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Model of an unstable system with a Coulomb interaction distorted at short distances

Boris O Kerbikov

Published Nov 1, 1983• 0 citations
OpenAlex

Abstract

We present an exactly soluble model of an unstable system with nuclear and Coulomb interactions. The system considered consists of two pairs of particles capable of transforming into each other. The interaction in the heavy-particle channel consists of the Coulomb potential and a short-range separable potential. The off-diagonal interaction which is responsible for the transition between the two pairs of particles is also taken in a separable form. The particles in the decay channel are free. Analytic expressions are obtained for the T matrix of the pure nuclear and nuclear-Coulomb scattering lengths. The equation which describes the shift of the atomic S levels in the heavy-particle channel is analyzed.

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We present an exactly soluble model of an unstable system with nuclear and Coulomb interactions. The system considered consists of two pairs of particles capable of transforming into each other. The interaction in the heavy-particle channel consists of the Coulomb potential and a short-range separable potential. The off-diagonal interaction which is responsible for the transition between the two pairs of particles is also taken in a separable form. The particles in the decay channel are free. Analytic expressions are obtained for the T matrix of the pure nuclear and nuclear-Coulomb scattering lengths. The equation which describes the shift of the atomic S levels in the heavy-particle channel is analyzed.
We study charged particles in three dimensions interacting via a short-range potential in addition to the Coulomb potential. When the Bohr radius and the scattering length are much larger than the potential range, low-energy physics of the system becomes independent from details of the short-range potential. We develop the zero-range theory to describe such universal physics in terms of the Bohr radius and the scattering length by generalizing the Bethe-Peierls boundary condition, which is then applied to two charged particles to reveal their bound states and resonances. Infinite resonances are found for a repulsive Coulomb potential, one of which turns into a bound state with increasing inverse scattering length, whereas infinite bound states exist for an attractive Coulomb potential with no resonances at any scattering length. The zero-range theory is also applied to three equally charged particles at infinite scattering length under the variational Born-Oppenheimer approximation. We find that the effective potential between two heavy particles induced by a light particle is an inverse-square attraction at distances shorter than the Bohr radius, leading to infinite deep bound states, whereas shallow ones successively turn into resonances with increasing Coulomb repulsion.
An expression for the short-range effective interaction potential of “quasinuclei” is derived based on the “pure” Coulomb interaction model. This model represents the equilibrium Coulomb system (CS) of interacting electrons and identical nuclei using the adiabatic approximation for nuclei and an arbitrarily strong (in general) interaction for the electronic subsystem (degenerate or nondegenerate). Based on general properties of the Coulomb interaction, it is shown that the Fourier component of the effective pair potential between “quasinuclei” is discontinuous at the wave vector q = 0 in the case of the weak interaction between electronic and nuclear subsystems. This discontinuity is important for Bose-condensed systems such as Hell and rarefied alkali-metal gases at temperatures lower than that of the Bose condensation transition when a macroscopic number of quasiparticles with momentum q = 0 exist. It is shown that there can be a spectral gap for single-particle excitations, which disappears in the normal state.
We study a three-body system, formed by a light particle and two identical heavy dipoles, in two dimensions in the Born-Oppenheimer approximation. We present the analytic light-particle wave function resulting from an attractive zero-range potential between the light and each of the heavy particles. It expresses the large-distance universal properties which must be reproduced by all realistic short-range interactions. We calculate the three-body spectrum for zero heavy-heavy interaction as a function of light to heavy mass ratio. We discuss the relatively small deviations from Coulomb estimates and the degeneracies related to radial nodes and angular momentum quantum numbers. We include a repulsive dipole-dipole interaction and investigate the three-body solutions as functions of strength and dipole direction. Avoided crossings occur between levels localized in the emerging small and large-distance minima, respectively. The characteristic exchange of properties such as mean-square radii are calculated. Simulation of quantum information transfer is suggested. For large heavy-heavy-particle repulsion all bound states have disappeared into the continuum. The corresponding critical strength is inversely proportional to the square of the mass ratio, far from the linear dependence from the Landau criterion.
The asymptotics of total scattering cross sections for a three-body quantum system with two heavy particles are studied when the masses tend to infinity. It is assumed that the initial channel is a two-cluster channel such that a light particle and a heavy one form a bound state and the relative velocity of another heavy particle to the pair is fixed. It is shown that the limits exist and they can be expressed in terms of transition probabilities which appear in the scattering theory for the charge transfer models.
The state of an atomic-molecular system near its stability threshold with regard to the detachment of one of the particles is studied. The decay of the system upon a decrease in the charge of a binding particle, as well upon an increasasymmetry of masses of like-charged particles, is considered. A special variational principle that allows one to directly calculate the threshold state of the system without repeatedly calculating its energy for different values of masses and charges of particles is used. With the motion of all particles fully taken into account, the threshold states of two-electron atoms with different nuclear charges and of atomic-molecular systems corresponding to the attachment of a positron of variable mass to a neutral atom are calculated. On the basis of calculation of quantum-mechanical expectation values of the kinetic energy of particles and the potential energy of their interaction, the rearrangement of the wave function upon passage of the system through the decay threshold is examined. The threshold characteristics of a purely adiabatic system containing infinitely heavy particles are considered separately.
Moshinsky introduced an exactly soluble model of a two-electron atom consisting of two spin-12\frac{1}{2}21​ particles interacting via harmonic forces and moving in a harmonic-oscillator potential. Here, the exact ground-state density $\ensuremath{\rho}(r)$ is related to the (also analytically known) Hartree-Fock density ${\ensuremath{\rho}}_{\mathrm{HF}}(r).$ The generalization to the off-diagonal matrix $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is then effected, this being related to the idempotent ${\ensuremath{\gamma}}_{\mathrm{HF}}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})/2.$ This exact information on this ``model atom'' prompts us to propose an approximate form of $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ for the He-like ions, the H2{\mathrm{H}}_{2}H2​ molecule and, in general, all two-electron systems. $\ensuremath{\gamma}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})$ is constructed solely from the exact $\ensuremath{\rho}(\mathbf{r})$ and its Hartree-Fock counterpart. Some detailed treatment of the two-electron Hookean atom with spring constant k=1/4k=1/4k=1/4 (atomic units) is also presented.