On Tuesday, June 5, 2018 at 9:05:13 AM UTC, Bruce wrote: > > From: <[email protected] <javascript:>> > > > On Tuesday, June 5, 2018 at 7:03:28 AM UTC, Bruce wrote: >> >> From: <[email protected]> >> >> >> On Tuesday, June 5, 2018 at 1:18:29 AM UTC, Bruce wrote: >>> >>> From: <[email protected]> >>> >>> >>> Remember that the analysis I have given above is schematic, representing >>> the general progression of unitary evolution. It is not specific to any >>> particular case, or any particular number of possible outcomes for the >>> experiment. >>> >>> Bruce >>> >>> *OK. For economy we can write, ** (|+>|e+> + |->|e->), where e stands >>> for the entire universe other than the particle whose spin is being >>> measured. What is the status of the interference between the terms in this >>> superposition? For a quantum superposition to make sense, there must be >>> interference between the terms in the sum. At least that's my understanding >>> of the quantum principle of superposition. But the universe excluding the >>> particle being measured seems to have no definable wave length; hence, I >>> don't see that this superposition makes any sense in how superposition is >>> applied. Would appreciate your input on this issue. TIA, AG* >>> >>> >>> A superposition is just a sum of vectors in Hilbert space. If these >>> vectors are orthogonal there is no interference between them. >>> >> >> *As a graduate student, in one of those standard problems, I seem to >> recall solving for the wf of some system using the SWE, and then expanding >> the solution using an orthonormal set of eigenfunctions as the basis (or >> maybe it was claimed there exists such an expansion). Are you saying there >> is no interference between the basis eigenvectors? TIA, AG* >> >> >> Basis vectors need not be orthogonal. But if you choose and orthonormal >> set then the individual basis vectors do not interfere, though the >> superposition made up of such a set may be thought of as an interference >> between them. But unless you take the product of this superposition with >> something else, you do not have interference cross terms. This is similar >> to the basis problem I referred to earlier. If we have a basis of |dead> >> and |alive>, being mutually orthogonal, then the basis vectors >> (|dead>+|alive>) and (|dead>-|alive>) are also orthonormal. But this basis >> is not stable under decoherence, so the environmental states corresponding >> to this basis do interfere to produce the stable |e_dead> and |e_alive> >> states. >> >> Thus, if the orthogonal basis you choose is such that the interaction of >> each basis vector with the environment leads to orthogonal environment >> vectors, then there is no interference between these environmental states >> -- this is how classical states emerge. >> >> Bruce >> > > * So in the case of the S Cat, the superposition, ( |alive> |undecayed> + > |dead>|decayed> ) , does NOT imply the cat is simultaneously alive and > dead because the states in this superposition are orthogonal? If that's > your conclusion, and if I am not misreading the discussions of this problem > incorrectly, most, if not all of the texts which discuss this problem are > completely misleading. Is that the situation? AG* > > > I am not sure exactly what you are wanting to discover in this discussion. >
*One objective was to convince myself whether the wf you have written for decoherence makes any sense. Originally I thought one needed mutual interference of all components for it to be viable. I doubted whether each component interferes with the others in your proposed wf because the |e> wave functions have no well defined deBroglie wave lengths (which I thought were necessary for a valid quantum superposition). Now I am not sure about any of this. Maybe you can assess the status of my confusion. But first read what I wrote below about the S Cat. TIA, AG* > From my perspective, interference is due to cross terms between states, so > if there are no cross terms, such as if the states are orthogonal, then > there is no interference. > *OK. AG* > In the cat example, the cat is not both dead and alive simultaneously. If > someone wants to claim that, then they are saying that the live and dead > states are not actually orthogonal, so the original state can be > re-established by the interference between live and dead cats. However, > classical states do not interfere because the classical dead state is > orthogonal to the alive state. > *How do you know they're orthogonal? AFAIK, Alive or Dead do NOT have mathematical representations, so we can DEFINE them as orthogonal or not, depending on how we define their inner products. But regardless of whether they're orthogonal or not, the cat is in a superposition involving Alive and Dead states in the form I wrote earlier, the standard wf for this problem. But any vector which is a sum of other vectors, can be said to manifest all vectors in the sum simultaneously. So even without interference, the cat's wf implies the cat is in two states simultaneously, Alive and Dead. So, in conclusion, it doesn't seem to matter whether these two states interfere with each other or not, to wind up with a cat which is simultaneously alive and dead (for some duration in Schrodinger's thought experiment). AG* > In MWI, the term FAPP occurs here, because MWIers claim that the > interference terms never vanish completely. > *I don't do windows and I don't do MWI. AG* > So probably the issue comes down to how you react to FAPP orthogonality. > If that means that the states can still interfere, then they are not > actually orthogonal. If they can't interfere to reconstruct the original > state, then decoherence has led to genuinely orthogonal states. > *I don't understand this issue well enough to intelligently comment, other than to say that for the S Cat paradox, interference or not doesn't seem to matter. The fact that the cat is in a superposition implies it's alive and dead simultaneously (as explained above). AG * > > 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.
