On Thursday, January 10, 2019 at 11:07:51 AM UTC, Bruno Marchal wrote: > > > On 9 Jan 2019, at 07:58, [email protected] <javascript:> wrote: > > > > On Monday, January 7, 2019 at 11:37:13 PM UTC, [email protected] wrote: >> >> >> >> On Monday, January 7, 2019 at 2:52:27 PM UTC, [email protected] wrote: >>> >>> >>> >>> On Sunday, January 6, 2019 at 10:45:01 PM UTC, Bruce wrote: >>>> >>>> On Mon, Jan 7, 2019 at 9:42 AM <[email protected]> wrote: >>>> >>>>> On Saturday, December 8, 2018 at 2:46:41 PM UTC, [email protected] >>>>> wrote: >>>>>> >>>>>> On Thursday, December 6, 2018 at 5:46:13 PM UTC, [email protected] >>>>>> wrote: >>>>>>> >>>>>>> On Wednesday, December 5, 2018 at 10:13:57 PM UTC, >>>>>>> [email protected] wrote: >>>>>>>> >>>>>>>> On Wednesday, December 5, 2018 at 9:42:51 PM UTC, Bruce wrote: >>>>>>>>> >>>>>>>>> On Wed, Dec 5, 2018 at 10:52 PM <[email protected]> wrote: >>>>>>>>> >>>>>>>>>> On Wednesday, December 5, 2018 at 11:42:06 AM UTC, >>>>>>>>>> [email protected] wrote: >>>>>>>>>>> >>>>>>>>>>> On Tuesday, December 4, 2018 at 9:57:41 PM UTC, Bruce wrote: >>>>>>>>>>>> >>>>>>>>>>>> On Wed, Dec 5, 2018 at 2:36 AM <[email protected]> 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 > > > > The *global* phase angle is arbitrary: Psi = e^phi Psi. > > The relative phase angle is not arbitrary: you can distinguish all states > up + e^phi down, when phi varies. > > All this follows from the Born rule. > > Bruno > What about the case where the superposition is a sum of many eigenstates? How do you calculate the phase angle of each eigenstate? I don't see how Born's rule helps. AG
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