Nonrelativistic Motion of Particles in Strongly BoundSStates

Abstract

We consider, for some different kinds of potentials, the mathematical question: Can particles in the ground state be strongly bound and still move nonrelativistically? As is well known, this is possible for a properly chosen square well. We show, however, that this is not possible for a Yukawa potential, nor for a purely attractive superposition of Yukawa potentials, nor for a Coulomb potential. For an exponential potential this is possible; however, the criterion for nonrelativistic motion of two particles of mass $M$ in an exponential potential of range $m^{-1\mathrm{}}$ is ${\left(m{M}^{-1\mathrm{}}\right)}^{\frac{1}{3}}\ll 1$, rather than $m{M}^{-1\mathrm{}}\ll 1$ as might be expected naively. The arguments used are elementary, and rely on exact solutions to soluble problems.