Tighter Uncertainty and Reverse Uncertainty Relations

We prove a few novel state-dependent uncertainty relations for product as well the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Roberson-Schr"odinger uncertainty relation and other ones existing in the current literature. Also, we derive state dependent upper bound to the sum and the product of variances using the reverse Cauchy-Schwarz inequality and the Dunkl-Williams inequality. Our results suggest that not only we cannot prepare quantum states for which two incompatible observables can have sharp values, but also we have both, lower and upper limits on the variances of quantum mechanical observables at a fundamental level.