Michel Bawin

Regularization of the singular inverse square potential in quantum mechanics with a minimal length

We study the problem of the attractive inverse square potential in quantum mechanics with a generalized uncertainty relation. Using the momentum representation, we show that this potential is regular in this framework. We solve analytically the s-wave bound states equation in terms of Heuns functions. We discuss in detail the bound states spectrum for a specific form of the generalized uncertainty relation. The minimal length may be interpreted as characterizing the dimension of the system.

Neutral atom and a charged wire: From elastic scattering to absorption

We solve the problem of a neutral atom interacting with a charged wire, giving rise to an attractive 1/r^2 potential in two dimensions. We show how a suitable average over all possible self-adjoint extensions of the radial Schroedinger Hamiltonian eventually leads to the classical formula for absorption of the atom, a formula shown to be in agreement with a recent experiment.

Dirac-Foldy term and the electromagnetic polarizability of the neutron

We reconsider the Dirac-Foldy contribution {mu}{sup 2}/m to the neutron electric polarizability. Using a Dirac equation approach to neutron-nucleus scattering, we review the definitions of Compton continuum ({bar {alpha}}), classical static ({alpha}{sub E}{sup n}), and Schr{umlt o}dinger ({alpha}{sub Sch}) polarizabilities and discuss in some detail their relationship. The latter {alpha}{sub Sch} is the value of the neutron electric polarizability as obtained from an analysis using the Schr{umlt o}dinger equation. We find in particular {alpha}{sub Sch}={bar {alpha}}{minus}{mu}{sup 2}/m, where {mu} is the magnitude of the magnetic moment of a neutron of mass m. However, we argue that the static polarizability {alpha}{sub E}{sup n} is correctly defined in the rest frame of the particle, leading to the conclusion that twice the Dirac-Foldy contribution should be added to {alpha}{sub Sch} to obtain the static polarizability {alpha}{sub E}{sup n}. {copyright} {ital 1997} {ital The American Physical Society}

Neutron charge radius and the Dirac equation

We consider the Dirac equation for a finite-size neutron in an external electric field. We explicitly incorporate Dirac-Pauli form factors into the Dirac equation. After a non-relativistic reduction, the Darwin-Foldy term is cancelled by a contribution from the Dirac form factor, so that the only coefficient of the external field charge density is

Strong Coulomb Coupling in the Todorov Equation

e/6 r^2_{En}

Nucleon-nucleon scattering and the Kadyshevsky equation

, i. e. the root mean square radius associated with the electric Sachs form factor . Our result is similar to a recent result of Isgur, and reconciles two apparently conflicting viewpoints about the use of the Dirac equation for the description of nucleons.

Renormalization of the singular attractive 1/r4 potential

A positronium-like system with strong Coulomb coupling, considered in its pseudoscalar sector, is studied in the framework of relativistic quantum constraint dynamics with the Todorov choice for the potential. Case’s method of self-adjoint extension of singular potentials, which avoids explicit introduction of regularization cut-offs, is adopted. It is found that, as the coupling constant α increases, the bound state spectrum undergoes an abrupt change at the critical value α=αc=1/2. For α>αc, the mass spectrum displays, in addition to the existing states for α<αc, a new set of an infinite number of bound states concentrated in a narrow band starting at mass W=0; all the states have indefinitely oscillating wave functions near the origin. In the limit α→αc from above, the oscillations disappear and the narrow band of low-lying states shrinks to a single massless state with a mass gap with the rest of the spectrum. This state has the required properties to represent a Goldstone boson and to signal spontane...

Strong Coulomb Coupling in Relativistic Quantum Constraint Dynamics

Abstract We investigate a three-dimensional relativistic wave equation given by Kadyshevsky. This equation has been applied recently to nucleon-nucleon and to pion-nucleon scattering. We solve the Kadyshevsky equation for phenomenological nucleon-nucleon potentials for uncoupled s-waves. The resulting phase shifts differ from the Lippmann-Schwinger phase shifts by 7% or less for superpositions of two and three Yukawa-like potentials. Our relativistic corrections differ from those obtained by other authors with the Bethe-Salpeter equation in the ladder approximation and the Blankenbecler-Sugar-Logunov-Tavkhelidze equation. We also find that the coupling constants must be changed by about 4% in order to reproduce the non-relativistic phase shifts at low energy. On the one hand, our results indicate that the Kadyshevsky equation is a bad approximation to the Bethe-Salpeter equation for the nucleon-nucleon problem. Alternatively, since the Bethe-Salpeter equation in the ladder approximation is not fundamental, they show that relativistic corrections crucially depend on the relativistic model chosen.

Virtual pair creation in nuclear dirac phenomenology

We study the radial Schrodinger equation for a particle of mass m in the field of a singular attractive g 2 / r 4 potential with particular emphasis on the bound-states problem. Using the regularization method of Beane et al. fPhys. Rev. A 64, 042103 s2001dg, we solve analytically the corresponding “renormalization-group flow” equation. We find in agreement with previous studies that its solution exhibits a limit cycle behavior and has infinitely many branches. We show that a continuous choice for the solution corresponds to a given fixed number of bound states and to low-energy phase shifts that vary continuously with energy. We study in detail the connection between this regularization method and a conventional method modifying the short-range part of the potential with an infinitely repulsive hard core. We show that both methods yield bound-states results in close agreement even though the regularization method of Beane et al. does not include explicitly any new scale in the problem. We further illustrate the use of the regularization method in the computation of electron bound states in the field of neutral polarizable molecules without dipole moment. We find the binding energy of s-wave polarization bound electrons in the field of C60 molecules to be 17 meV for a scattering length corresponding to a hard-core radius of the size of the molecule radius s,3.37 A d. This result can be further compared with recent two-parameter fits using the Lennard-Jones potential yielding binding energies ranging from 3 to 25 meV.

A quasipotential approach to low energy relativistic corrections

We study, in the framework of relativistic quantum constraint dynamics, the bound state problem of two oppositely charged spin 1/2 particles, with masses m1 and m2, in mutual electromagnetic interaction. We search for the critical value of the coupling constant α for which the bound state energy reaches the lower continuum, thus indicating the instability of the heavier particle or of the strongly coupled QED vacuum in the equal mass case. Two different choices of the electromagnetic potential are considered, corresponding to different extensions of the substitution rule into the nonperturbative region of α: (i) the Todorov potential, already introduced in the quasipotential approach and used by Crater and Van Alstine in Constraint Dynamics; (ii) a second potential (potential II), characterized by a regular behavior at short distances. For the Todorov potential we find that for m2>m1 there is always a critical value αc of α, depending on m2/m1, for which instability occurs. In the equal mass case, instability is reached at αc=1/2 with a vanishing value of the cutoff radius, generally needed for this potential at short distances. For potential II, on the other hand, we find that instability occurs only for m2>2.16 m1.