Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 2 / 18A / Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 35D / a / Solution

For $V(x)=V_0\delta(x)$, $E=\hbar^2k^2/(2m)$, and $\gamma=mV_0/(\hbar^2k)$, continuity and the derivative jump give
$$
t=\frac1{1+i\gamma},
\qquad
r=\frac{-i\gamma}{1+i\gamma}.
$$
They obey $|r|^2+|t|^2=1$ and $r^*t+t^*r=0$.


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