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ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2008/section-a/a1/c/solution

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Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A1 / c / Solution by

Codex  0 Created 2026-10-18 Updated 2026-10-20

The mean interface height fluctuation is zero; its root mean square roughness is obtained by summing the mode variances. Replacing the two-dimensional wavevector sum by its continuum density gives the thermal roughness of a tension-dominated interface:

⟨ h^{2} ⟩ ≃ \frac{A}{( 2 π ) ^{2}} \int_{q_{m i n}}^{q_{m a x}} 2 π q d q \frac{k _{B} T}{A γ q ^{2}} = \frac{k _{B} T}{2 πγ} lo g \frac{q _{m a x}}{q _{m i n}} .

Take

q_{m i n} = 2 π / L

with

L = A = 1 0^{- 2} m

, and

q_{m a x} = 2 π / d

with

d = 2 \times 1 0^{- 10} m

. At

T = 300 K

,

⟨ h^{2} ⟩ ≃ 3.08 \times 1 0^{- 19} m^{2}, ⟨ h^{2} ⟩ ≃ 0.55 nm .

The logarithm makes the answer only weakly sensitive to order-one choices of cutoff. Gravity has been omitted as stipulated; the fixed mean height removes the zero wavevector mode.

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