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