigenvalues. But any realistic
macroscopic system is only going to conserve energy
approximately. I think energy eigenvalues are found in atoms and
maybe molecules. But larger systems (C60 Bucky balls?) tend to
emit and absorb photons that localize them in a position basis.
I am glad you said "a position basis" and not "the position basis"
-- a mistake that is frequently made. Position is an operator in a
high dimensional Hilbert space, and there are an infinite number of
possible bases for this space, each corresponding to a different
operator in the space. Which one of these operators (and bases) is
"the" position basis? The answer from decoherence theory is that it
is the basis that is stable against environmental decoherence. But,
as I pointed out in a post on the 'Entanglement' thread, this is
defined by the operator that commutes with the interaction
Hamiltonian. However, the interaction Hamiltonian is usually
defined in terms of point particle interactions, so commutes with
the position operator because it contains that operator itself. So
that particular definition of the stable basis is circular -- any
chosen operator in the position Hilbert space would fit the bill
provided it was used for both the position measurement and the
interaction Hamiltonian.
But is it a vicious circle? Aren't all the position bases going to
be physically equivalent?
Well, yes. Insofar as you can describe any vector in a linear space
in terms of any of the possible bases. But no. Not all of these
descriptions are the same -- what is given by the eigenvalues of one
operator will be a superposition of the eigenvalues of another
operator. In terms of position measurements, we get single dots on
the screen in the basis consisting of delta functions for positions
along the line.
I don't see that. Suppose I did a Fourier transform of the basis
consisting little bins across the screen. The indeed each spot on the
screen will be represented by a superposition of Fourier components,
but it will still be a spot in that representation. And the
Schroedinger eqn solution for the interference pattern on the screen
will also be a superposition of Fourier components.
So you are saying that there is no preferred basis problem? What do you
think the problem is?
Bruce
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