Yu. F. Smirnov

Generalized Morse potential: Symmetry and satellite potentials

We study in detail the bound-state spectrum of the generalized Morse potential (GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schrodinger equation with the Laplace equation on the hyperboloid and the Schrodinger equation for the Poschl-Teller potential, we explain the exact solvability of the problem by an symmetry algebra, and obtain an explicit realization of the latter as . We prove that some of the generators connect among themselves wavefunctions belonging to different GMPs (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.

Tomographic representation of spin and quark states

We present a short review of the general principles of constructing tomograms of quantum states. We derive a general tomographic reconstruction formula for the quantum density operator of a system with a dynamical Lie group. In the reconstruction formula, the multiplicity of irreducible representation in Clebsch–Gordan decomposition is taken into account. Various approaches to spin tomography are discussed. An integral representation for the tomographic probability is found and a contraction of the spin tomogram to the photon-number tomography distribution is considered. The case of SU(3) tomography is discussed with the examples of quark states (related to the simplest triplet representations) and octet states.

Baryons in O(4) and the vibron model

The structure of the reported excitation spectra of the light unflavored baryons is described in terms of multi-spin valued Lorentz group representations of the so called Rarita-Schwinger (RS) type (K/2, K/2)* [(1/ 2,0)+ (0,1/2)] with K=1,3, and 5. We first motivate legitimacy of such pattern as fundamental fields as they emerge in the decomposition of triple fermion constructs into Lorentz representations. We then study the baryon realization of RS fields as composite systems by means of the quark version of the U(4) symmetric diatomic rovibron model. In using the U(4)/ O(4)/ O(3)/ O(2) reduction chain, we are able to reproduce quantum numbers and mass splittings of the above resonance assemblies. We present the essentials of the four dimensional angular momentum algebra and construct electromagnetic tensor operators. The predictive power of the model is illustrated by ratios of reduced probabilities concerning electric de-excitations of various resonances to the nucleon.

P-Matrix and J-Matrix Approaches: Coulomb Asymptotics in the Harmonic Oscillator Representation of Scattering Theory

Abstract The relation between the R- and P-matrix approaches and the harmonic oscillator representation of the quantum scattering theory (J-matrix method) is discussed. We construct a discrete analogue of the P-matrix that is shown to be equivalent to the usual P-matrix in the quasiclassical limit. A definition of the natural channel radius is introduced. As a result, it is shown to be possible to use a well-developed technique of R- and P-matrix theory for calculation of resonant states characteristics, scattering phase shifts, etc., in the approaches based on harmonic oscillator expansions, e.g., in nuclear shell-model calculations. The P-matrix is used also for formulation of the method of treating Coulomb asymptotics in the scattering theory in oscillator representation.

An exactly soluble nuclear model with SUq(3) supset SUq(2) supset SOq(2) symmetry

The q-deformed version of a two-dimensional toy interacting boson model (IBM) with the symmetry SUq(3) supset SUq(2) supset SOq(2) is constructed. Energy spectra and transition matrix elements are calculated, the latter being found to be much more sensitive to q-deformation than the former. Arguments in favour of the q-generalization of the full IBM are given.

q -analogue of the Krawtchouk and Meixner orthogonal polynomials

Abstract The comparative analysis of Krawtchouk polynomials on a uniform grid with Wigner D-functions for the SU(2) group is done. As a result the partnership between corresponding properties of the polynomials and D-functions is established giving the group-theoretical interpretation of the properties of the Krawtchouk polynomials. In order to extend such an analysis to the quantum groups SUq(2) and SUq(1, 1), q-analogues of Krawtchouk and Meixner polynomials of a discrete variable are studied in detail as solutions of the finite-difference equation of hypergeometric type on the nonuniform grid x(s) = q2s. However, on this grid there are two kinds of Krawtchouk and Meixner polynomials, characterized by different weight functions in the orthogonality relation. As a first step to such analysis the simpler polynomials of the first kind are considered. The total set of characteristics of these polynomials (orthogonality condition, normalization factor, recurrent relation, the explicit analytic expression, the Rodrigues formula, the difference derivative formula, various particular cases and values) is calculated.

Supermultiplets and relativistic problems: II. The Bhabha equation of arbitrary spin and its properties

In 1945 Bhabha was probably the first to discuss the problem of a free relativistic particle with arbitrary spin in terms of a single linear equation in the four-momentum vector p , but substituting the matrices of Dirac by other ones. He determined the latter by requiring that their appropriate Lorentz transformations lead to their formulation in terms of the generators of the O(5) group. His program was later extensively amplified by Krajcik, Nieto and others. We returned to this problem because we had an ab-initio procedure for deriving a Lorentz-invariant equation of arbitrary spin and furthermore could express the matrices appearing in them in terms of ordinary and what we called sign spins. Our procedure was similar to that of the ordinary and isotopic spin in nuclear physics that give rise to supermultiplets, hence the appearance of this word in the title. In the ordinary and sign spin formulation it is easy to transform our equation into one linear in both the p and some of the generators of O(5). We can then obtain the matrix representation of our equation for an irrep (n1n2/ of O(5) and find, through a similarity transformation, that for the irrep mentioned the particle satisfying our equation will have, in general, several spins and masses determined by a simple algorithm.

Supermultiplets and relativistic problems: I. The free particle with arbitrary spin in a magnetic field

Equations for relativistic particles for arbitrary spin have been of interest since Dirac original work for spin , but they involved either bothersome constraints or start with as many Dirac equations as are required to get the derived spin from its original value. We first show that it is possible to have just one equation involving s and s matrices that give possibilities up to for the spin. We then decompose the s and s into direct products of ordinary spin matrices and a new type of them that we call sign spin. The problem reduces then to one in terms of the generators of a U(4) group entirely similar to the one in the spin - isospin theory of nuclear physics and hence the name of supermultiplets in the title. Using then the techniques of the latter we discuss the problem of a free particle in a magnetic field for n = 1,2 and 3 or equivalently eigenvalues for spins 0,, 1 and , and the energies are given as solutions of elementary algebraic equations.

Supermultiplets and relativistic problems: III. The non-relativistic limit for a particle of arbitrary spin in an external field

In previous papers, with the same series title, an ab-initio procedure was developed for deriving a Lorentz invariant equation with arbitrary spins. This equation is linear in the four momentum , and its coefficients are matrices that can be expressed in terms of ordinary spin and what we called sign spin. In the present paper we consider this equation in an external field which implies just replacing by and discuss the cases when (, being the frequency of the oscillator), and , corresponding respectively to harmonic oscillator potential and a constant magnetic field. By using an appropriate complete set of states, with part of them characterized by the irreps of the chain of groups where the subscripts s and t respectively stand for the ordinary and sign spin, the problem can be formulated in a matrix representation whose diagonalization gives the energy spectrum. For simplicity we shall only consider the symmetric representation of for which s = t, and our interest is focussed on the case when the external field is weak, which gives the non-relativistic limit, and where a perturbation analysis can be applied. We show that the expected non-relativistic result can be obtained only when the sign spin projection takes its maximum value, i.e. when all individual states contributing to the final one correspond to positive energies. In the case of constant magnetic field, we obtain the gyromagnetic ratio consistent with other derivations.

Projection operators for Lie algebras, Superalgebras, and quantum algebras

The essence of the method of projection operators is considered using the suq(2) quantum algebra as an example. As an application the general analytical formula for the q—Clebsch‐Gordan coefficients is derived. The generalization of this method onto arbitrary Lie algebras (superalgebras and quantum algebras) is discussed shortly.