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.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.

Jan 1, 19880 citations

Hyperspherical Approach to Quantal Three-body Theory

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.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.

Jan 1, 20125 citations

Adiabatic hyperspherical study of triatomic helium systems

Hiroya Suno, Brett D. Esry

The $^{4}\mathrm{He}_{3}$ system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic $^{4}\mathrm{He}$ system. The bound state energies of the $^{4}\mathrm{He}$ trimer are computed as well as the $^{4}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two $^{4}\mathrm{He}$ and one $^{3}\mathrm{He}$ atoms, and compute the spectrum of the isotopic trimer $^{4}\mathrm{He}_{2}^{3}\mathrm{He}$ , the $^{3}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.The $^{4}\mathrm{He}_{3}$ system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic $^{4}\mathrm{He}$ system. The bound state energies of the $^{4}\mathrm{He}$ trimer are computed as well as the $^{4}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two $^{4}\mathrm{He}$ and one $^{3}\mathrm{He}$ atoms, and compute the spectrum of the isotopic trimer $^{4}\mathrm{He}_{2}^{3}\mathrm{He}$, the $^{3}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.

arXiv:0809.5111Physical Review ADec 1, 200883 citations

Three-body Scattering in One Dimension: a Geometric Version of an Adiabatic Approach

Pavel Kurasov, Melnikov, Yuri

Jan 1, 19891 citation

The

^{4}\mathrm{He}_{3}

system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic

^{4}\mathrm{He}

system. The bound state energies of the

^{4}\mathrm{He}

trimer are computed as well as the

^{4}\mathrm{He}+^{4}\mathrm{He}_{2}

elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two

^{4}\mathrm{He}

and one

^{3}\mathrm{He}

atoms, and compute the spectrum of the isotopic trimer

^{4}\mathrm{He}_{2}^{3}\mathrm{He}

, the

^{3}\mathrm{He}+^{4}\mathrm{He}_{2}

elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.