= Solution
Assume a real, time-independent potential with the same zero <asymptote> on both sides, so $k=\sqrt{2mE}/\hbar$ is the same incident and transmitted wave number. For unit incoming amplitudes,
$$
\psi_L\sim\begin{cases}e^{ikx}+r_Le^{-ikx},&x\to-\infty,\\t_Le^{ikx},&x\to+\infty,\end{cases}
\qquad
\psi_R\sim\begin{cases}t_Re^{-ikx},&x\to-\infty,\\e^{-ikx}+r_Re^{ikx},&x\to+\infty.\end{cases}
$$
Multiplying the two <Schrödinger equations> crosswise and subtracting shows that the <Wronskian> $W=\psi_L\psi_R'-\psi_L'\psi_R$ is constant. Its two asymptotic values are $-2ikt_R$ and $-2ikt_L$, hence $t_L=t_R$.
For a real potential the <probability current> $j=(\hbar/m)\operatorname{Im}(\psi^*\psi')$ is conserved, giving $|r_L|^2+|t_L|^2=|r_R|^2+|t_R|^2=1$. Therefore
$$
\boxed{T_L=T_R=|t|^2,\qquad R_L=R_R=1-T.}
$$
The reflection \i[amplitudes] can have different <phases>. Reality or <self-adjointness> is essential for the reflection-probability conclusion: a complex absorbing potential need not have equal left and right reflectances. Equal asymptotic <speeds> are also used in identifying transmission with the squared amplitude.
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