Adiabatic hyperspherical study of three-particle systems

Paolo Barletta, Alejandro Kievsky

Javier Botero

In an effort to unravel the dependence of the dynamics of three-bound-particle systems on the relative masses of the particles, an adiabatic treatment in hyperspherical coordinates is used to study the ground state and some doubly excited states of Ps('-) (positronium negative ion) and other three-particle systems, including H('-) and M('-) (muonium negative ion). Two different methods for the solution of the adiabatic eigenvalue equation are presented; one uses a "prediagonalized" basis of hyperspherical harmonics, and the other uses an asymptotic basis which consists of hydrogenic functions for the positronium. Potential curves for Ps('-) and M('-) in the ('1)S('e), ('1)P('o), ('3)P('e), ('3)P('o) and ('1)D('e) states and for H('-) in the ('1)P('o) states are presented. A method for calculating the probability density as a function of the hyperspherical coordinates (R,(alpha),(theta)(,12)) is introduced and used for the P states of Ps('-). This helps to understand the different symmetries and to distinguish among them. A remarkable linear dependence of the binding energy of the ground state and of the doubly excited states on the reduced mass of the atom, discovered in this study, is presented and discussed.

Zheng Zhen

The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.

Jia Wang, Written Jia Wang, Chris H. Greene +9 more

Hyperspherical coordinates provide a systematic way of describing three-body systems. Solving three-body Schrödinger equations in an adiabatic hyperspherical representation is the focus of this thesis. An essentially exact solution can be found numerically by including nonadiabatic couplings using either a slow variable discretization or a traditional adiabatic method. Two different types of three-body systems are investigated: (1) rovibrational states of the triatomic hydrogen ion H+3 and (2) ultracold collisions of three identical bosons.

S. P. Merkur′ev