  Modifications of Heisenberg's uncertainty relations have been proposed in the
literature which imply a minimum position uncertainty. We study the low energy
effects of the new physics responsible for this by examining the consequent
change in the quantum mechanical commutation relations involving position and
momenta. In particular, the modifications to the spectrum of the hydrogen atom
can be naturally interpreted as a varying (with energy) fine structure
constant. From the data on the energy levels we attempt to constrain the scale
of the new physics and find that it must be close to or larger than the weak
scale. Experiments in the near future are expected to change this bound by at
least an additional order of magnitude.
