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Fourier Transform of Normal Distribution

$$f(t)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{t-\mu }{\sigma }\right)}^2}$$

Uncertainty Principle

For $f(t)\in L^2(\mathbb{R})$, the distribution of propability density of $f(t)$ and $\hat{f}(\omega )$ are $$\frac{|f(t){|}^2}{\Vert f{\Vert }^2},\frac{|\hat{f}(\omega ){|}^2}{2\pi \Vert \hat{f}{\Vert }^2}$$