What is the Fourier Transform of the spatial portion of $\Psi(x,t)=A\exp(-b|x-2|)\exp(-i\omega t)$?

I'm not sure how to do it for the "spatial portion". I've only done Fourier transforms for functions of a single variable $f(x)$. I tried using the exponential Fourier transform $F(k)$ on it, but the integral seems to be impossible to solve. Any help/guidance/answer will be much appreciated.

Thank you very much!

I searched many times in many ways for how I would do the Fourier transform for the "spatial" portion, which I guess is like a 'partial Fourier transform with respect to x', and didn't find anything. My question here is the only relevant result on the internet at the moment it seems!

  • $\begingroup$ I think the idea here is that the time dependence is just an overall phase, so the "shape" of the wavefunction is not changing in time. Therefore, if you want to know the Fourier transform of the wavefunction in space, just Fourier transform the part that depends on $x$. $\endgroup$ – DanielSank Oct 7 '19 at 14:57
  • $\begingroup$ But how do you integrate 𝐴exp(−𝑏|𝑥−2|) ? $\endgroup$ – Arthur Karapetov Oct 7 '19 at 15:12
  • $\begingroup$ WP 207. Review your Cauchy distribution. $\endgroup$ – Cosmas Zachos Oct 7 '19 at 15:24
  • $\begingroup$ @ArthurKarapetov Just split the integral up. Remember, $|x-2|$ is just $x-2$ for $x>2$ and $-x+2$ for $x<2$ $\endgroup$ – BioPhysicist Oct 7 '19 at 15:28
  • $\begingroup$ Ok thank you very much! $\endgroup$ – Arthur Karapetov Oct 7 '19 at 15:49

The Fourier transform of the spatial portion of $\psi(x,t)$ simply means take the Fourier tansform with respect to $x$ and leave $t$ as a parameter, i.e.

$$ \tilde{\Psi}(k,t) = \int_{-\infty}^\infty dx\;e^{-i x k} \Psi(x,t)\;. $$

For the $\Psi(x,t)$ in your question the integral should be strait forward

  • $\begingroup$ Thank you, but I don't see how it's straight forward. What you wrote is what I tried but couldn't go further after simplifying the exponentials. $\endgroup$ – Arthur Karapetov Oct 7 '19 at 15:10
  • $\begingroup$ @ArthurKarapetov At this point you're asking how to do an integral, so you should probably make a new post (on the Math site) showing your work and asking a specific question about where you're stuck. $\endgroup$ – DanielSank Oct 7 '19 at 15:31

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