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

Open Physics

Four particle harmonic-oscillator transformation brackets and their simplifications for special values of the mass-ratio parameters

Saulius Mickevičius, Darius Germanas, R.K. Kalinauskas

Published Jan 7, 2012• 2 citations
Publisher

Abstract

Abstract We present a new independent scheme of SO(3) group transformations suitable for the N particle system, composed of N − 1 and 1 particle subsystems, where N − 1 particles have their own intrinsic clusterization. The simple expressions for corresponding four-particle harmonic-oscillator transformation brackets are presented, as well as their simplifications for the special values of mass ratio parameters d = 0, d → ∞ and d 1 = 0, d 1 → ∞.

DOI
10.2478/s11534-011-0115-3

Related papers

Nonspurious harmonic oscillator states in single particle coordinates

Akiva Novoselsky, Jacob Katriel

A procedure for the construction of nonspurious harmonic oscillator wave functions with arbitrary permutational symmetry has recently been proposed. The resulting wave functions are expressed in terms of normalized Jacobi coordinates, and involve a new type of harmonic oscillator coefficients of fractional parentage. A simple algorithm to transform these states from the Jacobi coordinates to the single particle coordinates is presented. This is a generalization to an arbitrary number of particles of the harmonic oscillator transformation from center-of-mass and relative coordinates to single particle coordinates.

Journal of Mathematical Physics•May 1, 1990•1 citation

Transformation bracket for 2D hyperspherical harmonics and its applications to few-anyon problems

W. Y. Ruan, Bao Cheng-guang

The transformation bracket of hyperspherical harmonics with different sets of hyperspherical coordinates as arguments has been derived for an arbitrary number of particles with arbitrary masses in two space dimensions. The solution of the four-anyon problem is given as an example to illustrate its applications.

Journal of Mathematical Physics•Nov 1, 1997•13 citations

Common generating functions of complete harmonic oscillator wave functions and transformation brackets in D dimensions

Lorea Chaos‐Cador, E. Ley‐Koo

Abstract The separability of the Schrödinger equation for harmonic oscillators in D dimensions and in different coordinate systems (Cartesian, circular, spherical) makes possible the construction of common generating functions for the complete harmonic oscillator wave functions in the corresponding dimensions and coordinates. Explicit forms of such generating functions and their series expansions are presented, and one of their applications is illustrated through the evaluation of transformation brackets. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum Chemistry•Nov 6, 2003•19 citations

Independent Bases on the Spatial Wavefunction of Three-Identical-Particle Systems

Xianhui Liu

A simple and intuitional prescription for constructing the independent bases on the spatial wavefunctions of three-identical-particle systems is proposed. The basis functions for are given in the explicit expressions based on the harmonic oscillator model. The advantage of this prescription is that the operation is directly proceeded in the center of mass system and can be programmed for the arbitrary N .

Communications in Theoretical Physics•Jul 30, 1996•3 citations

Five-body Moshinsky brackets

Shuyuan Xiao, Xueli Mu, Zhixuan Deng +1 more

In variational calculations with harmonic oscillator wavefunctions as trial bases, the transformation coefficients that relate harmonic oscillator wavefunctions in two different sets of internal coordinates are convenient to the evaluation of some matrix elements. Here, we present the explicit expression of these transformation coefficients for five-body systems. These transformation coefficients can be collected in a matrix according to the quantum number N of harmonic oscillator shell and can be programmed for arbitrary N.

Journal of Mathematical Physics•Apr 1, 2015•5 citations

Configuration-Space Approach to the Four-Particle Problem

Werner Zickendraht

The configuration-space approach to the three-particle problem is generalized to the case of four particles. Special coordinates are defined which have simple symmetry properties with respect to the exchange of identical particles. The construction of a suitable orthogonal system is discussed. Some of these functions are given explicitly. It is pointed out that the use of this orthogonal system leads to a considerable simplification for a large number of four-particle problems, namely, the approximate reduction of the Schrödinger equation to a finite system of coupled differential equations for functions that depend on one variable only.

Journal of Mathematical Physics•Jan 1, 1969•55 citations
Abstract We present a new independent scheme of SO(3) group transformations suitable for the N particle system, composed of N − 1 and 1 particle subsystems, where N − 1 particles have their own intrinsic clusterization. The simple expressions for corresponding four-particle harmonic-oscillator transformation brackets are presented, as well as their simplifications for the special values of mass ratio parameters d = 0, d → ∞ and d 1 = 0, d 1 → ∞.
A procedure for the construction of nonspurious harmonic oscillator wave functions with arbitrary permutational symmetry has recently been proposed. The resulting wave functions are expressed in terms of normalized Jacobi coordinates, and involve a new type of harmonic oscillator coefficients of fractional parentage. A simple algorithm to transform these states from the Jacobi coordinates to the single particle coordinates is presented. This is a generalization to an arbitrary number of particles of the harmonic oscillator transformation from center-of-mass and relative coordinates to single particle coordinates.
Abstract The separability of the Schrödinger equation for harmonic oscillators in D dimensions and in different coordinate systems (Cartesian, circular, spherical) makes possible the construction of common generating functions for the complete harmonic oscillator wave functions in the corresponding dimensions and coordinates. Explicit forms of such generating functions and their series expansions are presented, and one of their applications is illustrated through the evaluation of transformation brackets. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
In variational calculations with harmonic oscillator wavefunctions as trial bases, the transformation coefficients that relate harmonic oscillator wavefunctions in two different sets of internal coordinates are convenient to the evaluation of some matrix elements. Here, we present the explicit expression of these transformation coefficients for five-body systems. These transformation coefficients can be collected in a matrix according to the quantum number N of harmonic oscillator shell and can be programmed for arbitrary N.
The configuration-space approach to the three-particle problem is generalized to the case of four particles. Special coordinates are defined which have simple symmetry properties with respect to the exchange of identical particles. The construction of a suitable orthogonal system is discussed. Some of these functions are given explicitly. It is pointed out that the use of this orthogonal system leads to a considerable simplification for a large number of four-particle problems, namely, the approximate reduction of the Schrödinger equation to a finite system of coupled differential equations for functions that depend on one variable only.