A generalized Cauchy-Schwarz inequality is derived and applied to uncertainty relation in quantum mechanics. We see a modification in the uncertainty relation and minimum uncertainty wave packet.
The famous Cauchy-Schwarz inequality is deduced from the fundamental unequal relationship in different spaces in the nature.Considering the linear Hermitian property of the state physical operators on the Hilbert space,the uncertainty relation is derived by applying the inequality to the quantum system of microcosm.The significance of the uncertainty relation which is the objective report of the wave-particle duality of the microscopic particle and the basis for keeping the stability of atoms and even the whole world is discussed.