# 7.3: Normal Modes for a One-dimensional Chain

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

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:

$\omega(k)=2 \sqrt{\frac{K}{m}}\left|\sin \left(\frac{1}{2} k a\right)\right|$

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.