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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 4

Near $x=\lambda$,
$$
1-\frac{\lambda^2}{x^2}=\frac{2(x-\lambda)}{\lambda}+O((x-\lambda)^2).
$$
With
$$
t=\left(\frac2\lambda\right)^{1/3}
\frac{x-\lambda}{\epsilon^{2/3}},
$$
the leading equation is $y_{tt}-ty=0$. Decay for $t>0$ selects $y=C\operatorname{Ai}(t)$. The <Airy turning-point connection formula> gives on the oscillatory side
$$
y\sim C' p(x)^{-1/2}
\sin\left(\frac1\epsilon\int_x^\lambda p(s)\,ds+\frac\pi4\right).
$$
For this to be proportional to the inner expression from part (a), the two phases must differ by an integer multiple of $\pi$. Thus the <WKB quantization condition> is
$$
\boxed{\int_1^\lambda\left(\frac{\lambda^2}{x^2}-1\right)^{1/2}dx
=\epsilon\left(n+\frac34\right)\pi,
\qquad n=0,1,2,\ldots.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 31C / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 31C / b / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 31C / b / Solution

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