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I am trying to calculate the fourier transform of $$\left(\frac{1}{\sigma \sqrt{2\pi}}\right)^{1/2}e^{\frac{-x^2}{4\sigma^2}}$$

Attempt: $\vert\psi(x) \vert^2 = \frac{1}{\sqrt{2\pi \sigma}}e^{-\frac{x^2}{2\sigma^2}}$

Now $\hat{\psi}(k) = \frac{1}{\sigma2\pi}\int_{\mathbb{R}}e^{ikx}e^{-\frac{x^2}{2\sigma^2}} =\frac{1}{\sigma2\pi}\int_{\mathbb{R}} e^{ikx-\frac{x^2}{2\sigma^2}}$

I don't know how to simplify this into $\delta$ expression.

  • https://mathworld.wolfram.com/FourierTransformGaussian.html – K.defaoite Mar 10 '23 at 10:47
  • I tried the question on Wolfram Alpha and the solution was $$\sqrt[4]{2\pi}\sqrt{1/\sigma}\delta(k)e^{-\frac{x^2}{4\sigma^2}}$$ but there were no explicit steps. –  Mar 10 '23 at 19:57
  • I have solved the problem –  Mar 10 '23 at 20:41

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