Uncertainty product for a subensemble of particles

The Heisenberg uncertainty principle derives from an assumption that all physically permissible location valuesx are included within the ensemble of observations. By contrast, we consider here a case where observation is made over a finite subinterval of the permissiblex values. The resulting uncertainty product forx andk _x (wave momentum) is then inconsistent with the Heisenberg principle, as might be expected.The author gratefully acknowledges the encouragement and insights into this subject provided by Professor Mark J. Beran of Tel Aviv University.