Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Hyperspherical Approach to Quantal Three-body Theory

Jia Wang, Written Jia Wang, Chris H. Greene, Prof John Bohn, Eric Cornell, Chris H. Greene, Michal Tarana, Chen Zhang, John Papaioannou, Nathan Morrison, Stephen Ragole, Victor Colussi

Published Jan 1, 2012• 5 citations
OpenAlex

Abstract

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.

Related papers

Calculation of loosely bound levels for three-body quantum systems using hyperspherical coordinates with a mapping procedure

Viatcheslav Kokoouline, Françoise Masnou-Seeuws

In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer $^{4}\mathrm{He}_{3}$, isotopomers of the $\mathrm{H}_{3}{}^{+}$ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.

Physical Review A•Jan 10, 2006•38 citations

Green’s functions and the adiabatic hyperspherical method

Seth T. Rittenhouse, Nirav P. Mehta, Chris H. Greene

We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in $d$ dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three particles with $s$-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semianalytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom loss due to three-body recombination for a three-component Fermi gas of $^{6}\mathrm{Li}$ atoms is presented.

arXiv:1005.2911•Physical Review A•Aug 12, 2010•40 citations

Hyperspherical calculations of ultralow-energy collisions in Coulomb three-body systems

Yueming Zhou, Shinichi Watanabe, Oleg I. Tolstikhin +1 more

Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the $R$-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.

Physical Review A•Sep 28, 2015•6 citations

Adiabatic hyperspherical representation for the three-body problem in two dimensions

José Paulo D'Incao, Brett D. Esry

We explore the three-body problem in two dimensions using the adiabatic hyperspherical representation. We develop the main equations in terms of democratic hyperangular coordinates and determine several symmetry properties and boundary conditions for both interacting and noninteracting solutions. From the analysis of the three-body effective potentials, we determine the threshold laws for low-energy three-body recombination, collision-induced dissociation, as well as inelastic atom-diatom collisions in two dimensions. Our results show that the hyperspherical representation can offer a simple and conceptually clear physical picture for three-body process in two dimensions which is also suitable for calculations using finite-range two-body interactions supporting a number of bound states.

arXiv:1404.0324•Physical Review A•Oct 16, 2014•17 citations

Search for weakly bound states of the He-He-Be triatomic system

Yong Li, Hongwei Song, Qingdong Gou +2 more

We search for the existence of weakly bound He-He-Be molecules. The He-He-Be molecule is treated as a three-body system. By using hyperspherical coordinates, the Schr\"odinger equation for the triatomic system is solved in the adiabatic approximation. A bound state is found for each of the $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{9}\mathrm{Be}$, $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}^{9}\mathrm{Be}$, and $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}^{9}\mathrm{Be}$ trimers.

Physical Review A•Feb 5, 2009•8 citations

Asymptotic Adiabatic Hyperspherical Potential Curves for Three Body Systems.

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.

•Jan 1, 1988•0 citations
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.
In view of modelization of experiments involving cold atoms and molecules, we develop a method that allows us to calculate weakly bound levels of triatomic molecules. The method combines (1) the hyperspherical coordinates to describe interparticle motion in the three-body system, (2) the solution of the Schr\"odinger equation in two steps: determination of adiabatic states for a fixed hyper-radius and then solution of a set of coupled hyper-radial equations using the slow variable representation of Tolstikhin et al. [J. Phys. B: At. Mol. Opt. Phys. 29, L389 (1996)], (3) and a mapping procedure that reduces considerably the number of basis functions needed to represent wave functions of weakly bound levels. We apply the method to the three different systems: the helium trimer 4He3^{4}\mathrm{He}_{3}4He3​, isotopomers of the H3+\mathrm{H}_{3}{}^{+}H3​+ ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.
We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in ddd dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three particles with sss-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semianalytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom loss due to three-body recombination for a three-component Fermi gas of 6Li^{6}\mathrm{Li}6Li atoms is presented.
Quantum mechanical calculations of ultralow-energy collision of Coulomb three-body systems in the hyperspherical elliptic coordinates are presented. The nonadiabatic coupling between the hyperradius and hyperangular variables is treated with the slow-variable discretization method in combination with the RRR-matrix propagation technique. For scattering state calculations, the two-dimensional matching procedure using Gailitis's method [M. Gailitis, J. Phys. B 9, 843 (1976); C. Noble and R. Nesbet, Comput. Phys. Commun. 33, 399 (1984)] is implemented to determine the boundary conditions between the internal and the asymptotic wave functions. This method is proved to be very efficient and gives very accurate results. Taking advantage of this method, we accurately calculate the scattering phase shifts and the scattering lengths of Coulomb three-body systems with mass ratio varying over several orders of magnitudes. We observed jumps of the scattering length from $\ensuremath{-}\ensuremath{\infty}$ to $\ensuremath{\infty}$ at certain mass ratios and monotonic decreases between two jumps. These are closely related to the binding energy of the highest bound state through Levinson's theorem [N. Levinson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 25, 9 (1949)]. Our calculations provide a comprehensive perspective to the scattering length from the variation of mass ratio of Coulomb three-body systems.
We explore the three-body problem in two dimensions using the adiabatic hyperspherical representation. We develop the main equations in terms of democratic hyperangular coordinates and determine several symmetry properties and boundary conditions for both interacting and noninteracting solutions. From the analysis of the three-body effective potentials, we determine the threshold laws for low-energy three-body recombination, collision-induced dissociation, as well as inelastic atom-diatom collisions in two dimensions. Our results show that the hyperspherical representation can offer a simple and conceptually clear physical picture for three-body process in two dimensions which is also suitable for calculations using finite-range two-body interactions supporting a number of bound states.
We search for the existence of weakly bound He-He-Be molecules. The He-He-Be molecule is treated as a three-body system. By using hyperspherical coordinates, the Schr\"odinger equation for the triatomic system is solved in the adiabatic approximation. A bound state is found for each of the $^{3}\mathrm{He}\text{\ensuremath{-}}^{3}\mathrm{He}\text{\ensuremath{-}}^{9}\mathrm{Be}$, $^{3}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}^{9}\mathrm{Be}$, and $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}\text{\ensuremath{-}}^{9}\mathrm{Be}$ trimers.
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.