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One dimension scattering problem

  1. Apr 28, 2012 #1
    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.

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



    3. 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.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 28, 2012 #2
    You can derive these conditions by matching both the wavefunction and its derivative at [itex]x=a[/itex] (we generally require these functions to be continuous everywhere, when possible).
     
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