 Solution

ID: past-exam-of-the-chemistry-course-of-the-year-3-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2a/section-b/23/c/i/solution

The nontotally symmetric longitudinal modes are the two

Σ_{u}^{+}

coordinates

t

and

a

. Both have

q_{3} - q_{2} = 0

, so the middle spring is not extended. Hence its spring constant cannot alter their energies:

D t = 0, D a = \frac{2 k}{m} a, ω_{t} = 0, ω_{a} = 2 k / m .

These results are exact for every positive

γ

, not just perturbative. The zero mode expresses overall translation invariance.

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