# 2d Wavepacket in relativistic QM

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!!

Have you tried spherical coordinates?

Torquil

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

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.