1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook