M. Kibler

Statistical mechanics of a q-deformed boson gas

Abstract This paper is concerned with statistical properties of a noninteracting q -boson system. The accent is put on D = 2 and D = 3 dimensions. We first give the mean occupation numbers for the energy levels. We subsequently calculate expressions for the total energy, the specific heat, the entropy and the equation of state. The results differ from the standard ones for ordinary bosons (but converge to them as q → 1). In particular, we find that Bose condensation may occur in D = 2 dimensions when q ≠1.

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, the unitary group and the Pauli group

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and is analyzed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.

Experimental and calculated energy levels of Sm2+:BaClF

New investigations on the optical properties of divalent samarium in barium chlorofluoride single crystals have been performed. As a result, an energy level diagram including 42 Stark components of the 7FJ and 5DJ multiplets of the 4f6 configuration has been deduced from the excitation and fluorescence spectra. Parameter optimizations involving combined spin–orbit and crystal‐field interactions have been achieved from the experimental Stark levels of the 7F ground term. An interpretation of the fitted crystal‐field parameters has been developed in the framework of an additive ligand‐field model. The irreducible tensor method applied to the chain O3⊆C∞v⊆C4v⊆C2v is used throughout this paper.

Motion of a particle in a Coulomb plus Aharonov-Bohm potential

Abstract The motion of a particle in a Coulomb plus Aharonov-Bohm potential is investigated from a classical and a quantum mechanical viewpoint. The quantum bound states are derived by using the KS transformation. The grouping of certain levels is explained via the introduction of an SU (2) dynamical algebra. All classical finite ttrajectories are found to be periodic.

Statistical mechanics of qp-bosons in D dimensions

Abstract This paper is concerned with statistical properties of a gas of qp -bosons without interaction. Some thermodynamical functions for such a system in D dimensions are derived. Bose-Einstein condensation is discussed in terms of the parameters q and p . Finally, the second-order correlation function of a gas of photons is calculated.

Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems

We introduce a one-parameter-generalized oscillator algebra (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter κ. We define an (Hamiltonian) operator associated with and examine the degeneracies of its spectrum. For the finite (when κ < 0) and the infinite (when κ ≥ 0) representations of , we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated-generalized oscillator algebra , where s denotes the truncation order. We construct two types of temporally stable states for (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of ). Two applications are considered in this paper. The first concerns physical realizations of and in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Poschl–Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.

On the q-analogue of the hydrogen atom

The discrete spectrum of a q-analogue of the hydrogen atom is obtained from a deformation of the Pauli equations. As an alternative, the spectrum is derived from a deformation of the four-dimensional oscillator arising in the application of the Kustaanheimo-Stiefel transformation to the hydrogen atom. A model of the 2s-2p Dirac shift is proposed in the context of q-deformations.

The k -Fermions as Objects Interpolating Between Fermions and Bosons

In the recent years, the theory of deformations, mainly in the spirit of quantum groups and quantum algebras, has been the subject of considerable interest in statistical physics. More precisely, deformed oscillator algebras have proved to be useful in parasiatistics(connected to irreducible representations, of dimensions greater than 1, of the symmetric group), in anyonic statistics(connected to the braid group) that concerns only particles in (one or) two dimensions, and in q-deformed statisticsthat may concern particles in arbitrary dimensions. In particular, the q-deformed statistics deal with: (i) q-bosons (which are bosons obeying a q-deformed Bose-Einstein distribution), (ii) q-fermions (which are fermions obeying a q-deformed Fermi-Dirac distribution), and (iii) quons (with qsuch that q k = 1, where k∈ ℕ \ {0,1}) which are objects, refered to as k-fermions in this work, interpolating between fermions (corresponding to k= 2) and bosons (corresponding to k→ ∞).

Ionic and paramagnetic energy levels algebra

Abstract The algebra of coupling and recoupling coefficients relative to a (finite) subgroup G of the special unitary group in two dimensions SU (2) is developed in this paper. Following the work of Schonfeld on the cubic field, and the work of Flato on the trigonal and tetragonal fields, the f symbols are defined from the Clebsch-Gordan coefficients (j1j2m1m2 ∥ j1j2j3m3) expressed in the jΓγ scheme where Γγ stands for the γ-row of the irreducible representation Γ of G. The definition of the irreducible tensorial sets under the group G, leads to the Wigner-Eckart theorem. Properties of the f coefficients and of the related f are obtained from the properties of the 3 - j Wigners symbols for SU (2). Utilizing the technique developed by Racah, the recoupling coefficients W and X are calculated as function of f . Numerical values of the fs for the cubic and tetragonal groups (electronic configurations d2, f2) are given in the appendix.

Fractional supersymmetric quantum mechanics as a set of replicas of ordinary supersymmetric quantum mechanics

A connection between fractional supersymmetric quantum mechanics and ordinary supersymmetric quantum mechanics is established in this Letter.