Hans de Vries
Not surprisingly, The average radius of the electron's orbit at Z=137 becomesLocalization of relativistic particles in strong attractive potentials may lead to some peculiar (though not necessarily unphysical) effects. For example, imagine an atomic nucleus with charge Z and corresponding attractive Coulomb potential for electrons. The lowest state available for electrons has energy E. If you increase the nuclear charge, the energy E would increase, and at some point (I think this happens at Z > 137) this energy becomes larger than the energy [itex] 2mc^2 [/itex] required to create one electron-positron pair. As far as I know, such conditions have not been achieved experimentally. However, the current understanding about what happens next is this: The electron-positron pair, indeed, gets created. The electron gets attached to the nucleus, thus effectively reducing its charge to Z-1, and the positron is repelled to infinity.
Note that such pair creation requires a strong external potential. However, if are you simply localizing a free particle in empty space, then there is no reason to expect creation of pairs.
the Compton radius. Compressing a wavefunction into a space smaller as the
Compton radius requires an energy (from external fields or scattering momentum)
which is larger as the rest mass energy and therefor leads to particle production.