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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Analysis of N -anyon spectrum in hyperspherical coordinates

W. Y. Ruan

Published Nov 15, 1997• 0 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.56.6258

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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.
An exact closed form of solution to the hyperradial Schrödinger equation is constructed for any general case comprising any hypercentral power and inverse-power potential. The hypercentral potential depends only on the hyperradius, which itself is a function of Jacobi relative coordinates that are functions of particle positions ( r 1 , r 2 ,…, r N ). This article is mainly devoted to the dernonstrat of the fact that any ψ of the form ψ = powerseries×exp(polynomial) = [ f ( x )exp ( g ( x ))] is potentially a solution of the Schrödinger equation, where the polynomial g( x ) is an ansatz depending on the interaction potential.
Within the hyperspherical framework, the solution of the time-independent Schr\"odinger equation for a nnn-particle system is divided into two steps: the solution of a Schr\"odinger-type equation in the hyperangular degrees of freedom and the solution of a set of coupled Schr\"odinger-type hyperradial equations. The solutions to the former provide effective potentials and coupling matrix elements that enter into the latter set of equations. This paper develops a theoretical framework to determine the effective potentials, as well as the associated coupling matrix elements, for few-body systems with finite angular momentum L=1L=1L=1 and negative and positive parity $\ensuremath{\Pi}$. The hyperangular channel functions are expanded in terms of explicitly correlated Gaussian basis functions, and relatively compact expressions for the matrix elements are derived. The developed formalism is applicable to any nnn; however, for $n\ensuremath{\ge}6$, the computational demands are likely beyond present-day computational capabilities. A number of calculations relevant to cold-atom physics are presented, demonstrating that the developed approach provides a computationally efficient means to solving four-body bound and scattering problems with finite angular momentum on powerful desktop computers. Details regarding the implementation are discussed.
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.
A derivation of the quantum mechanical wave equation for the three body problem expressed in hyperspherical coordinates is presented. The coordinates, due to Smith and Whitten and later modified by Johnson [B. R. Johnson, J. Chem. Phys. 73, 5051 (1980)], are used in this study. The analysis is presented from a point of view that emphasizes the role of the three-dimensional configuration space that is associated with these coordinates. Three types of motion are analyzed: full three dimensional; motion restricted to a plane surface; and zero angular momentum motion.