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7.3: Normal Modes for a One-dimensional Chain

( \newcommand{\kernel}{\mathrm{null}\,}\)

The matrix A is all zeros except for 2 on the diagonal and −2 on the superdiagonal. But this doesn’t really help us solve the problem. The solution comes from physical insight, not mathematical trickery!

Dispersion relation:

ω(k)=2Km|sin(12ka)|

Meaning of term “dispersion relation”:

Start with an arbitrary wave packet, break it up into Fourier components.

Each such component moves at a particular speed.

After some time, find how all the components have moved, then sew them back together.

The wave packet will have changed shape (usually broadened. . . dispersed).

Remember that we haven’t done any statistical mechanics in this section, nor even quantum mechanics. This has been classical mechanics!


This page titled 7.3: Normal Modes for a One-dimensional Chain is shared under a CC BY-SA license and was authored, remixed, and/or curated by Daniel F. Styer.

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