**Theorem 1** (see [Bru, Wil00]) Suppose that $f(t)$ belongs to $L^2(\mathbb{R})$, and denote $\hat f(\omega)$ its Fourier transform of $f$. Then the Heisenberg relation is $$\int (t-a)^2\abs{f(t)}^2 dt \cdot \int (\omega-b)\abs{\hat f(\omega)}^2 d\omega \gtrsim \norm{f}_{L^2}^2,$$ where $a,b$ are arbitrary real numbers. The uncertainty principle expresses, roughly speaking, that if a function is confined to a small time interval, then its Fourier transform cannot be confined to a small frequency interval. For example, the Dirac function $\delta(x)$ is confined to the origin, but its Fourier transform is $1$, spreading over the whole real line. === Relation with Quantum Mechanics **The Uncertainty Relation:** If $A$ and $B$ are Hermitian or anti-Hermitian operator, then we must have the following inequality: $$\abs{\langle \alpha|A^2|\alpha\rangle \langle \alpha|B^2|\alpha\rangle } \geq \f{1}{4}\abs{\langle \alpha|[A,B]|\alpha\rangle}^2,$$ for any state $|\alpha\rangle$. **//Proof://** Using the Cauchy-Schwarz inequality, we can immediately obtain the result. If we let $A: f(x)\mapsto xf(x)$, and $B: f(x)\mapsto \p_x f(x)$, then we can easily verify that $[A,B]=\mathbb{1}$ (see [Chr03, Chapter 6]). Therefore, the uncertainty relation shows **Theorem 1** in the special case when $a=b=0$ with $|\alpha\rangle = f(x)$. === References [Bru] N. G. De Bruijn, [[http://alexandria.tue.nl/repository/freearticles/597595.pdf|Uncertainty Principles in Fourier Analysis]] [Wil00] B. Williamson, [[http://users.cecs.anu.edu.au/~williams/uncertainty.pdf|The Uncertainty Principle]] [Chr03] J. G. Christensen, [[http://jens.math.tufts.edu/texts/masterth.pdf|Uncertainty Principles - Master’s Thesis in Mathematics]]