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

Write $({\varphi }_q(0),0{)}^T=C_B{e}_B+C_A{e}_A$. The equations $C_A=-\beta C_B$ and ${\varphi }_q(0)=C_B(1-\alpha \beta )$ giveSince $\alpha \beta =-c^2{q}^2/a^2$,Therefore $G_{\varphi }=f_1{e}^{-{\lambda }_{B}t}+f_2{e}^{-{\lambda }_{A}t}$. At $q=0$, conservation makes ${\varphi }_0$ 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.