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Physical Review A

Loosely bound states of three particles

Zheng Zhen, J. H. Macek

Published Aug 1, 1988• 31 citations
Publisher

Abstract

The adiabatic hyperspherical technique is applied to a three-body system interacting via a short-range potential. Analytical expressions for the effective hyper-radial potential have been obtained without explicit use of the two-body potential. Possible loosely bound states of the system have been analyzed quantitatively and the result confirms the prediction by V. Efimov (Yad. Fiz. 10, 107 (1969) [Sov. J. Nucl. Phys. 10, 62 (1970)]). At the same time, some new characteristics of the three-body system are obtained.

DOI
10.1103/physreva.38.1193

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The adiabatic hyperspherical technique is applied to a three-body system interacting via a short-range potential. Analytical expressions for the effective hyper-radial potential have been obtained without explicit use of the two-body potential. Possible loosely bound states of the system have been analyzed quantitatively and the result confirms the prediction by V. Efimov (Yad. Fiz. 10, 107 (1969) [Sov. J. Nucl. Phys. 10, 62 (1970)]). At the same time, some new characteristics of the three-body system are obtained.
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.
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 developed a method to calculate positions and widths of three-body resonances. The method combines the hyperspherical adiabatic approach, slow variable discretization method [O. I. Tolstikhin et al., J. Phys. B 29, L389 (1996)], and a complex absorbing potential. The method can be used to obtain resonances having short-range or long-range wave functions. In particular, we have applied the method to obtain very shallow three-body Efimov resonances for a model system [E. Nielsen et al., Phys. Rev. A 66, 012705 (2002)].
The adiabatic hyperspherical method is used to investigate the spectrum of the helium trimer and to explore effects on the spectrum of varying the helium masses as well as the interaction strength. When a realistic helium dimer potential is adopted, only two isotopic combinations are observed to have three-body bound states: 4He3^{4}\mathrm{He}_{3}4He3​ and 4He23^{4}\mathrm{He}_{2}^{3}4He23​He. By artificially adjusting the dimer potential, we also determine the range of interaction strengths for which halo states can result. The existence of Efimov states for both physical and unphysical systems is also examined. \textcopyright{} 1996 American Institute of Physics.
We have used the hyperspherical adiabatic representation to describe the system of three identical bosons in a spin stretched state interacting through an attractive 1∕r1∕r1∕r potential. A proposal has been forwarded to enable the experimental realization of such a system in cold trapped atoms using extremely off-resonant laser fields [Phys. Rev. Lett., 84, 5687 (2000)]. We have obtained the effective potentials, channel functions, and nonadiabatic couplings for this gravitylike interaction, allowing us to calculate the ground-state energy with accuracy that substantially improves upon previous results. We have similarly calculated the energies for the first four 0+{0}^{+}0+ excited states. These results show that the simple adiabatic hyperspherical approximation offers an accurate description for such a system. Further, since the effective potentials have a long-range 1∕R1∕R1∕R attraction, the properties of the excited states follow from the well-known results for Rydberg states. We have also analyzed the possible geometries and vibrational modes for this system.