On Thursday, June 14, 2018 at 7:29:50 AM UTC-5, Bruce wrote: > > From: Lawrence Crowell <[email protected] <javascript:>> > > On Wednesday, June 13, 2018 at 11:18:10 PM UTC-5, Bruce wrote: >> >> From: Brent Meeker <[email protected]> >> >> >> 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. >> >> >> I don't think that is correct. The preferred basis is selected as the >> eigenvectors of the operator that commutes with the interaction >> Hamiltonian. If you choose a different basis for the Hilbert space, even by >> a simple rotation of your present basis, you are going to get eigenvectors >> (and eigenvalues) of a different operator. Since this operator must also be >> dominant in the interaction Hamiltonian, the physics is necessarily going >> to be different. A different position basis is going to result in more than >> different places on the screen for the spots. >> >> Bruce >> > > I would agree, and that you are invoking the Hamiltonian segues into what > I wrote yesterday. I can set an apparatus to measure the spin of an > electron in any orientation. > > > That is true; but that is just making a choice about what to measure -- > equivalent to the choice of whether to measure the position or momentum of > a free particle. These measurements are mutually exclusive, but they do not > set the measurement basis. When you use a S-G magnet to measure the spin > projection of a spin-half particle you chose an orientation, but the actual > measurement that gives you the required result is a position measurement -- > whether the particle emerges in the up or down channel. That is why this > was originally referred to as "space quantization". > > 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. LC > I think people get trapped into thinking that our usual delta-function > basis for either position or energy is the only possible basis, because > that is the only basis in which we are able to measure anything. But that > itself is just a consequence of einselection to a preferred basis -- > attempting to measure in some other basis is not a position or energy > measurement as we know it, and the eigenfunctions of the alternative > operators decohere into our known basis extremely rapidly. But the fact > that the usual basis is ubiquitous, made so by decoherence, does not > explain why it is that basis, rather than some other basis, which is stable > against decoherence. We could easily choose another basis, and as I pointed > out to Brent, the split into separate branches on the MWI would be very > different in a different basis. Eigenfunctions and probabilities would be > different with a different basis, so the physics would be different. The > real question is "Why is physics the way it is? It could easily have been > different." > > Bruce > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
