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On the 1D Coulomb Klein–Gordon equation

2007; Institute of Physics; Volume: 40; Issue: 5 Linguagem: Inglês

10.1088/1751-8113/40/5/010

1751-8121

G. Barton,

Advanced Mathematical Physics Problems

For a single particle of mass m experiencing the potential −α/|x|, the 1D Klein–Gordon equation is mathematically underdefined even when α ≪ 1: unique solutions require some physically motivated prescription for handling the singularity at the origin. The procedure appropriate in most cases is to soften the singularity by means of a cutoff. Here we study the bound states of spin-zero particles in the potential −α/(|x| + R), extending the nonrelativistic results of Loudon (1959 Am. J. Phys. 27 649) to allow for relativistic effects, which become appreciable and eventually dominant for small enough mR: they are totally different from conclusions based hitherto on mathematically simple-seeming matching conditions on the wavefunction at x = 0. For realizable R, all relativistic effects remain very small; but with mR decreasing to order α2 the ground-state energy E decreases through zero, and soon after that mR reaches a finite critical value below which E becomes complex, signalling a breakdown of the single-particle theory. At this critical point of the curve E(mR) the Klein–Gordon norm changes sign: the curve has a lower branch describing a bound antiparticle state, with positive energy −E, which exists for mR between the critical and some higher value where E reaches −m. Though apparently unanticipated in this context, similar scenarios are in fact familiar for strong short-range potentials (1D or 3D), and also for strong 3D Coulomb potentials with α of order unity.