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phonon derivation

In this derivation of phonon solutions, everywhere, we are forcefully assuming the wavelike characteristics along the length of the chain. While all we can deduce for finding out the fundamental frequencies is that the solution will be periodic in time, and solution should be of the form $\exp(i\omega t)$, I am not getting how the derivers arrive directly at $\exp(ikx - i\omega t)$. That one is in Kittel.

In the figure below, Here also, somehow, $q$ has been deliberately has been involved linking it to distance. PS: $n$ is a measure of distance along the chain.

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  • $\begingroup$ Ansatz? The better way would probably be to derive the general solutions by diagonalizing the matrix form of these equations. $\endgroup$ – CuriousOne Sep 4 '14 at 6:35
  • $\begingroup$ Yes. The matrix form can be achieved by substituting $x_n$ with $\exp(iwt)$. The problem is that will give only $\omega$ solutions. My doubt is, how could they assume the the periodicity along length of chain in the exponential. I think, all we can predict is that eigen frequencies exist, and solution will be periodic in time. $\endgroup$ – Harshfi6 Sep 4 '14 at 6:45
  • $\begingroup$ I agree with that. The solution presented in phenomenological physics books like Kittel is not a complete solution theory of these equations. It does not prove that the solutions they give are all the solutions that exist. Personally I never liked the Kittel book much... probably because I had to learn it like a bible rather than for understanding to pass the test in that class. As for the solutions... of course there are non-periodic solutions to these equations. You can find some of them by linear superposition of harmonic solutions with non-rational ratios between individual frequencies. $\endgroup$ – CuriousOne Sep 4 '14 at 7:10
  • $\begingroup$ Irrational ratios is interesting. Can you cite a source for it, for a detailed solution to this problem and problems like these? $\endgroup$ – Harshfi6 Sep 4 '14 at 7:24
  • $\begingroup$ I don't have a document that I could cite about the complete solution theory of these equations... I hope that some of the theoreticians can help you with that. Good luck! $\endgroup$ – CuriousOne Sep 4 '14 at 7:37
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There is nothing wrong with looking for plane-wave like solutions of the form $A \exp (i (\omega t - k x) )$. Given the linearity of the equations, and as @ignacio pointed out the fact that the $\exp (i k x_n)$ form a basis of solutions, you can write a more general solution as a combination of these plane waves. This solution isn't necessarily periodic (think of a propagating wavepacket peaked at a certain position in space, for example).

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For an infinite chain with periodic boundary conditions, you have $n$ translation symmetry. This means you can look for a basis of solutions whose $n$ dependence is $e^{ikn}$. The boundary conditions constrain you to $k=m\frac{2\pi}N$ with $m=0$..$N-1$

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