> On 9 Jan 2019, at 07:58, agrayson2...@gmail.com wrote:
> 
> 
> 
> 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 
> <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 







> 
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