On Monday, January 7, 2019 at 11:37:13 PM UTC, agrays...@gmail.com wrote: > > > > On Monday, January 7, 2019 at 2:52:27 PM UTC, agrays...@gmail.com wrote: >> >> >> >> On Sunday, January 6, 2019 at 10:45:01 PM UTC, Bruce wrote: >>> >>> On Mon, Jan 7, 2019 at 9:42 AM <agrays...@gmail.com> wrote: >>> >>>> On Saturday, December 8, 2018 at 2:46:41 PM UTC, agrays...@gmail.com >>>> wrote: >>>>> >>>>> On Thursday, December 6, 2018 at 5:46:13 PM UTC, agrays...@gmail.com >>>>> wrote: >>>>>> >>>>>> On Wednesday, December 5, 2018 at 10:13:57 PM UTC, >>>>>> agrays...@gmail.com wrote: >>>>>>> >>>>>>> On Wednesday, December 5, 2018 at 9:42:51 PM UTC, Bruce wrote: >>>>>>>> >>>>>>>> On Wed, Dec 5, 2018 at 10:52 PM <agrays...@gmail.com> wrote: >>>>>>>> >>>>>>>>> On Wednesday, December 5, 2018 at 11:42:06 AM UTC, >>>>>>>>> agrays...@gmail.com wrote: >>>>>>>>>> >>>>>>>>>> On Tuesday, December 4, 2018 at 9:57:41 PM UTC, Bruce wrote: >>>>>>>>>>> >>>>>>>>>>> On Wed, Dec 5, 2018 at 2:36 AM <agrays...@gmail.com> wrote: >>>>>>>>>>> >>>>>>>>>>>> >>>>>>>>>>>> *Thanks, but I'm looking for a solution within the context of >>>>>>>>>>>> interference and coherence, without introducing your theory of >>>>>>>>>>>> consciousness. Mainstream thinking today is that decoherence does >>>>>>>>>>>> occur, >>>>>>>>>>>> but this seems to imply preexisting coherence, and therefore >>>>>>>>>>>> interference >>>>>>>>>>>> among the component states of a superposition. If the >>>>>>>>>>>> superposition is >>>>>>>>>>>> expressed using eigenfunctions, which are mutually orthogonal -- >>>>>>>>>>>> implying >>>>>>>>>>>> no mutual interference -- how is decoherence possible, insofar as >>>>>>>>>>>> coherence, IIUC, doesn't exist using this basis? AG* >>>>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> I think you misunderstand the meaning of "coherence" when it is >>>>>>>>>>> used off an expansion in terms of a set of mutually orthogonal >>>>>>>>>>> eigenvectors. The expansion in some eigenvector basis is written as >>>>>>>>>>> >>>>>>>>>>> |psi> = Sum_i (a_i |v_i>) >>>>>>>>>>> >>>>>>>>>>> where |v_i> are the eigenvectors, and i ranges over the >>>>>>>>>>> dimension of the Hilbert space. The expansion coefficients are the >>>>>>>>>>> complex >>>>>>>>>>> numbers a_i. Since these are complex coefficients, they contain >>>>>>>>>>> inherent >>>>>>>>>>> phases. It is the preservation of these phases of the expansion >>>>>>>>>>> coefficients that is meant by "maintaining coherence". So it is the >>>>>>>>>>> coherence of the particular expansion that is implied, and this has >>>>>>>>>>> noting >>>>>>>>>>> to do with the mutual orthogonality or otherwise of the basis >>>>>>>>>>> vectors >>>>>>>>>>> themselves. In decoherence, the phase relationships between the >>>>>>>>>>> terms in >>>>>>>>>>> the original expansion are lost. >>>>>>>>>>> >>>>>>>>>>> Bruce >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> I appreciate your reply. I was sure you could ascertain my error >>>>>>>>>> -- confusing orthogonality with interference and coherence. Let me >>>>>>>>>> have >>>>>>>>>> your indulgence on a related issue. AG >>>>>>>>>> >>>>>>>>> >>>>>>>>> Suppose the original wf is expressed in terms of p, and its >>>>>>>>> superposition expansion is also expressed in eigenfunctions with >>>>>>>>> variable >>>>>>>>> p. Does the phase of the original wf carry over into the >>>>>>>>> eigenfunctions as >>>>>>>>> identical for each, or can each component in the superposition have >>>>>>>>> different phases? I ask this because the probability determined by >>>>>>>>> any >>>>>>>>> complex amplitude is independent of its phase. TIA, AG >>>>>>>>> >>>>>>>> >>>>>>>> The phases of the coefficients are independent of each other. >>>>>>>> >>>>>>> >>>>>>> When I formally studied QM, no mention was made of calculating the >>>>>>> phases since, presumably, they don't effect probability calculations. >>>>>>> Do >>>>>>> you have a link which explains how they're calculated? TIA, AG >>>>>>> >>>>>> >>>>>> I found some links on physics.stackexchange.com which show that >>>>>> relative phases can effect probabilities, but none so far about how to >>>>>> calculate any phase angle. AG >>>>>> >>>>> >>>>> Here's the answer if anyone's interested. But what's the question? How >>>>> are wf phase angles calculated? Clearly, if you solve for the >>>>> eigenfunctions of some QM operator such as the p operator, any phase >>>>> angle >>>>> is possible; its value is completely arbitrary and doesn't effect a >>>>> probability calculation. In fact, IIUC, there is not sufficient >>>>> information >>>>> to solve for a unique phase. So, I conclude,that the additional >>>>> information >>>>> required to uniquely determine a phase angle for a wf, lies in boundary >>>>> conditions. If the problem of specifying a wf is defined as a boundary >>>>> value problem, then, I believe, a unique phase angle can be calculated. >>>>> CMIIAW. AG >>>>> >>>>>> >>>>>>>> Bruce >>>>>>>> >>>>>>> >>>> I could use a handshake on this one. Roughly speaking, if one wants to >>>> express the state of a system as a superposition of eigenstates, how does >>>> one calculate the phase angles of the amplitudes for each eigenstate? AG >>>> >>> >>> One doesn't. The phases are arbitrary unless one interferes the system >>> with some other system. >>> >>> Bruce >>> >> >> If the phases are arbitrary and the system interacts with some other >> system, the new phases presumably are also arbitrary. So there doesn't seem >> to be any physical significance, yet this is the heart of decoherence >> theory as I understand it. What am I missing? TIA, AG >> > > Also, as we discussed, the phase angles determine interference. If they > can be chosen arbitrarily, it seems as if interference has no physical > significance. AG >

Puzzling, isn't it? We have waves in Wave Mechanics. Waves interfere with each other, even if they're probability waves, and this is one of the core features of Wave Mechanics. So phase angles must relate to degrees of interference. But if the phase angles are arbitrary; ERGO, so is the interference; arbitrary and thus NOT well defined. What am I missing? TIA, AG > > > -- 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.