19: One-Dimensional Crystal Dynamics Last updated Dec 30, 2020 Save as PDF 18.2: Damped Driven Oscillator 19.1: The Model Page ID29525 Michael FowlerUniversity of Virginia ( \newcommand{\kernel}{\mathrm{null}\,}\) 19.1: The Model19.2: The Circulant Matrix- Nature of its Eigenstates19.3: Comparison with Raising Operators19.4: Finding the Eigenvectors19.5: Eigenvectors of the Linear Chain19.6: Allowed Wavenumbers from Boundary Conditions19.7: Finding the Eigenvalues19.8: The Discrete Fourier Transform19.9: A Note on the Physics of These Waves