Posted by Steve Simon on February 1, 2013, 6:26 pm, in reply to "Phase velocities in the 1D chain"
Really good question! (This would be a very nasty question to put on an exam!!)
To repeat the question, the definition of phase velocity is omega/k. But k is the same as k+2pi/a, so which phase velocity is right?
The answer here is similar to the answer to a question asked yesterday here
http://www-thphys.physics.ox.ac.uk/people/SteveSimon/members.boardhost.com/OxfordSolidState2013/msg/1359643978.html
The key here is to realize that the wave for the monatomic chain is only defined at positions x=na with n being an integer. In between these special lattice points, the wave is not defined. It is this feature that allows us to say that
e^{ikx} = e^{i (k + 2 pi/a) x}
This is true because x only takes values na. However, if you look between these lattice points, it makes a difference whether you are talking about k or k+2 pi/a. It is useful to look at the picture here
http://en.wikipedia.org/wiki/File:AliasingSines.svg
which shows the two different waves and shows how they take the same values at the special points x = na.
Recall now that phase velocity is the velocity at which the wave crests move (as compared the velocity at which the wave packet moves, which is the group velocity). There is a really nice animation here
http://en.wikipedia.org/wiki/Phase_velocity
Note that to find the crest of a wave, you need to look between lattice points, and thus it makes a difference whether you think of this as k or k+2pi/a.
So if you consider the time dependent waves, and you tracked the position of the crest of the wave, you would find that the wave k, moves at phase velocity omega/k and the crest of the wave k+2pi/a moves at phase velocity omega/(k+2pi/a) !
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