> On 9 Jan 2019, at 07:58, [email protected] 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 > <http://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 > > > > -- > 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] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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.
