Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 34B (Applications of Quantum Mechanics)

Projecting $H|\Psi ⟩=E|\Psi ⟩$ onto $⟨{\psi }_n|$ and using orthonormality givesThus, with $d=E_0+\alpha$,

The symmetric vector $(1,1,1)$ has eigenvalueLet $\omega =e^{2\pi i/3}$, so $1+\omega +{\omega }^2=0$. For either vectorthe other two components in each row sum to minus the component in that row. Both therefore have eigenvalueThe spectrum is consequentlywith normalized eigenvectors obtained by multiplying the three displayed vectors by $1/\sqrt{3}$.

Translation by one site multiplies a lattice wave by $e^{ika}$, so the coefficients of a finite periodic tight-binding ring arePeriodicity after three sites requiresChoosing representatives in the Brillouin zonegivesThe corresponding states areThey reproduce respectively the coefficient vectors $(1,1,1)$, $(1,\omega ,{\omega }^2)$ and $(1,{\omega }^2,\omega )$.

In position space,Henceup to the common normalization factor $1/\sqrt{3}$.

Under $x\mapsto x+a$, reindexing $m=n-1$ gives the same three terms modulo three sites. The term shifted by three sites is unchanged because ${\psi }_{m+3}={\psi }_m$ and $e^{3ika}=1$. Therefore

The one-dimensional Bloch theorem states that an eigenfunction in a potential of period $a$ can be chosen in the formIt applies because the three-atom ring is invariant under translation by one lattice spacing and obeys periodic boundary conditions.