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In short laserpulses there is a minimal product of the frequency width and the pulselength for Gaussian pulses $\tau \cdot \Delta\omega \geq4\ln2$ this is the fourier boundary. So I know it origins from the Heisenberg uncertainty principle, but how can I derive it?

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It doesn't originate from the Heisenberg uncertainty principle, it is the Heisenberg uncertainty principle itself!

I'll give you tips:

  • Write down the gaussian pulse expression $f(t)$
  • Calculate its variance with respect to time: $$ \langle t^2\rangle = \int_{-\infty}^\infty t^2 f(t) $$
  • Take the Fourier transform of $f(t)\longrightarrow \tilde{f}(\omega)$

  • Calculate the variance $\langle \omega^2\rangle$ of $\tilde{f}(\omega)$ with respect to $\omega$

  • Multiply the two variances $\langle \omega^2\rangle\langle t^2\rangle = \,?$. You should get $1/4$, so that if you take the square root you get $1/2$.

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