A particle of mass $m$ and energy $E$ moving in one dimension is incident from the left on a potential barrier $V(x)$ given by

$$V(x)= \begin{cases}V_{0} & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise }\end{cases}$$

with $V_{0}>0$.

In the limit $V_{0} \rightarrow \infty, a \rightarrow 0$ with $V_{0} a=U$ held fixed, show that the transmission probability is

$$T=\left(1+\frac{m U^{2}}{2 E \hbar^{2}}\right)^{-1}$$