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
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