[Pw_forum] phonon eigenvectors

[Eric Abel]

>From: Stefano Baroni <baroni at sissa.it>
>Reply-To: pw_forum at pwscf.org
>To: pw_forum at pwscf.org
>Subject: Re: [Pw_forum] phonon eigenvectors
>Date: Thu, 31 Aug 2006 22:28:04 +0200
>
>Dear Eric: I feel flattered (and a little guilty) for your taking me  so 
>seriously. Also, I may have had a role in starting the discussion  taken up 
>by Axel and Fernando which, to tell the truth, I do not  fully understand. 
>All in all, forgive my being (involuntarily!)  paternalistic, and let's 
>come to the real stuff!

No worries.  It's hard to portray tone over email.  I guess I assumed your 
tone from reading your responses to other postings (where the post-er was 
clearly not doing their homework) and assumed you thought the same of me...I 
just wanted to clear that up.

>
>>Question: What is the meaning of an eigenvector?
>>Answer:  An eigenvector tells how the atoms are displaced in the  
>>vibration.
>
>OK
>
>>Question: What is the imaginary part?
>>Answer: The imaginary part is a phase factor.
>
>Forgive my fussiness: its is *not* a phase factor. A non zero  imaginary 
>part is *due* to a non zero phase.
>
>>If one atom has a significant imaginary part with respect to the  rest, 
>>then then it's displacement in the vibration will be phase  shifted with 
>>respect to the others.
>
>OK - More precisely: a phase dfference between two different atomic  
>components of a same eigenvectors imply that the motions of the two  atoms 
>are out of phase (i.e. the velocities of the two may vanish at  different 
>times)
>
>>In the case of a completely imaginary eigenvector, the displacement  will 
>>be completely out of phase.
>
>This statement, instead, is meaningless. Ou of phase WITH RESPECT TO  WHAT?

Yeah, poorly chosed words here...I guess I redeem myself below.  I really 
did mean "out of phase with respect to an atom with say a completly real 
eigenvector".

>The only meaningful thing is the phase difference between the  eigenvector 
>components of two different atoms. An overall phase equal  for all the 
>atoms is simply equivalent to a shift of the origin of  time. CAN YOU SEE 
>THIS?

yes.

>
>Can you see the analogies with the well know statement that quantum  
>mechanical wavefunctions (a problem which is conceptually TOTALLY  
>different) are defined to within an overall phase factor?

again, yes.

>
>>Question:  If the imaginary part is a phase factor, then what does  it 
>>mean if all of the components are imaginary?
>>Answer:  First answer: nothing.  If all of the atoms are "out of  phase", 
>>then they are "in phase" with respect to eachother,  therefore, having a 
>>completely imaginary or completely real  eigenvector should be equivalent.
>
>VERY GOOD. YOU GOT IT!
>
>>Question:  Great.  If this is the case, then why does this  eigenvector 
>>come up imaginary when all of the other vectors come up  real?
>>Answer:  Hmmm.  That's a good one.  There must be some reason that  the 
>>code chooses an imaginary eigenvector for this mode...time to  get help on 
>>this one.  We are fresh out of answers.
>
>Very good, Erich. You came to the point. The answer is deceivingly  simple. 
>If the the phase is *physically* irrelevant, how could a  physically sound 
>mathematical algorithm choose it? Answer (deeper  than it may sound at 
>first): AT RANDOM! Actual algorithms may not be  really random, but it 
>wouldn't harm if they were!
>
>>And with this I come first to the pwscf discussion archive.  I  didn't 
>>find any discussion of imaginary eigenvectors with real  eigenvalues.
>
>Hey, hey, hey! Slow down! Real eigenvalues are a consequence of the  
>Hermiticity of the dynamical matrix and have nothing to do with the  
>eigenvector being or not real. A Hermitean matrix may or may not have  
>complex eigenvectors. If the matrix is real, the phases can alwayes  (*not* 
>"must": you see the difference?) in such a way that the  eigenvectors are 
>real. If the matrix is not real (still being  Hermitean), I do not know: 
>eigenvectors are in general complex (i.e.  no overall phase can be chosen  
>so as to make all of their components  real), but they may also be in some 
>particular instance real ...

OK, that makes sense.

>
>>I looked to the literature, with the same result.  There comes a  time 
>>when one asks himself questions to which he doesn't have the  answers, 
>>then it comes time to discuss with his peers.  With that I  come to you 
>>with my question:
>>
>>Why did the code "choose" to make this displacement imaginary.  Is  this 
>>simply an artifact of the matrix diagonalization,
>
>YES
>
>>or is there some physical implications to this?
>
>NO
>
>>Thank you for your time and patience.
>
>Thank you for challenging them! ;-)
>(I am not kidding: it's refreshing and instructive for us to help, no  less 
>that it is for you to be assisted - hopefully ;-)
>
>Stefano
>
>---
>Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center -  Trieste
>[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)
>
>Please, if possible, don't  send me MS Word or PowerPoint attachments

Amen to that...although, Openoffice does quite well at opening these :)

>Why? See:  http://www.gnu.org/philosophy/no-word-attachments.html
>
>
>