Past exam of the quantum condensed matter course of the University of Cambridge / 2014 / 1 / b

Adopt the usual orthogonal tight-binding approximation, so the three localized spatial states are orthonormal. Each site has two neighbors, with the same real hopping matrix element. For coefficients $c_n$, the Schrödinger equation at site $n$ isThe constant-coefficient state has $c_1=c_2=c_3$, giving energy $E_0+2t$. Its normalization therefore yieldsTo verify that it is the ground state, the three discrete ring momenta are $\theta =0,\pm 2\pi /3$, with $c_n\propto e^{i\theta (n-1)}$ and energies $E_0+2t\cos \theta$. The other two states both have energy $E_0-t$. Since $t<0$, $E_0+2t<E_0-t$ and the gap is $3|t|$. This is the three-site tight-binding ring spectrum.

The spatial ground state is unique and has equal probability $1/3$ on every site; its common overall phase is arbitrary. Spin would supply a separate twofold degeneracy when no spin-dependent terms are included, but does not change the requested spatial state. If the localized functions were instead nonorthogonal raw atomic orbitals, their overlap matrix would be additional data needed for normalization and a generalized eigenvalue problem.