On 6/13/2018 3:53 PM, Bruce Kellett wrote:
From: *Brent Meeker* <[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]>
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?
No. There's a preferred basis in which this "world" and it's spots on
the screen, is spanned by basis vectors which are orthogonal to the
basis vectors of the "worlds" in which the spots are in different places
on the screen. But in each world there are different (not necessarily
position) bases, but they describe the same physics.
Brent
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