19.6: Allowed Wavenumbers from Boundary Conditions

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

$$k_{n}=\dfrac{2 \pi}{N a} n=\dfrac{2 \pi}{L} n$$

with $n$ an integer.

The circulant structure of the matrix has determined the eigenvectors, but not the eigenvalues $\omega_{n}$