Hierarchy of local minimum solutions of Heisenberg's uncertainty principle

Local minimum solutions to Heisenberg's uncertainty principle are studied in detail. Of particular concern is obtaining an optimal procedure to squeeze the uncertainty in one canonical operator while introducing, in a well-defined mathematical sense, the least possible increase in the uncertainty of the relevant conjugate operator. The result is a hierarchy of 'minimum uncertainty (μ-) wavelets', which are intimately associated with the 'Hermite distributed approximating functionals' [D. K. Hoffman and D. J. Kouri, Phys. Rev. Lett. 85, 5263 (2000)]. The μ wavelets can be used with any quantum state to squeeze the uncertainty of a particular observable. Several specific examples are given and some possible applications are discussed