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A wave packet and a Delta potential

  1. Feb 22, 2017 #1
    1. The problem statement, all variables and given/known data
    I want to plot using Mathematica a wave packet entering a delta potential ##V(x) = s\delta(x) ## (s is the strength) but I need to get the physics right first and I'm having trouble with a a few parts. I need to compute the integral ## \int_{0}^{\infty} e^{-i\omega t}\phi_{\omega} f(\omega) d\omega ## to get the linear superpositions and get the wave packet but Mathematica can't handle it and I can't compute it by hand.
    I'm taking ## f(\omega) = e^{-\omega^2/\alpha^2}## (I assume to make this a gaussian wave packet).
    2. Relevant equations
    ##\phi_{\omega} (x) = e^{ikx} + R_{\omega}e^{-ikx}, x<0 \\
    \phi_{\omega}(x) =T_{\omega}e^{ikx}, x>0 \\
    k^2 = 2mE/\hbar \\
    \omega = E/\hbar ##

    3. The attempt at a solution
    I found the Reflection and Transmission coefficients:
    ## R_{\omega} = \frac{1}{\frac{\hbar^2ik}{sm}-1} \\
    T_{\omega} = \frac{1}{1-\frac{sm}{\hbar^2ik}} ##
    from the continuity conditions and also from integrating the Schrodinger equation.

    From what I understand I think I then need to compute the integral as stated above and plot this against time, but I can't get Mathematica to solve it and I was told it could be computed exactly so I must be messing something up.

    Any help would be greatly appreciated thank you.
     
  2. jcsd
  3. Feb 27, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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