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Transmission Coefficient through a delta potential

  1. Jan 30, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider an uni-dimensional scattering by a delta function at the origin, given by the potential ##V(x) = g \delta(x)##, with ##g>0##.

    Using the following result, with ##G(x)## being the green function:

    $$\Psi (x) = e^{ikx} + g \dfrac{2m}{\hbar}\dfrac{G(x)}{1-2mgG(0)/\hbar^2} = e^{ikx} + g \dfrac{2m}{\hbar 2ik}\dfrac{e^{ikx}}{1-2mgG(0)/\hbar^2}$$

    Find the transmission coefficient through the potential.

    2. Relevant equations

    $$T = \dfrac{J_t}{J_i}$$

    $$J=\dfrac{\hbar}{2mi}(\Psi^\star\dfrac{d\Psi}{dx} - \Psi \dfrac{d \Psi ^\star}{dx})$$

    3. The attempt at a solution

    I know I am supposed to use the equations I wrote above, but I don't know what I should consider as the ##\Psi_i## and ##\Psi_t##. Any help in that regard?
     
    Last edited: Jan 30, 2015
  2. jcsd
  3. Jan 30, 2015 #2

    vela

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    The i means incident, and the t means transmitted, right? What's the physical situation in this problem? The incident wave is coming toward the potential from one side, and the transmitted wave is the part that gets through. Which side is the the incident wave coming from?
     
  4. Jan 30, 2015 #3
    I forgot to mention that ##G(x) = \dfrac{1}{2ik}e^{ikx}##

    You're correct, ##i## means incident and ##t## means transmitted. I don't know how to answer your other queries though. I do not know what's the physical situation in this problem, but I suppose you're right in saying that a wave is coming from one side and gets transmitted and reflected. Again, there's no info about the physical situation, but I'm gonna say that the wave comes from the left (##x<0##) and gets transmitted (##x>0##).
     
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