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.
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.
Non-relativistic quantum chemical calculations of the particle mass, m±2, corresponding to the dissociation threshold in a range of Coulomb three-particle systems of the form m±1m±2m±3, are performed variationally using a series solution method with a Laguerre-based wavefunction. These masses are used to calculate an accurate stability boundary, i.e., the line that separates the stability domain from the instability domains, in a reciprocal mass fraction ternary diagram. This result is compared to a lower bound to the stability domain derived from symmetric systems and reveals the importance of the asymmetric (mass-symmetry breaking) terms in the Hamiltonian at dissociation. A functional fit to the stability boundary data provides a simple analytical expression for calculating the minimum mass of a third particle required for stable binding to a two-particle system, i.e., for predicting the bound state stability of any unit-charge three-particle system.
We present recent works [15,16] on the thermodynamic limit of quantum Coulomb systems. We provide a general method which allows to show the existence of the limit for many different systems. Ordinary matter is composed of electrons (negatively charged) and nuclei (positively charged) interacting via Coulomb forces. The potential between two particles of charges z and z ′ located at x and x ′ in R 3 is zz ′ |x − x ′ |. There are two difficulties which occur when trying to describe systems composed of electrons and nuclei. Both have to do with the physical problem Talk given by M.L. at QMath10, 10th Quantum Mathematics International Conference, Moeciu, Romania, September 10–15, 2007. c○2008 by the authors. This work may be reproduced, in its entirety, for non-commercial purposes. 2 of stability of quantum systems. The first is due to the singularity of 1/|x | at 0: it is necessary to explain why a particle will not rush to a particle of the opposite charge. One of the first major triumphs of the theory of quantum mechanics is the explanation it gives of the stability of the hydrogen atom (and the complete description of its spectrum) and of other microscopic quantum Coulomb systems, via the uncertainty principle. Stability means that the total energy of the considered system cannot be arbitrarily negative. If there was no such lower bound to the energy it would be possible in principle to extract an infinite amount of energy. One often refers to this kind of stability as stability of the first kind [19,20]. If we denote by E(N) the ground state energy of the system under consideration, for N particles stability of the first kind can be written E(N)> −∞. (1) In proving (1) for Coulomb systems, a major role is played by the uncertainty principle which for nonrelativistic systems is mathematically expressed by the critical Sobolev embedding H 1 (R 3) ֒ → L 6 (R 3). The latter allows to prove Kato’s inequality
The question of stability of a given quantum system made up of charged particles is of fundamental interest in atomic, molecular, and nuclear physics. In this work, the stability for the negatively charged positronium (Ps)-like ions or the three-body system ( Z e + , e − , e − ) with Yukawa potentials is studied using correlated exponential wavefunctions based on the Ritz variational method. We obtained the critical screening parameter μ C as a function of the continuously varied nuclear charge Z , the critical nuclear charge Z C as a function of the screening parameter μ , and the ionization energies in terms of the screening parameter μ and Z . The critical nuclear charge for the bare Coulomb system ( Z e + , e − , e − ) obtained using 700-term correlated exponential wavefunctions is in accord with the reported results. The ionization energy, μ C , and Z C for the Yukawa system ( Z e + , e − , e − ) exhibit interesting behaviors. The present study describes the possible nonexistence of Borromean binding as well as Efimov states. The possible existence of quasi-bound resonances states for the negatively charged screened Ps-like ions is briefly discussed.
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.
A concept of kinetic energy in quantum mechanics is analyzed. Kinetic energy is a non-zero positive value in many cases of bound states, when a wave function is a real-valued one and there are no visible motion and flux. This can be understood, using expansion of the wave function into Fourier integral, that is, on the basis of virtual plane waves. The ground state energy of a hydrogen atom is calculated in a special way, regarding explicitly all the terms of electrostatic and kinetic energies. The correct values of the ground state energy and the radius of decay are achieved only when the electrostatic energies of the electron and the proton (self-energies) are not taken into account. This proves again that self-action should be excluded in quantum mechanics. A model of a spherical ball with uniformly distributed charge of particles is considered. It is shown that for a neutral ball (with compensated electric charge) the electrostatic energy is a non-zero negative value in this model. This occurs because the self-energy of the constituting particles should be subtracted. So it shown that the energy of the electric field does not have to be of a positive value in any imaginable problem.
A Hookean model of a three-body problem for particles with arbitrary masses and charges where two of them interact with each other through a Coulomb potential and with the third through a harmonic potential is presented. It is shown that a condition relating the masses to the harmonic coupling constants must be satisfied in order to render this problem separable. A general exact analytic solution written in terms of the relative interparticle coordinates is given as well as general expressions for the total and binding energies of this three-body system. We apply these results to examine electronic, muonic, antiprotonic, and pionic families of non-Born-Oppenheimer Hookean systems. The first contains the atoms or atomic ions: ${\mathrm{Ps}}^{\ensuremath{-}}({e}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{H}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{D}}^{\ensuremath{-}}({d}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{T}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, $^{4}\mathrm{He}(h{e}^{+2}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, and the following molecular ions: ${\mathrm{Ps}}_{2}^{+}({e}^{\ensuremath{-}}{e}^{+}{e}^{+})$, ${\mathrm{H}}_{2}^{+}({e}^{\ensuremath{-}}{p}^{+}{p}^{+})$, $\mathrm{H}{\mathrm{D}}^{+}({e}^{\ensuremath{-}}{d}^{+}{p}^{+})$, ${\mathrm{HT}}^{+}({e}^{\ensuremath{-}}{t}^{+}{p}^{+})$, ${\mathrm{DT}}^{+}({e}^{\ensuremath{-}}{d}^{+}{t}^{+})$, ${\mathrm{D}}_{2}^{+}({e}^{\ensuremath{-}}{d}^{+}{d}^{+})$, ${\mathrm{T}}_{2}^{+}({e}^{\ensuremath{-}}{t}^{+}{t}^{+})$. The muonic and antiprotonic families are similar to the electronic ones except that the species are formed replacing ${e}^{\ensuremath{-}}$ by ${\ensuremath{\mu}}^{\ensuremath{-}}$ or ${p}^{\ensuremath{-}}$. The pionic family comprises exotic atoms containing at least one pion. We also apply these results to two-electron three-dimensional spherical quantum dots and for these systems we examine the effect of electronic correlation, particularly on the singlet-triplet transitions and on the collective motion of the electrons and center of mass leading to ``floppy''dynamics.