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For a parity-symmetric one-dimensional potential, the even and odd parity states diagonalize the One-dimensional S-matrix:For a real potential, unitarity further gives $|S_{++}|=|S_{--}|=1$ at real positive energy.

For incidence from the left, normalize the asymptotic wavefunction byFor incidence from the right, define $r^{\prime },t^{\prime }$ analogously with the directions reversed. Conservation of the probability current givesApplying the same conserved sesquilinear current to the two scattering solutions givesThese are exactly the column orthonormality relations $S^{†}S=I$ for the stated One-dimensional S-matrix, so $S$ is unitary.

For a wave incident from the left, define the reflection and transmission amplitudes byReflection invariance gives the same amplitudes for incidence from the right. If the incoming amplitudes from the left and right are ordered first, and the outgoing amplitudes toward the left and right second, the One-dimensional S-matrix is

For $V(x)=V_0\delta (x)$, the wavefunction is continuous and its derivative has the jumpThe scattering ansatz givesWithwe obtain the scattering by a delta potential amplitudesThey satisfyThereforeso the scattering matrix is unitary.