On Long-Range Potentials

Marc Baus

Per Chr. Hemmer, Joel L. Lebowitz

Mohsen Razavy

Wen-Du Li, Wu-Sheng Dai

Inspired by general relativity, we suggest an approach for long-range potential scattering. In scattering theory, there is a general theory for short-range potential scattering, but there is no general theory for long-range potential scattering. This is because the scattering boundary conditions for all short-range potentials are the same, but for different long-range potentials, they are different. In this paper, by introducing tortoise coordinates, we convert long-range potential scattering to short-range potential scattering. This allows us to deal with long-range potential scattering as short-range potential scattering. An explicit expression of the scattering wave function for long-range potential scattering is presented, in which the scattering wave function is represented by the tortoise coordinate and the scattering phase shift. We show that the long-range potential scattering wave function is just the short-range potential scattering wave function with a replacement of a common coordinate by a tortoise coordinate. The approach applies not only to scattering but also applies to bound states. Furthermore, in terms of tortoise coordinates, we suggest a classification scheme for potentials. We also discuss the duality between tortoise coordinates.

H. Wang, P. L. Gould, William C. Stwalley

This paper reports on a comprehensive study of the long-range interaction of the K39(4s)+39K(4p) asymptotic system. We present a detailed discussion of the R-dependent angular momentum couplings and correlation between the Hund’s case (a) and case (c) molecular states. Analytical expressions for the 16 adiabatic Hund’s case (c) long-range potential curves are derived including the higher order dispersion forces and the effects of retardation. Experimentally, six Hund’s case (c) long-range molecular states (0u+, 1g, and 0g− dissociating to the 4 2S1/2+4 2P3/2 asymptote and 0u+, 1g, and 0g− to the 4 2S1/2+4 2P1/2 limit) are observed with rovibrational resolution by photoassociative spectroscopy of ultracold K39 atoms in a high density magneto-optical trap (MOT). Among the six observed long-range states, the upper 0g− “pure long-range” state has negligible short-range chemical exchange contributions and the measured molecular binding energies (v=0–26) are used to precisely determine the long-range potential constants of the 4s+4p interaction. We determine: C3Π=8.436(14) a.u., C3Σ=16.872(28) a.u., C6Π=6272(94) a.u., and C6Σ=9365(141) a.u.. Molecular constants for the three special pure long-range states, the 0g− and 1u (dissociating to the 4 2P3/2 limit and with potential minimum) and the 1u (dissociating to the 4 2P1/2 and with potential maximum), are reported. The internal consistency of the theoretical model used in this work is confirmed by the excellent agreement between the long-range potential curve of the 1g state obtained in present work (from the 0g− state) and the long-range portion of the RKR potential curve of the 1 1Πg state previously determined by conventional molecular spectroscopy. The radiative lifetime of the K 4p state derived from the dipole–dipole interaction constant C3Π is also in excellent agreement with a recent fast-beam measurement.