Energy Spectrum of Anyon in the Coulomb Field (original) (raw)
Classical dynamics of anyons and the quantum spectrum
Physical Review A, 1993
In this paper we show that (a) all the known exact solutions of the problem of Nanyons in oscillator potential precisely arise from the collective degrees of freedom, (b) the system is pseudo-integrable ala Richens and Berry. We conclude that the exact solutions are trivial thermodynamically as well as dynamically.
Approximate ground state of a confined Coulomb anyon gas in an external magnetic field
Physical Review B, 2003
We derive an analytic, albeit approximate, expression for the ground state energy of N Coulomb interacting anyons with fractional statistics ν, 0 ≤ |ν| ≤ 1, confined in a two-dimensional well (with characteristic frequency ω 0) and subjected to an external magnetic field (with cyclotron frequency ω c). We apply a variational principle combined with a regularization procedure which consists of fitting a cutoff parameter to existing exact analytical results in the noninteracting case, and to numerical calculations for electrons in quantum dots in the interacting case. The resulting expression depends upon parameters of the system |ν|, N, ω 0 , r 0 , a B and ω c , where r 0 represents a characteristic unit length and a B the Bohr radius. Validity of the result is critically assessed by comparison with exact, approximate, and numerical results from the literature.
A semi-classical approximation to the three-anyon spectrum
1992
Using a semi-classical approximation whmh g~ves exact results for two anyons, the spectrum of three anyons bound m a harmomc oscdlator potentml ts obtained. The energy levels are approximately hnear as functions of the stattsUcal parameter and fall into two classes, with slopes _+ I and _+ 3 respectively.
Two anyons with Coulomb interaction in a magnetic field
Physics Letters B, 1992
We have computed numerically the energy spectrum of two anyons with Coulomb (1/r) interaction in a uniform magnetic field. Both non-singular and singular energy eigenfunctions are considered. We solve a second order difference equation with boundary conditions by a new, simple and efficient numerical method.
Quantum solution for the one-dimensional Coulomb problem
Physical Review A, 2011
The one-dimensional hydrogen atom has been a much studied system with a wide range of applications. Since the pioneering work of Loudon [R. Loudon, Am. J. Phys. 27, 649 (1959).], a number of different features related to the nature of the eigenfunctions have been found. However, many of the claims made throughout the years in this regard are not correct-such as the existence of only odd eigenstates or of an infinite binding-energy ground state. We explicitly show that the one-dimensional hydrogen atom does not admit a ground state of infinite binding energy and that the one-dimensional Coulomb potential is not its own supersymmetric partner. Furthermore, we argue that at the root of many such false claims lies the omission of a superselection rule that effectively separates the right side from the left side of the singularity of the Coulomb potential.
Energy Spectrum of the Dirac Oscillator in the Coulomb Field
2006 International Conference on Mathematical Methods in Electromagnetic Theory
The spectrum of the Dirac oscillator perturbed by the Coulomb potential is considered. The Regge trajectories for its bound states are obtained with the method of ħ-expansion. It is shown that the split of the degenerate energy levels of the Dirac oscillator in the Coulomb field is approximately linear in the coupling constant.
Ground-state properties of one-dimensional anyon gases
Physical Review A, 2008
We investigate the ground state of the one-dimensional interacting anyonic system based on the exact Bethe ansatz solution for arbitrary coupling constant (0 ≤ c ≤ ∞) and statistics parameter (0 ≤ κ ≤ π). It is shown that the density of state in quasi-momentum k space and the ground state energy are determined by the renormalized coupling constant c ′ . The effect induced by the statistics parameter κ exhibits in the momentum distribution in two aspects: Besides the effect of renormalized coupling, the anyonic statistics results in the nonsymmetric momentum distribution when the statistics parameter κ deviates from 0 (Bose statistics) and π (Fermi statistics) for any coupling constant c. The momentum distribution evolves from a Bose distribution to a Fermi one as κ varies from 0 to π. The asymmetric momentum distribution comes from the contribution of the imaginary part of the non-diagonal element of reduced density matrix, which is an odd function of κ. The peak at positive momentum will shift to negative momentum if κ is negative.
Classical and Quantum Mechanics of Anyons
New Research in Quantum Physics, 2003
We review aspects of classical and quantum mechanics of many anyons confined in an oscillator potential. The quantum mechanics of many anyons is complicated due to the occurrence of multivalued wavefunctions. Nevertheless there exists, for arbitrary number of anyons, a subset of exact solutions which may be interpreted as the breathing modes or equivalently collective modes of the full system. Choosing the three-anyon system as an example, we also discuss the anatomy of the so called "missing" states which are in fact known numerically and are set apart from the known exact states by their nonlinear dependence on the statistical parameter in the spectrum.
Anyon spectra and the third virial coefficient
Nuclear Physics B, 1993
The third virial coefficient of a gas of anyons in a harmonic potential is studied using two different approximations to obtain the three-anyon spectrum. A semiclassical (weak-coupling) approximation which provides us with the overall behavior of the spectrum leads to a divergent third virial coefficient. As a second approximation the low-lying three-anyon spectrum at the fermion point is constructed up to second order in the statistical parameter. The analogue spectrum at the boson point follows by symmetry. In this case the results are consistent with a finite third virial coefficient. The semiclassical approximation is also applied to the four-anyon spectrum and can be readily generalized to many anyons. Within their range of validity, both types of approximate spectra are in complete agreement with previously obtained exact numerical spectra.
Anyons, monopole and Coulomb problem
1997
The monopole systems with hidden symmetry of the two-dimensional Coulomb problem are considered. One of them, the "charge-charged magnetic vortex" ("charge-Z 2-dyon) with a half-spin, is constructed by reducing the quantum circular oscillator with respect to the action of the parity operator. The other two systems are constructed by reduction from the two-dimensional complex space. The first system is a particle on the sphere in the presence of the exterior constant magnetic field (generated by Dirac's monopole located in its center). This system is dual to the massless (3+1)-dimensional particle with fixed energy. The second system represents the particle on the pseudosphere in the presence of exterior magnetic field and is dual to the massive relativistic anyon.
The one-dimensional hydrogen atom revisited
Canadian Journal of Physics, 2006
The one dimensional Schrödinger hydrogen atom is an interesting mathematical and physical problem to study bound states, eigenfunctions and quantum degeneracy issues. This 1D physical system gave rise to some intriguing controversy over more than four decades. Presently, still no definite consensus seems to have been reached. We reanalyzed this apparently controversial problem, approaching it from a Fourier transform representation method combined with some fundamental (basic) ideas found in self-adjoint extensions of symmetric operators. In disagreement with some previous claims, we found that the complete Balmer energy spectrum is obtained together with an odd parity set of eigenfunctions. Closed form solutions in both coordinate and momentum spaces were obtained. No twofold degeneracy was observed as predicted by the degeneracy theorem in one dimension, though it does not necessarily have to hold for potentials with singularities. No ground state with infinite energy exists since the corresponding eigenfunction does not satisfy the Schrödinger equation at the origin. PACS number(s): 03.65.Ge 03.65.-w *
Anyons, monopoles, and Coulomb problem
Physics of Atomic Nuclei
The monopole systems with hidden symmetry of the two-dimensional Coulomb problem are considered. One of them, the "charge-charged magnetic vortex" ("charge-Z 2-dyon) with a half-spin, is constructed by reducing the quantum circular oscillator with respect to the action of the parity operator. The other two systems are constructed by reduction from the two-dimensional complex space. The first system is a particle on the sphere in the presence of the exterior constant magnetic field (generated by Dirac's monopole located in its center). This system is dual to the massless (3+1)-dimensional particle with fixed energy. The second system represents the particle on the pseudosphere in the presence of exterior magnetic field and is dual to the massive relativistic anyon.
Linear-optical dynamics of one-dimensional anyons
2020
We study the dynamics of bosonic and fermionic anyons defined on a one-dimensional lattice, under the effect of Hamiltonians quadratic in creation and annihilation operators, commonly referred to as linear optics. These anyonic models are obtained from deformations of the standard bosonic or fermionic commutation relations via the introduction of a non-trivial exchange phase between different lattice sites. We study the effects of the anyonic exchange phase on the usual bosonic and fermionic bunching behaviors. We show how to exploit the inherent Aharonov-Bohm effect exhibited by these particles to build a deterministic, entangling two-qubit gate and prove quantum computational universality in these systems. We define coherent states for bosonic anyons and study their behavior under two-mode linear-optical devices. In particular we prove that, for a particular value of the exchange factor, an anyonic mirror can generate cat states, an important resource in quantum information proces...
Semi-Classical Quantization of the Many-Anyon System
1992
We discuss the problem of N anyons in harmonic well, and derive the semi-classical energy spectrum as an exactly solvable limit of the many-anyon Hamiltonian. The relevance of our result to the solution of the anyon-gas model is discussed.
The one-dimensional hydrogen atom in momentum representation
European Journal of Physics, 1987
A momentum representation treatment of the one-dimensional hydrogen atom with Coulomb interaction is presented. We obtain the energy levels and bound-state eigenfunctions for the problem and discuss the general properties of the solution. The energy spectrum is discrete and given by the Balmer formula. We show that, despite the symmetry of the potential. it is impossible to find eigenfunctions with definite parity for the system.
On the one-dimensional Coulomb problem
Physics Letters A, 2009
We analyse the one-dimensional Coulomb problem (1DCP) pointing out some mistaken beliefs on it. We show that no eigenstates of even or odd parity can represent states of the system. The 1DCP exhibits a sort of spontaneous breaking of parity. We also show that a superselection rule operates in the system. Such rule explains some of its peculiarities. We build the superpotential associated to the 1DCP.
The European Physical Journal D, 2011
In this paper we present an exact analytic solution of the Schrödinger equation both in the discrete and continuous spectra for the combination of a 2D Coulomb potential and the Aharonov-Bohm flux. We analyze the influence of the Aharonov-Bohm flux on the energy spectrum of such a system and show that its presence leads to the broadening of the electron density in the bound states with the given value of the principal quantum number. We have shown that the scattering phase shift, which determines the S-matrix, can be represented as a sum of the Aharonov-Bohm scattering phase, first obtained by Henneberger, and a "modified" 2D Coulomb phase. We have noticed, that the Aharonov-Bohm scattering phase has a full analogy with the "quantum defect" for such a system. We have shown also, that the presence of the Aharonov-Bohm flux affects the radiation spectrum of the electron in this case, and this fact is demonstrated by calculations of the corresponding oscillator strengths. The explicit analytic expression for the scattering cross section on such a system is found in the frame of the eikonal approach. Obtained formula contains the two exact limiting cases, namely, the "pure" 2D Coulomb scattering as well as the "pure" Aharonov-Bohm effect. The mutual influence of a 2D Coulomb potential and the Aharonov-Bohm flux is also discussed.