 Solution

$\rho =|\psi {|}^2$ and $j=(\hslash /m)\mathrm{Im}({\psi }^{\ast }{\psi }_x)$. Schrödinger’s equation and its conjugate give ${\rho }_t+j_x=0$; a stationary state has $j_x=0$. Continuity of $\psi ,\psi \prime$ at $0,a$ and matching plane waves giveso $F=1/3$ and $G=a\sqrt{3m}/\hslash$. Transmission is minimised when $G\sqrt{U_0}=\pi /2+n\pi$, i.e. $U_0=(\pi /2+n\pi {)}^2/G^2$.