Hyperspherical Asymptotics of a System of Four Charged Particles

Abstract

We present a detailed analysis of the charged four-body system in hyperspherical coordinates in the large hyperradial limit. In powers of R −1 for any masses and charges, the adiabatic Hamiltonian is expanded to third order in the dimer-dimer limit and to first order in the particle-trimer limit.

Low-energy scattering properties of ground-state and excited-state positronium collisions

“…where R is the hyperradial coordinate, Ω represents the hyperangular coordinates, µ is the hyperspherical reduced mass, T Ω is the hyperangular kinetic energy, and V int (R, Ω) is the interaction potential. For relatively straightforward definitions of R, µ, T Ω and V int (R, Ω), see [25]. The Hamiltonian is quasi-separable in the hyperradial and hyperangular coordinates, which leads to the standard ansatz for the solutions of Eq.…”

“…[|n 1 l 1 (13) |n 2 l 2 (24) |ρ 13ρ24ρ3 + (−1) l3+S++S− |n 1 l 1 (24) |n 2 l 2 (13) |ρ 24ρ13ρ3 + (−1) S+ |n 1 l 1 (23) |n 2 l 2 (14) |ρ 23ρ14ρ3 where the n's and l's represent the principal and angular momentum quantum numbers for the two bound positronium atoms, |ρ 13ρ24ρ3 represents a coupled spherical harmonic (see [25,27]), and S 13 and S 24 are the spins of the positronium atoms. The states given by Eqs.…”

“…The system being studied is the collision between two positronium atoms. This system consists of two electrons and two positrons and the Hamiltonian of such a system can be written in Cartesian coordinates as [24,25],…”

“…The spin-orbit interaction between the two Ps atoms in s-orbitals is zero for total orbital angular momentum quantum number L = 0, considered in this study. For a description of the diverse ways this system can fragment, the four-body Hamiltonian is conveniently represented in hyperspherical coordinates [24][25][26]:…”

“…where R is the hyperradial coordinate, Ω represents the hyperangular coordinates, µ is the hyperspherical reduced mass, T Ω is the hyperangular kinetic energy, and V int (R, Ω) is the interaction potential. For relatively straightforward definitions of R, µ, T Ω and V int (R, Ω), see [25].…”