Journal of Mathematical Physics

Harmonic functions and angular momentum for anyon systems

Jan Milewski, Tadeusz Lulek

Published Mar 1, 2001• 2 citations

Abstract

A method of construction of harmonic functions for the system of anyons on a plane is given. Spectra of angular momentum are derived. Anyonic statistics implies a restriction of the range of angular momentum which is stronger in comparison to the bosonic and fermion case.

DOI: 10.1063/1.1339220

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Journal of Mathematical Physics

Harmonic functions and angular momentum for anyon systems

Jan Milewski, Tadeusz Lulek

Published Mar 1, 2001• 2 citations

Publisher

Abstract

DOI: 10.1063/1.1339220

Related papers

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Geoffrey Hunter, Mohsen Emami-Razavi

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Four anyons in a harmonic well

M. Sporre, J. J. M. Verbaarschot, Ismail Zahed A

Physical review. B, Condensed matterSep 1, 199228 citations

The Fermion Spherical harmonics [

Y_\ell^{m}(θ,ϕ)

for half-odd-integer

\ell

and

m

- presented in a previous paper] are shown to have the same eigenfunction properties as the well-known Boson Spherical Harmonics [

Y_\ell^{m}(θ,ϕ)

for integer

\ell

and

m

]. The Fermion functions are shown to differ from the Boson functions in so far as the ladder operators

M_+

(

M_-

) that ascend (descend) the sequence of harmonics over the values of

m

for a given value of

\ell

, do not produce the expected result {\em in just one case}: when the value of

m

changes from

\pm{1/2}

\mp{1/2}

; i.e. when

m

changes sign; in all other cases the ladder operators produce the usually expected result including anihilation when a ladder operator attempts to take

m

outside the range:

-\ell\le m\le +\ell

. The unexpected result in the one case does not invalidate this scalar coordinate representation of spin angular momentum, because the eigenfunction property is essential for a valid quantum mechanical state, whereas ladder operators relating states with different eigenvalues are not essential, and are in fact known only for a few physical systems; that this coordinate representation of spin angular momentum differs from the abstract theory of angular momentum in this respect, is simply an interesting curiosity. This new representation of spin angular momentum is expected to find application in the theoretical description of physical systems and experiments in which the spin-angular momentum (and associated magnetic moment) of a particle is oriented in space, since the orientation is specifiable by the spherical polar angles,

θ

and

ϕ

We study a many-body system consisting of

N

identical anyons in an external harmonic-oscillator potential. A partial set of exact solutions is obtained by using Jacobi coordinates. For completing the full investigation, we employ a perturbative method to compute those spectra, which have not been analytically obtained. Near the boson limit, we point out that the requirement of the boson being hard core is needed for performing the perturbative calculations, and the first-order

N

-anyon ground-state energies are explicitly presented by using a proper regulation procedure. Up to third perturbative order from the fermion limit, the three-anyon ground-state energy corrections are explicitly presented. Finally we compute the first-order three-anyon ground-state wave function near the fermion limit.