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2d Wavepacket in relativistic QM

  1. Mar 1, 2010 #1
    Dear users,

    I wonder if there is anybody who can give me a hint on how to handle the following situation:

    In the 2+1 dimensional Klein-Gordon equation with coordinates (t,x,y), I use as initial condition for [tex]\Psi(x,0)[/tex] a spherically symmetric Gaussian. The relativistic dispersion relation is of course [tex]\omega^2=k_x^2+k_y^2[/tex].

    I can now Fourier transform [tex]\Psi(x,0)[/tex] to [tex]\Phi(k,0)[/tex], no problem.

    But when I want to calculate [tex]\Psi(x,t)[/tex], by doing a second Fourier transformation, I get stuck because of the square root of the dispersion relation.

    In other words, I can't solve the integral [tex]\int_{-\infty}^{+\infty} dk_x dk_y \exp\left( i k_x x+i k_y y - i \sqrt{k_x^2+k_y^2}t -\alpha^2(k_x^2+k_y^2)/2\right) [/tex].

    If anybody could give me a hint on this, I would be very thankful and happy, I already spent way too much time on this!

    Thanks very much!!
     
  2. jcsd
  3. Mar 1, 2010 #2
    Have you tried spherical coordinates?

    Torquil
     
  4. Mar 2, 2010 #3
    Yes, I have tried polar coordinates, but I ended up with a fairly complicated angle expression in the exponent which resulted in some Bessel function after angle integration, and then I could not proceed.

     
  5. Mar 11, 2010 #4
    Yeah, pretty sure you will end up convolving with Bessel functions, given the generality of the initial data. Remember that Bessel functions of order n+1/2 can be written in closed form.

    This thread might help... it was for the 3+1dim massless case however. I had the idea at the time about seeing how exp(-r^2) would evolve in time, and then since d/dx commutes with the wave operator this would give closed form solutions to the wave equation with initial conditions corresponding to all the Hermite functions.

    https://www.physicsforums.com/showthread.php?t=269345

    Best of luck

    Dave
     
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