Re: Coherent states of a superposition

```
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 <javascript:>> 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
>>>>>>>> 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
t

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