We study the uncertainty relation in the product form of variances and obtain some new uncertainty relations with weight, which are shown to be tighter than those derived from the Cauchy–Schwarz inequality.
We prove a few state-dependent uncertainty relations for the product as well as the sum of variances of two incompatible observables. These uncertainty relations are shown to be tighter than the Robertson-Schr\"odinger uncertainty relation and other ones existing in the current literature. Also, we derive a 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 can we not prepare quantum states for which two incompatible observables can have sharp values, but also we have both the lower and the upper limits on the variances of quantum mechanical observables at a fundamental level.
A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.