= Solution
For incidence from the left, normalize the asymptotic wavefunction by
$$
\psi(x)\sim e^{ikx}+r e^{-ikx}\quad(x\to-\infty),
\qquad
\psi(x)\sim t e^{ikx}\quad(x\to+\infty).
$$
For incidence from the right, define $r',t'$ analogously with the directions reversed. Conservation of the <probability current> gives
$$
|r|^2+|t|^2=1,\qquad |r'|^2+|t'|^2=1.
$$
Applying the same conserved sesquilinear current to the two scattering solutions gives
$$
t'^*r+r'^*t=0.
$$
These are exactly the column orthonormality relations $S^\dagger S=I$ for the stated <One-dimensional S-matrix>, so $S$ is unitary.
Back to article page
