以波函数的规范化模平方积分作为概率密度函数，我们给出了在L2意义下的位移函数与速度函数的方差乘积有正下界的海森伯不等式；并用傅里叶变换的微分性质、Plancherel等式以及Cauchy-Schwarz不等式作了证明. 另外，Hardy不确定性原理表明可积函数和它的傅里叶变换不能同时迅速衰减，其最优的衰减方式是取高斯函数形式达到等式；基于Phragmen-Lindelof定理（无界区域上的最大模原理），给出了Hardy不确定性的复分析方法证明；最后我们给出了推广的Morgan不等式和Beurling不确定性. 

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.