$$\require{cancel}$$

7.3: Normal Modes for a One-dimensional Chain

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!