19.6: Allowed Wavenumbers from Boundary Conditions
- Page ID
- 29524
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The usual way of representing a wave on a line in physics is to have displacement proportional to \(e^{i k x}\), and \(k\) is called the wavenumber. For our discretized system, the displacement parameter for the \(n^{\text {th }}\) atom, at position na, would therefore be proportional to \(e^{i k n a}\).
But we know this is an eigenvector of a circulant, so we must have \(e^{i N k a}=1\), and the allowed values of \(k\) are
\[k_{n}=\frac{2 \pi}{N a} n=\frac{2 \pi}{L} n\]
with \(n\) an integer.
The circulant structure of the matrix has determined the eigenvectors, but not the eigenvalues \(\omega_{n}\)