= Solution
Translation by one site multiplies a lattice wave by $e^{ika}$, so the coefficients of a <finite periodic tight-binding ring> are
$$
c_n=\frac1{\sqrt3}e^{ikna}.
$$
Periodicity after three sites requires
$$
e^{3ika}=1.
$$
Choosing representatives in the <Brillouin zone>
$$
-\frac\pi a\leq k<\frac\pi a
$$
gives
$$
\boxed{k=0,\quad \frac{2\pi}{3a},\quad-\frac{2\pi}{3a}}.
$$
The corresponding states are
$$
\boxed{|\Psi_k\rangle=\frac1{\sqrt3}
\sum_{n=0}^2e^{ikna}|\psi_n\rangle}.
$$
They reproduce respectively the coefficient vectors $(1,1,1)$, $(1,\omega,\omega^2)$ and $(1,\omega^2,\omega)$.
Back to article page
