Abstract: Taking the square of normalized modulus for the wave function of quantum mechanics as the density function of probability, the Heisenberg inequality with positive lower bound for a product of the variance of displacement function and velocity function is described. It is proved by the derivative property of Fourier transform, Plancherel lemma and Cauchy-Schwarz inequality. Hardy uncertainty principle shows that an integrable function and its Fourier transform can not rapid attenuation at the same time. The Gauss function with a negative power is to achieve the best way for the Hardy uncertainty. We apply the Phragmen-Lindelof theorem (unbounded region on the maximum modulus principle) and the argument of complex analysis to prove the Hardy uncertainty principle. In addition, we also provide some generalizations of Hardy inequality, such as the Morgan inequality and the Beurling uncertainty principle.

Key words:  Fourier Transform, Heisenberg uncertainty, Hardy uncertainty, Morgan inequality, Beurling uncertainty principle