Abstract
The left-hand side of the Heisenberg variance uncertainty relation of the spatial and momentum probability densities assigned to a wave function is the product of the corresponding standard deviations. This uncertainty relation can only be applied if both standard deviations are finite. We show that except of the well-known Cauchy wave packets there are a number of other interesting wave functions whose spatial standard deviations diverge. Taking these functions as eigenfunctions of Hamiltonians with potentials we get a set of potentials which have a form of single or double potential wells. If the spatial standard deviation of a wave function is infinite then it is appropriate to take for the position and momentum uncertainties their information entropies and use the corresponding entropic uncertainty relation.
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