Quantum Mechanics | Part IB, 2003

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

V(x)={V00xa0 otherwise V(x)= \begin{cases}V_{0} & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise }\end{cases}

with V0>0V_{0}>0.

In the limit V0,a0V_{0} \rightarrow \infty, a \rightarrow 0 with V0a=UV_{0} a=U held fixed, show that the transmission probability is

T=(1+mU22E2)1T=\left(1+\frac{m U^{2}}{2 E \hbar^{2}}\right)^{-1}

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