Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 34B / b / 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} = \frac{1}{3} e^{ikna} .

Periodicity after three sites requires

e^{3 ika} = 1.

Choosing representatives in the Brillouin zone

- \frac{π}{a} \leq k < \frac{π}{a}

gives

k = 0, \frac{2 π}{3 a}, - \frac{2 π}{3 a} .

The corresponding states are

∣ Ψ_{k} ⟩ = \frac{1}{3} n = 0 \sum 2 e^{ikna} ∣ ψ_{n} ⟩ .

They reproduce respectively the coefficient vectors

(1, 1, 1)

,

(1, ω, ω^{2})

and

(1, ω^{2}, ω)

.

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