V(x) is an arbitrary repulsive potential localized at a position along the x-axis as shown below.
The solution of the Schroedinger equation must be of the
ψ(x) = A eikx + B e-ikx for x < -a,
ψ(x) = C eikx + D e-ikx for x > a.
and assume A and B are arbitrary complex numbers. Use
conservation of flux to show that
|S11|2 + |S21|2 = |S12|2 + |S22|2 = 1,
S11S12* + S21S22* = 0.
Show that S is a unitary matrix.
Flux conservation requires
that |A|2 - |B|2 = |C|2 - |D|2,
or |A|2 + |D|2 = |C|2 + |B|2.
(Flow into the region where V(x) is not zero equals flow out of the region where V(x) is not zero.)
C = S11A + S12 D. B = S21A + S22D.
|C|2 + |B|2 = |S11|2|A|2 + |S12|2|D|2 + S11S12*AD* + S11*S12A*D + |S21|2|A|2 + |S22|2|D|2 + S21S22*AD* + S21*S22A*D
= (|S11|2 + |S21|2)|A|2 + (|S12|2 + |S22|2)|D|2 + (S11S12* + S21S22*)AD* + (S11*S12 +S21*S22)A*D
= |A|2 + |D|2.
This implies that
|S11|2 + |S21|2 = 1, |S22|2 + |S12|2 = 1, S11S12* + S21S22* = 0.
S is unitary.