2d Wavepacket in relativistic QM

  • Thread starter pitfall
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  • #1
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Main Question or Discussion Point

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!!
 

Answers and Replies

  • #2
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Have you tried spherical coordinates?

Torquil
 
  • #3
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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.

Have you tried spherical coordinates?

Torquil
 
  • #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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