2d Wavepacket in relativistic QM

[pitfall]

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 $$\Psi(x,0)$$ a spherically symmetric Gaussian. The relativistic dispersion relation is of course $$\omega^2=k_x^2+k_y^2$$.

I can now Fourier transform $$\Psi(x,0)$$ to $$\Phi(k,0)$$, no problem.

But when I want to calculate $$\Psi(x,t)$$, 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 $$\int_{-\infty}^{+\infty} dk_x dk_y \exp\left( i k_x x+i k_y y - i \sqrt{k_x^2+k_y^2}t -\alpha^2(k_x^2+k_y^2)/2\right)$$.

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!

 

[torquil]

Have you tried spherical coordinates?

Torquil

 

[pitfall]

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

 

[schieghoven]

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