Bound Quantum Systems Research Articles

Power spectrum and form factor in random diagonal matrices and integrable billiards

Triggered by a controversy surrounding a universal behavior of the power spectrum in quantum systems exhibiting regular classical dynamics, we focus on a model of random diagonal matrices (RDM), often associated with the Poisson spectral universality class, and examine how the power spectrum and the form factor get affected by two-sided truncations of RDM spectra. Having developed a nonperturbative description of both statistics, we perform their detailed asymptotic analysis to demonstrate explicitly how a traditional assumption (lying at the heart of the controversy) – that the power spectrum is merely determined by the spectral form factor – breaks down for truncated spectra. This observation has important consequences as we further argue that bounded quantum systems with integrable classical dynamics are described by heavily truncated rather than complete RDM spectra. High-precision numerical simulations of semicircular and irrational rectangular billiards lend independent support to these conclusions.

BOOM: Rainbow’s Gravity as the Source of Light Refraction

On the fully Variable Light Speed (VSL) universe derived by alliance of Mach inertia principle and Planck’s quantized natural units and generalized Minkowski metric, it is extracted the fundamental equation of variable light speed called here Alpha Genesis Prime. Then the fundamental gravity is defined as a vector to justify the light speed variation accordingly. Gravitational vector via a tangential component does set the light speed to be matched with the Alpha Genesis Prime and according to the Pythagorean Theorem; it has also a component normalized to the light velocity. Interestingly by the gravity divided to the tangent and normal components relative to the light velocity we argue the Snell’s law of the light refraction and we find that dependency of the refractive index to the volume density is not fundamental but fundamentally the refractive index follows gravitational potential in bound quantum systems. The gravity is enhanced in bound quantum systems extended from Femto-scale (Nucleuses and strong nuclear force), Micro-gravity (fundamental rainbow’s gravity), the galaxies and clusters as the large scale bound quantum systems, up to the observable universe which the variable gravitational G is Newtonian (constant G in short cosmic time). The refraction of the light is sourced potentially by Rainbow’s gravity in bound quantum systems which the potential integration domain is limited to the wavelength of the photons. Quantum mechanically the photons are force carrier in the range of their wavelength to enhance gravitational G inasmuch as large that atoms to bend the photons similar to the enhanced gravity in the nucleuses in the range of hadron wavelengths. In fact reestablish of the Newtonian mechanics yields to the fundamental gravity which is identical with the refraction of the light and we find that the mechanical potential of the light’s refraction is the gravity.

Ultrafast manipulation of the weakly bound helium dimer

Controlling the interactions between atoms with external fields opened up new branches in physics ranging from strongly correlated atomic systems to ideal Bose and Fermi gases and Efimov physics. Such control usually prepares samples that are stationary or evolve adiabatically in time. On the other hand, in molecular physics external ultrashort laser fields are employed to create anisotropic potentials that launch ultrafast rotational wave packets and align molecules in free space. Here we combine these two regimes of ultrafast times and low energies. We apply a short laser pulse to the helium dimer, a weakly bound and highly delocalized single bound state quantum system. The laser field locally tunes the interaction between two helium atoms, imparting an angular momentum of $2\hbar$ and evoking an initially confined dissociative wave packet. We record a movie of the density and phase of this wave packet as it evolves from the inside out. At large internuclear distances, where the interaction between the two helium atoms is negligible, the wave packet is essentially free. This work paves the way for future tomography of wave packet dynamics and provides the technique for studying exotic and otherwise hardly accessible quantum systems such as halo and Efimov states.

Hyper-Poisson Photon Statistics

The statistical distribution for the number of bosons in a specified subsystem of a finite-dimensional multilevel system is obtained and investigated under conditions for which the total number of particles is a random variable characterized by the Poisson distribution. The resulting hyper-Poisson distribution is determined by the confluent hypergeometric function and contains as its particular limiting cases the Poisson distribution, as well as the single-mode and multimode thermal distributions. The developed model can be efficiently used to describe the thermal characteristics of bounded quantum systems, as well as in the problems of statistical reconstruction of optical quantum states, processes, and detectors.

Toward the multi-component quantum theory of atoms in molecules: a variational derivation

The general formalism of an extended quantum theory of atoms in molecules (QTAIM) dealing with the multi-component quantum systems, composed of various types of quantum particles, is disclosed in this contribution. This novel methodology, termed as the multi-component QTAIM (MC-QTAIM), is able to deal with non-adiabatic ab initio wavefunctions extracting atoms in molecules quantifying their properties. It can also be applied to elucidate the AIM structure of exotic species and bound quantum systems consisting of fundamental elementary particles like positrons and muons. The formalism is based on the previously disclosed density combination idea that is extended to derive the multi-component subsystem hypervirial theorem as well as the extended subsystem energy functional. Through the extended subsystem variational procedure, inspired from Schrodinger’s original variational principle, the surface terms containing the flux of the current property densities are derived. Accordingly, the extended Gamma field is introduced during this variational procedure that is used as the basic scalar field in the topological analysis yielding atoms in molecules and their real space boundaries. The Gamma field is central to the MC-QTAIM, replacing the usual one-electron density employed in the orthodox QTAIM and corresponding topological analysis. Through the multi-component hypervirial theorem, various regional theorems are derived which are then used to quantify the mechanical properties of atoms in molecules; these include the force, virial, torque, power, continuity and current theorems. In order to demonstrate the capability of the formalism, isotopically asymmetric hydrogen molecules, HD, HT and DT as well as YX systems (Y = 6Li, 7Li; X = H, D, T) composed of electrons and two different nuclei, all treated equally as quantum waves instead of clamped particles, are analyzed within context of the MC-QTAIM. The resulting computational analysis demonstrates that the MC-QTAIM is able to yield reasonable topological structures similar to those observed previously for diatomic species within context of the orthodox QTAIM. The asymmetrical nature of these species, inherent in their non-Born–Oppenhiemer wavefunctions, manifests itself clearly in the MC-QTAIM analysis yielding two distinguishable atomic basins with different properties. These differences are rationalized generally by the observed electron transfer from one basin to the other. Finally, some possible future theoretical extensions are considered briefly.

Extending the Four-Body Problem of Wolfes to Non-Translationally Invariant Interactions

We propose and solve exactly the Schr\"odinger equation of a bound quantum system consisting in four particles moving on a real line with both translationally invariant four particles interactions of Wolfes type \cite{Wolf74} and additional non translationally invariant four-body potentials. We also generalize and solve exactly this problem in any $D$-dimensional space by providing full eigensolutions and the corresponding energy spectrum. We discuss the domain of the coupling constant where the irregular solutions becomes physically acceptable

Graviatoms with de Sitter Interior

We present a graviatom with de Sitter interior as a new candidate to atomic dark matter generically related to a vacuum dark energy through its de Sitter vacuum interior. It is a gravitationally bound quantum system consisting of a nucleus represented by a regular primordial black hole (RPBH), its remnant or gravitational vacuum soliton G-lump, and a charged particle. We estimate probability of formation of RPBHs and G-lumps in the early Universe and evaluate energy spectrum and electromagnetic radiation of graviatom which can in principle bear information about a fundamental symmetry scale responsible for de Sitter interior and serve as its observational signatures.

Spectra of nonlocally bound quantum systems

We discuss a class of nonlinear and nonlocal models for the dynamics of a composite quantum system. The models in question depend on the following constituents: on two subsystem Hamiltonians (denoted by H and Ĥ), an analytic function (f), and a real parameter (s). As demonstrated elsewhere before, the stationary states can be described in these models fairly explicitly. In this article, we build upon that result, and discuss the topological as well as statistical characteristics of the spectra. Here, we concentrate on the special case f = log. It turns out that an energy spectrum of the nonlocally bound system substantially differs from that of its components. Indeed, we show rigorously that, if H is the harmonic oscillator and Ĥ is completely degenerate with one energy level, then the energy spectrum of the composite system has the topology of the Cantor set (for s > 2). In addition, we show that, if H is replaced by the logarithm of the harmonic oscillator, then the spectrum consists of finitely many intervals separated by gaps (for s sufficiently large). In the last case, the key analytic object is the series Σn−s. In particular, as an interesting offshoot, this structure furnishes a nontautological immersion of fundamental number-theoretic functions into the quantum formalism.

Going from classical to quantum description of bound charged particles II: Implications for the atomic physics

This paper is the continuation of the analysis of bound quantum systems started in part I (A.L. Kholmetskii, T. Yarman and O.V. Missevitch, Going from classical to quantum description of bound charged particles. I: Basic concepts and assertions), which is based on a novel approach to the transition from classical to quantum description of electrically bound charges, involving the requirement of energy-momentum conservation for the bound electromagnetic (EM) field, when the EM radiation is forbidden. It has been shown that the modified expression for the energy levels of hydrogenic atoms within such a pure bound field theory (PBFT) provides the same gross and fine structure of energy levels, like in the standard theory. At the same time, at the scale of hyperfine interactions, our approach, in general, does evoke some important corrections to the energy levels. Part of such corrections, like the spin-spin splitting in the hydrogen atom, is less than the present theoretical/experimental uncertainties in the evaluation of hyperfine contributions into the atomic levels. But the most interesting result is the appearance of a number of significant corrections (the 1S -2S interval and 1S spin-spin interval in positronium, 1S and 2S -2P Lamb shift in light hydrogenic atoms), which improve considerably the convergence between theoretical predictions and experimental results. In particular, the corrected 1S -2S interval and 1S spin-spin splitting in positronium practically eliminate the existing up-to-date discrepancy between theoretical and experimental data. The re-estimated classic 2S -2P Lamb shift as well as ground-state Lamb shift in the hydrogen atom lead to the proton charge radius r p = 0.841(6) fm (from 2S -2P Lamb shift), and r p = 0.846(22) fm (from 1S Lamb shift), which perfectly agrees with the latest estimation of proton size via the measurement of 2S -2P Lamb shift in muonic hydrogen, i.e. r p = 0.84184(67) fm. Finally, we consider the decay of bound muons in meso-atoms and achieve a quantitative agreement between experimental data and the results obtained through our approach.

Energy–momentum conservation in classical electrodynamics and electrically bound quantum systems

We call attention to the well-known fact that the electromagnetic (EM) field consists of bound and radiation components, and only their sum provides for the implementation of energy–momentum conservation for interacting classical charges. In this context, we then focus our attention on quantum systems of electrically bound charges, which do not radiate in the stationary energy states and thus their EM field comprises only the bound component. The non-applicability of Maxwell's equations to quantum mechanics does not permit, in general, ignoring the problem of recovering the energy–momentum conservation for such pure bound EM field systems, and we explore this problem within the Schrödinger–Dirac quantization scheme.

Neutron halo in deformed nuclei

Halo phenomena in deformed nuclei are investigated within a deformed relativistic Hartree Bogoliubov (DRHB) theory. These weakly bound quantum systems present interesting examples for the study of the interdependence between the deformation of the core and the particles in the halo. Contributions of the halo, deformation effects, and large spatial extensions of these systems are described in a fully self-consistent way by the DRHB equations in a spherical Woods-Saxon basis with the proper asymptotic behavior at a large distance from the nuclear center. Magnesium and neon isotopes are studied and detailed results are presented for the deformed neutron-rich and weakly bound nucleus $^{44}\mathrm{Mg}$. The core of this nucleus is prolate, but the halo has a slightly oblate shape. This indicates a decoupling of the halo orbitals from the deformation of the core. The generic conditions for the occurrence of this decoupling effects are discussed.

Almost-periodic time observables for bound quantum systems

It is shown that a canonical time observable may be defined for any quantum system having a discrete set of energy eigenvalues, thus significantly generalizing the known case of time observables for periodic quantum systems (such as the harmonic oscillator). The general case requires the introduction of almost-periodic probability operator measures (POMs), which allow the expectation value of any almost-periodic function to be calculated. An entropic uncertainty relation for energy and time is obtained which generalizes the known uncertainty relation for periodic quantum systems. While non-periodic quantum systems with discrete energy spectra, such as hydrogen atoms, typically make poor clocks that yield no more than 1 bit of time information, the anisotropic oscillator provides an interesting exception. More generally, a canonically conjugate observable may be defined for any Hermitian operator having a discrete spectrum.

Calibration of the Density-Gradient model by using the multidimensional effective-mass Schrödinger equation

A calibration method is developed for the electron effective mass in the Density-Gradient model. This method uses the two- and three-dimensional effective-mass Schrodinger equations, which are solved for bounded quantum systems. The electron effective mass is computed by fitting the electron concentration computed by using the Density-Gradient model to the electron concentration computed by using the Schrodinger equation. Results for strongly confined silicon system with (100), (110), and (111) crystallographic orientations are presented. It is shown that the effective mass varies with the shape and dimensions of the quantum box. In device simulations, one should use the value of m_n that corresponds to the right shape and dimensions of the confinement region in the device.

Band structure of a harmonically confined electron with an impenetrable boundary

AbstractWe study finite‐size effects of the spatially bounded quantum systems exemplified by a single‐electron quantum dot with a harmonic potential and an impenetrable boundary. A general solution of the corresponding Schrödinger equation is obtained and the unique special solution for any energy is derived from the normalization and boundary conditions. The classical‐mechanically allowable eigenenergies form the continuous spectrum or piecewise continuous bands with the minimum value being much less than the zero point energy of a free harmonic oscillator. As the increase of the confining size, the band widths reduce and the energies finally close to the discrete level of the free oscillator. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the conservative bound systems

It is well known that the Schrodinger equation is equivalent to the wave equation for conservative bound quantum systems. Consequently, the motion of such a system is represented—from a mathematical point of view—by the motion of the characteristic surface of the wave equation. In this paper we present a demonstration of the periodic motion of the characteristic surface. It results that the normal curves of the characteristic surface are periodic solutions of the Hamilton–Jacobi equation written for the same system. This leads to a direct connection between the periodic solutions of the Hamilton–Jacobi equation and the wave properties of the system. The constants of motion corresponding to the above periodic solutions of the Hamilton–Jacobi equation are identical to the eigenvalues of the Schrodinger equation. These properties are proved without any approximation and they are valid for all the values of the principal quantum number.

Quantum sensitive dependence

Wave functions of bounded quantum systems with time-independent potentials, being almost periodic functions, cannot have time asymptotics as in classical chaos. However, bounded quantum systems with time-dependent interactions, as used in quantum control, may have continuous spectrum and the rate of growth of observables is an issue of both theoretical and practical concern. Rates of growth in quantum mechanics are discussed by constructing quantities with the same physical meaning as those involved in the classical Lyapunov exponent. A generalized notion of quantum sensitive dependence is introduced and the mathematical structure of the operator matrix elements that correspond to different types of growth is characterized.

Basic Ideas and Concepts in Nuclear Physics: An Introductory Approach

KNOWING THE NUCLEUS: THE NUCLEAR CONSTITUENTS AND CHARACTERISTICS Global Nuclear Properties General Nuclear Radioactive Decay Properties and Transmutations NUCLEAR INTERACTIONS: STRONG, WEAK AND ELECTROMAGNETIC FORCES General Methods Alpha Decay: The Strong Interaction at Work Beta Decay: The Weak Interaction at Work Gamma decay: The Electromagnetic Interaction at Work NUCLEAR STRUCTURE: AN INTRODUCTION The Liquid Drop Model Approach: A Semi-Empirical Method The Simplest Independent Particle Model: The Fermi-Gas Model The Nuclear Shell Model NUCLEAR STRUCTURE: RECENT DEVELOPMENTS The Nuclear Mean Field: Single Particle Excitations and Global Nuclear Properties The Nuclear Shell Model: Including Residual Interactions Nuclear Physics of Very Light Nuclei Collective Modes of Motion Deformation in Nuclei Nuclear Physics at the Extremes of Stability: Weakly Bound Quantum Systems and Exotic Nuclei Deep Inside the Nucleus: Subnuclear Degrees of Freedom and Beyond Outlook: The Atomic Nucleus as Part of a Larger Structure Appendices Units and Conversion Between Various unit Systems Spherical Tensor Properties Second Quantization - An Introduction References

TEMPORAL QUANTUM CHAOS

We study the long-time behavior of bound quantum systems whose classical dynamics is chaotic and put forward two conjectures. Conjecture A states that the autocorrelation function C(t)=<Ψ(0)|Ψ(t)> of a delocalized initial state |Ψ(0)> shows characteristic fluctuations, which we identify with a universal signature of temporal quantum chaos. For example, for the (appropriately normalized) value distribution of S~|C(t)| we predict the distribution P(S)=(π/2)Se-πS2/4. Conjecture B gives the best possible upper bound for a generalized Weyl sum and is related to the extremely large recurrence times in temporal quantum chaos. Numerical tests carried out for numerous chaotic systems confirm nicely the two conjectures and thus provide strong evidence for temporal quantum chaos.

Quantum-classical correspondence via Liouville dynamics. II. Correspondence for chaotic Hamiltonian systems

We prove quantum-classical correspondence for bound conservative classically chaotic Hamiltonian systems. In particular, quantum Liouville spectral projection operators and spectral densities, and hence classical dynamics, are shown to approach their classical analogs in the h!0 limit. Correspondence is shown to occur via the elimination of essential singularities. In addition, applications to matrix elements of observables in chaotic systems are discussed. @S1050-2947~96!05212-2# The validity of quantum mechanics as a description of the macroscopic world is contingent upon the reduction of the laws of quantum mechanics to Newton’s laws in the limit where the characteristic actions of a system are large with respect to Planck’s constant @1#. Thus, diagonal and offdiagonal matrix elements must reduce to their classical analogs and quantum dynamics must reproduce the predictions of classical mechanics as h!0. Despite the fundamental importance of quantum-classical correspondence it has only been satisfactorily demonstrated @2‐5# in the very restrictive case of regular systems, i.e., systems that classically possess as many constants of the motion as degrees of freedom. Indeed some authors have suggested that bound quantum systems with a discrete quantum spectrum and a chaotic classical analog may violate the correspondence principle @6#. These doubts about the validity of the correspondence principle for chaotic systems stem from the difficulty of reconciling the quasiperiodic nature of bound state quantum dynamics with the chaotic nature of classical dynamics for the same Hamiltonian. The issue of correspondence for quantum systems whose classical analogs exhibit chaos ~irregular systems! is thus of great interest. Verification of correspondence should be distinguished from the development of semiclassical approximation methods. While semiclassical theories provide a natural starting point for an exploration of the classical limit their existence does not guarantee correspondence. For example, semiclassical theories for regular systems preceded the development of modern quantum mechanics @7#, but an understanding of correspondence for regular systems has only recently been achieved @2,3,5#. By comparison, attempts to develop semiclassical quantization rules for chaotic systems have had some success @8#, whereas the correspondence limit remains largely unexplored @9#. In this paper we demonstrate that the existing semiclassical theories of quantum dynamics for classically chaotic systems are sufficiently well developed to allow us to show that such systems do in fact approach their proper correspondence limits as Planck’s constant approaches zero. This completes the Liouville correspondence program outlined in the preceding paper @5#, and significantly extends the results of our study of quantum maps @10,11#, where we rigorously demonstrated that a nonchaotic quantum map dynamics can completely recover a fully chaotic classical dynamics in the limit h!0. The Liouville picture affords a means of gaining insight into the connections between quantum and classical mechanics @3,12,13#, and is a natural framework for studies of correspondence. As outlined in the preceding paper ~henceforth referred to as paper I !@ 5 #the essential ingredients for Liouville dynamics are eigenstates and eigenvalues of the Liouville operators in both mechanics. In particular, the dynamics is completely characterized by the Liouville eigenfunctions and eigenvalues or the spectral projectors once the class of allowed initial distributions is specified. Here we consider correspondence in chaotic systems from this Liouville perspective.

Cutoff in molecular harmonic-generation spectra resulting from classical chaotic dynamics.

The plateau in the harmonic-generation (HG) spectrum of a bound quantum system is first associated with classical chaotic dynamics. The molecular cutoff in the HG spectra was obtained at \ensuremath{\Omega}=m\ensuremath{\omega}, where \ensuremath{\omega} is the frequency of the time-periodic field and m is the area of the bound chaotic region in the phase space divided by 2\ensuremath{\pi}\ensuremath{\Elzxh}.