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)=2√Km|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!