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Is this analytically solvable?

  1. Feb 26, 2017 #1
    1. The problem statement, all variables and given/known data
    I am trying to solve integrals of the form:
    ## \int_{-\infty}^{\infty} \frac{k}{k-1} e^{itk^2}e^{ikx}e^{-(k-k_0)^2/\alpha^2} dk \\
    \int_{-\infty}^{\infty} \frac{1}{k-1} e^{itk^2}e^{ikx}e^{-(k-k_0)^2/\alpha^2} dk ##

    A bit of background:
    I'm trying to create an animation in Mathematica to show a gaussian wave packet passing through a delta potential. The fractions at the start of the integrals come from the transmission and reflection coeffecients such that
    ##\phi_k(x) = \left\{
    \begin{array}{ll}
    e^{ikx} + R_k e^{-ikx} & \quad x < 0 \\
    T_k e^{ikx} & \quad x > 0
    \end{array}
    \right.##
    so Rk and Tk are the fractions at the start then we multiply the phi(x) by a time dependent part and then the gaussian part and integrate for the linear superposition.


    2. Relevant equations
    The overall integral is
    ##\int_{-\infty}^{\infty} \phi_k(x) e^{-(k-k_0)^2/\alpha^2} e^{-i\hbar k^2 t/2m} dk \\
    R_k =\frac{1}{\frac{\hbar^2ik}{sm} -1} \\
    T_k =\frac{1}{1-\frac{sm}{\hbar^2ik}} ##

    where s is the strength of the delta potential such that:

    ##
    V(x) = s\delta(x) ##
    and I've set virtually every constant to 1 (except k0 and alpha which I plan to toggle once I get something).
    3. The attempt at a solution
    I'm told these integrals should be possible analytically but I can't see how. I've also tried using mathematica to compute it but it's having trouble with the function not converging on the interval. I've tried using NIntegrate too but I think the fact there's factors of x and t in the integrand is causing issues when trying to get a numerical approximation.

    It might be worth mentioning that I've managed to solve the integral for the the e^(ikx) part for x<0 using completing the square but I can't see how to do it with the fraction.

    Any help or guidance would be greatly appreciated thank you!
     
  2. jcsd
  3. Feb 28, 2017 #2
    It seems that Mathematica can calculate
    FourierTransform[ Exp[I t k^2] Exp[-(k - k0)^2/\[Alpha]^2] , k, x]
    and
    FourierTransform[k/(k - 1) , k, x]
    but it chokes on the whole thing.
    Maybe you can do something with the convolution theorem to combine them?
     
  4. Feb 28, 2017 #3

    Ray Vickson

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    Science Advisor
    Homework Helper

    Let ##f(k) = \exp(i t k^2 + i x k - (k-k_0)^2/ \alpha^2)##. If you can get ##\int_{\mathbb{R}} f(k) \, dk## and ##\int_{\mathbb{R}} k f /(k-1) \, dk##, then you an get ##\int_{\mathbb{R}} f/(k-1) \, dk## because ##1/(k-1) = (1-k + k)/(k-1) = k/(k-1) - 1.##
     
    Last edited: Feb 28, 2017
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