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

Take fixed $E>0$ and define $k=\sqrt{2mE}/\hslash$. For sufficiently large $V_0$, the barrier region has $\kappa =\sqrt{2m(V_0-E)}/\hslash$ and solves ${\psi }^{\prime \prime }={\kappa }^2\psi$. Solving this constant-coefficient equation gives the exact transfer relationIn the specified limit, $a=U/V_0$ and $\kappa a\to 0$, whileThe transfer matrix therefore tends to $\left(\begin{array}{cc}1& 0\\ \eta & 1\end{array}\right)$. As the two boundaries collapse to the origin, this proves continuity and the derivative jump for a delta potential:The vanishing exterior phase shift $e^{ika}\to 1$ introduces no further factor.

Write the incoming and reflected waves as $e^{ikx}+r{e}^{-ikx}$ on the left and the transmitted wave as $t{e}^{ikx}$ on the right. Matching givesThe denominator is nonzero for real $\eta$ and $k>0$, so the limiting matching system is well defined. The probability currents on either side have the same factor $\hslash k/m$, making the transmission probability $|t{|}^2$. Thus scattering by a delta potential givesThe limit retains a finite reflection probability despite the barrier's vanishing width because its integrated strength remains finite.