From: *Lawrence Crowell* <[email protected]
<mailto:[email protected]>>
On Thursday, June 14, 2018 at 7:29:50 AM UTC-5, Bruce wrote:
From: *Lawrence Crowell* <[email protected]>
I can't make a measurement of energy that is something other than
the eigenstates or the diagonal form of the Hamiltonian. Energy
is the physical quantity which defines the einselected basis that
is stable in a classical-(like) outcome or for the emergence of
classicality.
That is incorrect. If you are making a position measurement,
energy does not come into it. Certainly, for many physical system,
such as atoms and molecules, the energy eigenstates are what one
measures. But one measures these in the preferred energy basis,
which is quite similar to the preferred position basis. We are
used to a position basis with eigenstates as position delta
functions along the real line. The preferred energy basis is
similar, energy delta functions along the real line (remember that
we can get any real value as the result of a generic energy
measurement. Energies are quantized only for specific physical
systems.)
I was wondering if you might catch this. I needed more time to reflect
on this and left this open. It is true that the position measurement
does not involve a kinetic energy E = p^2/2m or E = sqrt(p^2 + m^2)
term. Things are not too mysterious with momentum measurements. Is
energy completely out of the loop? Remember that potential energy V =
V(x) in most cases. So in the double slit experiment what happens? The
photon or electron wave reaches the screen and interacts with it. This
interaction is going to be position dependent and I would argue this
potential energy is much larger than the kinetic energy V(x) >>
p^2/2m, and so in a decent approximation E = E(x). Again, this is not
the energy of the free particle, but what happens with the particle
interaction with the screen.
There can be more. In particular if the interaction is of the form V =
ipx, constants ignored. Since px = i/4[(a^†)^2 - a^2 + ħ] this is a
parametric amplification operator and it squeezes the state into the
position basis.
As a result I still think, though have not worked through anything,
that energy is somehow deeply involved with the einselection of states
and the emergence of a large scale classical world. As for below it is
not the case that we make spectral measurements of atoms or other
systems that are in a basis other than the diagonalization basis for
the eigenvalues measured.
I don't know why you think that energy is central to einselection. The
idea is that the einselected basis vectors correspond to an operator
that commutes with the interaction Hamiltonian. But that is just the
interaction Hamiltonian, not the full Hamiltonian which could be seen as
the energy operator. So this criterion applies independently for any
measured quantity, be it position, momentum, energy, or anything else.
These einselected bases are not related in any other way. As I have
pointed out, this criterion does not, of itself, tell us what the
einselected basis is -- we have to go to something else for this.
Bruce
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