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

## Abstract

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 Schroedinger 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 {sup 4}He{sub 3}, isotopomers of the H{sub 3}{sup +} ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.

- Authors:

- Department of Physics, University of Central Florida, Orlando, Florida 32816 (United States)
- (France)
- Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex (France)

- Publication Date:

- OSTI Identifier:
- 20786706

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 73; Journal Issue: 1; Other Information: DOI: 10.1103/PhysRevA.73.012702; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; BOUND STATE; COORDINATES; HELIUM; HYDROGEN IONS 3 PLUS; ISOTOPE EFFECTS; MAPPING; MATHEMATICAL SOLUTIONS; MOLECULES; NUCLEONS; SCHROEDINGER EQUATION; THREE-BODY PROBLEM; VIBRATIONAL STATES; WAVE FUNCTIONS

### Citation Formats

```
Kokoouline, Viatcheslav, Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex, and Masnou-Seeuws, Francoise.
```*Calculation of loosely bound levels for three-body quantum systems using hyperspherical coordinates with a mapping procedure*. United States: N. p., 2006.
Web. doi:10.1103/PHYSREVA.73.0.

```
Kokoouline, Viatcheslav, Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex, & Masnou-Seeuws, Francoise.
```*Calculation of loosely bound levels for three-body quantum systems using hyperspherical coordinates with a mapping procedure*. United States. doi:10.1103/PHYSREVA.73.0.

```
Kokoouline, Viatcheslav, Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex, and Masnou-Seeuws, Francoise. Sun .
"Calculation of loosely bound levels for three-body quantum systems using hyperspherical coordinates with a mapping procedure". United States.
doi:10.1103/PHYSREVA.73.0.
```

```
@article{osti_20786706,
```

title = {Calculation of loosely bound levels for three-body quantum systems using hyperspherical coordinates with a mapping procedure},

author = {Kokoouline, Viatcheslav and Laboratoire Aime Cotton, CNRS, Bat. 505, Campus d'Orsay, 91405 Orsay Cedex and Masnou-Seeuws, Francoise},

abstractNote = {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 Schroedinger 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 {sup 4}He{sub 3}, isotopomers of the H{sub 3}{sup +} ion, and finally a model three-body problem involving three nucleons. For all these systems, we show that the suggested method provides accurate results.},

doi = {10.1103/PHYSREVA.73.0},

journal = {Physical Review. A},

number = 1,

volume = 73,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2006},

month = {Sun Jan 15 00:00:00 EST 2006}

}