Is the S Matrix Unitary in One-Dimensional Scattering Problems?

[jaobyccdee]

1. The problem statement, all variables and given/known
A general one dimensional scattering problem could be characterized by an
(arbitrary) potential V (x) which is localized by the requirement that V (x) = 0
for |x|> a. Assume that the wave-function is
ψ (x) =

Ae^(ikx) + Be^(-ikx) x < -a
Ce^(ikx) + De^(-ikx) x > a
Relating the \outgoing" waves to the \incoming" waves by the matrix equation

C=S11A+ S12B
B=S21A+ S22D


show that
|S11|^2 + |S21|^2 = 1
|S12|^2 + |S22|^2 = 1
S11S12* + S21S22* = 0
Use this to show that the S matrix is unitary.

Homework Equations

I don't understand why C=S11A+S12B or B=S21A+S22D

The Attempt at a Solution

I calculate the flux for the incoming beam and the outgoing beam and set them equal, i get 2A^2 ik-2ikB^2=2C^2ik-2D^2ik i don't see how C and B can be expressed with only two other variables.

 

[Steely Dan]

I don't understand why C=S11A+S12B or B=S21A+S22D

You can derive these conditions by matching both the wavefunction and its derivative at x=a (we generally require these functions to be continuous everywhere, when possible).