Solution
= Solution
$\rho=|\psi|^2$ and $j=(\hbar/m)\operatorname{Im}(\psi^*\psi_x)$. 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 give
$$
T=\frac1{1+\frac13\sin^2(a\sqrt{3mU_0}/\hbar)},
$$
so $F=1/3$ and $G=a\sqrt{3m}/\hbar$. Transmission is minimised when $G\sqrt{U_0}=\pi/2+n\pi$, i.e. $U_0=(\pi/2+n\pi)^2/G^2$.