2d Wavepacket in relativistic QM

In summary: He tried polar coordinates, but ended up with a fairly complicated angle expression in the exponent which resulted in some Bessel function after angle integration. He has also tried spherical coordinates, but he is not sure if he will end up convolving with Bessel functions, given the generality of the initial data. Someone suggested solving the equation for psi(x,t) in terms of spherical coordinates using the wave equation, and Dave found a thread on Physics Forums that might help him.
  • #1
pitfall
2
0
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!
 
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  • #2
Have you tried spherical coordinates?

Torquil
 
  • #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.

torquil said:
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
 

1. What is a 2d wavepacket in relativistic quantum mechanics?

A 2d wavepacket in relativistic quantum mechanics refers to a mathematical description of a particle's wave-like behavior in a 2-dimensional space, taking into account the effects of special relativity. This allows for the prediction of the particle's position and momentum at different points in time.

2. How is a 2d wavepacket different from a 1d wavepacket?

A 2d wavepacket differs from a 1d wavepacket in that it takes into account the particle's motion in two dimensions, rather than just one. This means that the wavepacket will spread out in both the x and y directions, rather than just in the x direction as in a 1d wavepacket.

3. What is the significance of considering relativistic effects in a 2d wavepacket?

Relativistic effects in a 2d wavepacket are significant because they allow for the prediction of a particle's behavior at high speeds, where classical mechanics fails to accurately describe the particle's motion. By incorporating special relativity, a more accurate and comprehensive understanding of the particle's behavior can be achieved.

4. How is a 2d wavepacket described mathematically in relativistic quantum mechanics?

In relativistic quantum mechanics, a 2d wavepacket is described by a wavefunction, which is a complex-valued function that describes the probability amplitude of the particle at different points in the 2-dimensional space. This wavefunction evolves over time according to the relativistic Schrödinger equation, taking into account the particle's energy, momentum, and potential energy.

5. What are some real-world applications of studying 2d wavepackets in relativistic quantum mechanics?

The study of 2d wavepackets in relativistic quantum mechanics has many practical applications, such as in the development of new technologies like quantum computing and precision measurements. It also has implications in particle physics and cosmology, helping us to better understand the behavior of particles at high energies and in extreme environments such as black holes.

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