Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 3 / v

Solution

Write

(ϕ_{q} (0), 0)^{T} = C_{B} e_{B} + C_{A} e_{A}

C_{A} = - β C_{B}

and

ϕ_{q} (0) = C_{B} (1 - α β)

give

f_{1} = \frac{C _{B}}{ϕ _{q} ( 0 )} = \frac{1}{1 - α β}, f_{2} = \frac{α C _{A}}{ϕ _{q} ( 0 )} = \frac{- α β}{1 - α β} .

Since

α β = - c^{2} q^{2} / a^{2}

f_{1} (q) = 1 - \frac{c ^{2} q ^{2}}{a ^{2}} + O (q^{4}), f_{2} (q) = \frac{c ^{2} q ^{2}}{a ^{2}} + O (q^{4}) .

Therefore

G_{ϕ} = f_{1} e^{- λ_{B} t} + f_{2} e^{- λ_{A} t}

. At

q = 0

, conservation makes

ϕ_{0}

exactly constant. Any nonzero overlap with the fast mode would change it on the finite timescale

a^{- 1}

, so conservation requires

f_{2} \to 0

; the explicit

q^{2}

factor enforces this.

Solved by gpt-5.6-sol high.

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