Wave packets in the quantum theory of collisions

Two methodological difficulties of the quantum theory of collisions are considered. The first is the undesirable interference of the incident and scattered waves in the stationary approach to scattering. The second concerns the nonstationary approach to the theory of collisions of the type a + b → c + d. In order to calculate the cross section one uses the matrix element 〈cd|S|ab〉 of the S-matrix. The element is proportional to δ-function expressing the energy conservation. The corresponding probability |〈cd|S|ab〉|² contains δ², which is mathematically meaningless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the problems are resolved using wave packets of incident particles. The text was submitted by the author in English.     

An analytic representation of quantum field theory

The connection between a space of quadratically integrable functions of real variablesq and a Hilbert space of analytic functions of complex variablesz established byBargmann is used to introduce quantised field operators for which the -functions of the commutation relations inq-space are replaced by analytic kernel functions inz-space, and a reference to distributions can be avoided.Bargmann's representation is first somewhat modified, so that the derivative terms in the field equations retain their form in the new representation. Local interaction terms inq-space obtain a non-local appearance inz-space. The transition to a 4-dimensional formulation inz-space has to resort to a Euclidean metric. The equations can be derived directly by starting from an action integral inz-space, and applying a variational calculus in which variations are restricted to analytic functions. Explicit analytic expressions are given for free field propagators.     

Regular and chaotic quantum dynamics in atom-diatom reactive collisions

A new microirreversible 3D theory of quantum multichannel scattering in the three-body system is developed. The quantum approach is constructed on the generating trajectory tubes which allow taking into account influence of classical nonintegrability of the dynamical quantum system. When the volume of classical chaos in phase space is larger than the quantum cell in the corresponding quantum system, quantum chaos is generated. The probability of quantum transitions is constructed for this case. The collinear collision of the Li + (FH) → (LiF) + H system is used for numerical illustration of a system generating quantum (wave) chaos. The text was submitted by the authors in English.     

Generator coordinate representation of the time independent mean field theory of collisions

We show how matrix elements of theT-matrix can be easily estimated on a basis of Slater determinants, with a mean field approximation. Linear superpositions of these Slater determinants then generate plane waves, or distorted (Coulomb) waves. This provides physical matrix elements ofT.     

Stationary perturbation theory in the probability representation of quantum mechanics

In the probability representation of quantum mechanics, the eigenvalue problems in Hilbert space appear as *-genvalue equations. We show the possibility of employing the nondegenerate stationary perturbation method in the probability representation of quantum mechanics. The perturbed eigentomograms and the eigenvalues of energy are shown to be computed ab initio in terms of tomographic symbols of the operators involved.     

On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

Category of trees in representation theory of quantum algebras

New applications of categorical methods are connected with new additional structures on categories. One of such structures in representation theory of quantum algebras, the category of Kuznetsov-Smorodinsky-Vilenkin-Smirnov (KSVS) trees, is constructed, whose objects are finite rooted KSVS trees and morphisms generated by the transition from a KSVS tree to another one.     

Application of the method of semiclassical representation to quantum scattering theory

It is shown how the method of semiclassical representation [1] can be used to considerably simplify problems in the quantum theory of scattering. In a study of “feedback” — a mutual change in the quantum state and center-of-mass trajectory — the method enables one to separate variables: to consider the Schrödinger equation for the quantum state when the classical motion of the centers of mass is given, and to write a potential for the classial problem that does not depend on the quantum indices.     

On inelastic reactive collisions in kinetic theory of chemically reacting gas mixtures

A kinetic theory for a simple reversible reaction-characterized by a binary mixture of ideal gases whose constituents denoted by A and B undergo a reaction of the type A+A?B+B-is developed by considering the reactive collisions as inelastic ones. The geometry of the collision is taken into account in the line-of-centers differential cross section by allowing that a chemical reaction may occur only when the energy of the relative velocity in the direction of the line which joins the centers of the molecules at collision is larger than the activation energy. It is shown that the restitution coefficients: (i) depend explicitly on the reaction heat and on the relative translational energy in the direction of the line which joins the centers of the molecules during an inelastic collision; (ii) vanish when the reaction heat is zero; (iii) are larger or smaller than one depending on the direction of the reaction and on the sign of the reaction heat. First approximations to the distribution functions are determined from the system of Boltzmann equations for the last stage of a chemical reaction. It is shown that the deviations from the Maxwellian distribution functions and the production terms of the particle number densities: (i) vanish when the reaction heat is zero provided that the affinity is close to zero and (ii) are negative or positive depending on the sign of the reaction heat and on the direction of the reaction.     

Langevin representation of Coulomb collisions for bi-Maxwellian plasmas

Langevin model corresponding to the Fokker–Planck equation for bi-Maxwellian particle distribution functions is developed. Rosenbluth potentials and their derivatives are derived in the form of triple hypergeometric functions. The Langevin model is tested in the case of relaxation of the proton temperature anisotropy and implemented into the hybrid expanding box model. First results of this code are presented and discussed.     

On the representation theorem for quantum logic

The main assumption of Varadarajan's version of Piron's representation theorem for quantum logic, stating that the lattice under any finite element of the logic is a geometry of finite rank, is eliminated by means of more plausible assumptions, concerning the property of symmetry of the transition probability between pure states. It is also proved, that the quantum logic with symmetric transition probability is irreducible iff it is completely irreducible.     

Detailed symmetry selection rules for reactive collisions

A simple general method for obtaining selection rules for the ro-vibronic states of reactant and product molecules connected in a reactive collision is discussed. Neglecting only the coupling with nuclear spin, rather stringent restrictions are found to occur in systems involving three and more identical nuclei. Several radical and ion molecule reactions of current interest are used to illustrate this finding. Even more restrictive selection rules are found when assuming weak coupling (‘ incomplete exchange of identical nuclei ’) in the intermediate reaction complex. These include the well-known selection rules for inelastic collisions of molecules with several identical nuclei, but less trivial examples with chemical reaction are also presented. The symmetry corrections and nuclear spin statistics in statistical theories of scattering (including the prior distributions for the information-theoretic approach) are derived. Further applications are discussed briefly.     

A new quantum representation for canonical gravity and SU(2) Yang-Mills theory

Starting from Rovelli-Smolin's infinite-dimensional graded Poisson-bracket algebra of loop variables, we propose a new way of constructing a corresponding quantum representation. After eliminating certain quadratic constraints, we “integrate” an infinite-dimensional subalgebra of loop variables, using a formal group law expansion. With the help of techniques from the representation theory of semidirect-product groups, we find an exact quantum representation of the full classical Poisson-bracket algebra of loop variables, without any higher-order correction terms. This opens new ways of tackling the quantum dynamics for both canonical gravity and Yang-Mills theory.     

Watson representation for the amplitude of lepton-pair production in nucleus-nucleus collisions

An expression for the amplitude of lepton pair production in nucleus-nucleus collisions is obtained in terms of the amplitudes of lepton-nucleus scattering. By using this representation an approximate compact expression for this amplitude valid to terms of the order (Zα)⁵ is obtained.     

Helicity representation for deep inelastic collisions of heavy ions

Quantum-mechanical cross-section for the inelastic collisions characterized by large values of the angular momenta is analysed. For the case of a planar mechanism of the reaction the approximation of the small helicity is drawn.     

Application of nonperturbative quantum field theoretic methods to hadron-nucleus collisions: Background theory

An exposition of the background theory necessary for understanding the application of nonperturbative QFT methods (LSZ reduction formalism) to hadron-nucleus collisions, for example the derivation of pi-nucleus (πn) Low and Chew Low equations, is given. The many channels and complex targets on the one hand, and the quantized field interactions on the other, introduce subtleties not well covered in particle theory or potential scattering literature. By specializing the πn Low Equation we derive a “pi-nucleon Low equation in the nuclear medium.” The second main goal of the paper is to compare this equation with Dover and Lemmer's analogous equation arrived at by graphical arguments. This requires especially making explicit the analyticity properties left tacit in their work. In is concluded that the two are essentially different.     

Almost-adiabatic collisions of quantum systems

Expressions are derived for corrections resulting from a small variation of the “slow” internal degrees of freedom during an almost-adiabatic quantum collision. First-order non-adiabatic contributions to the scattering wave function and scattering T-matrix element are derived making use of (a) Wronskian surface terms, and (b) converging factors. Advantages of method (a) are indicated. Modifications of the expressions necessary to include a possible projectile intrinsic spin are described. A one-dimensional model illustrating the range of validity of the adiabatic approximation is presented.     

Angular momentum polarization in molecular collisions : Classical and quantum theory for measurements using resonance fluorescence

Inelastic or reactive collisions typically produce an anisotropic distribution of rotational angular momentum. An explicit and general treatment is given for the intensity and polarization of resonance fluorescence from molecules produced in such processes. Both classical and quantum results are expressed in terms of bipolar harmonics and state multipoles formed from linear combinations of density matrix elements. The treatment provides an inversion procedure for determining moments of the rotational angular momentum distribution ; twelve independent moments can be obtained. The combinations of angular momentum operators involved are even in eight of these moments and odd in four, with respect to reflection in a plane containing the initial and final relative velocity vectors. Measurements of the even moments require linearly polarized excitation and fluorescence, whereas measurements of the odd moments require circularly polarized excitation. The requisite experimental geometry and other practical aspects are discussed. In the three appendices are discussed the classical limits of transition intensities, a density matrix treatment of atom-rigid-rotor collisions, including analysis of state multipole symmetries ; and the coupling coefficients for parallel angular momenta.