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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 36D / Solution

Let

G_{n} = 2 πn / a

. The free states

∣ k ⟩

and

∣ k - G_{n} ⟩

are degenerate at the Bragg point

k = G_{n} /2 = nπ / a

. Since

⟨ k ∣ V ∣ k - G_{n} ⟩ = V_{n},

degenerate perturbation theory restricts the Hamiltonian near this crossing to

(ℏ^{2} k^{2} / (2 m) + V_{0} V_{n}^{*} V_{n} ℏ^{2} (k - G_{n})^{2} / (2 m) + V_{0}) .

Writing

k = nπ / a + κ

and diagonalizing gives the nearly-free electron dispersion near a one-dimensional band gap

E_{\pm} (k) = V_{0} + \frac{ℏ ^{2}}{2 m} [(\frac{nπ}{a})^{2} + κ^{2}] \pm (\frac{ℏ ^{2} nπκ}{ma})^{2} + ∣ V_{n} ∣^{2} .

Thus the two continuous bands avoid crossing, and at

κ = 0

their separation is

Δ E_{n} = 2∣ V_{n} ∣ .

This is the nearly-free electron model. The dispersion relation is the relation

E = E (k)

between energy and Bloch wavevector.

For the specified potential,

\frac{8}{3} V_{0} cos^{4} (\frac{2 π x}{a}) = V_{0} + \frac{4}{3} V_{0} cos (\frac{4 π x}{a}) + \frac{1}{3} V_{0} cos (\frac{8 π x}{a}) .

Hence

V_{\pm 2} = \frac{2}{3} V_{0}, V_{\pm 4} = \frac{1}{6} V_{0},

and all other nonconstant Fourier coefficients vanish. The gaps are therefore

n 24 crossing wavevectors k = \pm 2 π / a k = \pm 4 π / a gap width 4∣ V_{0} ∣/3 ∣ V_{0} ∣/3.

Their centre energies are respectively

V_{0} + \frac{2 ℏ ^{2} π ^{2}}{m a ^{2}}, V_{0} + \frac{8 ℏ ^{2} π ^{2}}{m a ^{2}} .

The extended-zone sketch consists of the free-particle parabolas shifted upward by

V_{0}

, with avoided crossings of these two widths at the listed Bragg points; elsewhere they meet to this order because the corresponding Fourier coefficients vanish.

Solved by gpt-5.6-sol high.

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