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

Journal of Mathematical Physics

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

W. Y. Ruan, Bao Cheng-guang

Published Nov 1, 1997• 13 citations
Publisher

Abstract

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.
DOI
10.1063/1.532156

Related papers

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

Numerical Solution of the Four-Anyon Problem

Ruan Wenying

By making use of the Talmi–Moshinsky transformation bracket, the Schrodinger equation for four anyons in configuration space has been solved variationally to obtain the eigenenergies and eigenfunctions. The low-lying states have been presented as functions of the statistical parameter. We illustrated how the bosonic states evolve continuously into the fermionic states.

Communications in Theoretical Physics•Jan 30, 1998•0 citations

Analysis of N -anyon spectrum in hyperspherical coordinates

W. Y. Ruan

The introduction of hyperspherical coordinates allows the separation of an $N$-anyon Schr\"odinger equation into a hyperradial and a hyperangular part in a harmonic well. While the hyperradial part is analytically solvable, we present a systematic procedure to solve the hyperangular equation. The inclusion of a perpendicular magnetic field and inverse-square-type particle-particle interaction does not complicate the problem. The interplay of exchange statistics and conventional particle-particle interaction is emphasized.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1997•0 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

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

Saulius Mickevičius, Darius Germanas, R.K. Kalinauskas

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 → ∞.

Open Physics•Jan 7, 2012•2 citations

Wave equations in momentum space

Maurice M. Lévy

Abstract The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Dec 7, 1950•56 citations
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
The introduction of hyperspherical coordinates allows the separation of an NNN-anyon Schr\"odinger equation into a hyperradial and a hyperangular part in a harmonic well. While the hyperradial part is analytically solvable, we present a systematic procedure to solve the hyperangular equation. The inclusion of a perpendicular magnetic field and inverse-square-type particle-particle interaction does not complicate the problem. The interplay of exchange statistics and conventional particle-particle interaction is emphasized.
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.
Abstract The non-relativistic and relativistic wave equations of a particle are solved in momentum space for different kinds of fields. It is assumed that the potential, when defined in co-ordinate space, is central; but the method applies also to more general forms of momentum space potentials. First, the general integral equations which correspond, in momentum space, to Schrödinger and Dirac equations are derived, and the angular variables separated. Then, a general method of solution of these integral equations is given, which lies mainly on a transformation into a four-dimensional hyperspace. These results are (relativistically and non-relativistically) applied rigorously to the Coulombian field (hydrogen atom) and to the scalar Yukawa potential. A more general case is further investigated. In this, the momentum space potential can be expanded in a series of powers of the modulus of the momentum (this last application involves a cut-off procedure). A mathematical appendix deals with some properties of the hyperspherical harmonics (especially the integral equation of these functions) and with the Fourier transform of some kinds of central potentials.