COULOMB SCATTERING IN NON-COMMUTATIVE QUANTUM MECHANICS

Abstract

Recently we formulated the Coulomb problem in a rotationally invariant NC configuration space specified by NC coordinates x_i, i = 1, 2, 3, satisfying commutation relations [x_i, x_j ] = 2iλε_ijkx_k (λ being our NC parameter). We found that the problem is exactly solvable: first we gave an exact simple formula for the energies of the negative bound states E^λ_n < 0 (n being the principal quantum number), and later we found the full solution of the NC Coulomb problem. In this paper we present an exact calculation of the NC Coulomb scattering matrix S^λ_j (E) in the j-th partial wave. As the calculations are exact, we can recognize remarkable non-perturbative aspects of the model: 1) energy cut-off — the scattering is restricted to the energy interval 0 < E < E_crit = 2/λ²; 2) the presence of two sets of poles of the S-matrix in the complex energy plane — as expected, the poles at negative energy E^I_λn = E^λ_n for the Coulomb attractive potential, and the poles at ultra-high energies E^II_λn = E_crit − E^λ_n for the Coulomb repulsive potential. The poles at ultra-high energies disappear in the commutative limit λ→0.