Independent Particle Motion Research Articles

Quantum mechanics emerging from complex Brownian motions

The connection between the Schrödinger equation and Einstein's diffusion theory based on the Brownian motion of independent particles is well known. However, in contrast to diffusion theory, quantum mechanics theory has suffered controversial interpretations due to the counterintuitive concept of wavefunction. Here, while we confirm there is no difference in the mathematical form of these two equations, we derive the imaginary version of displacement. Using the diffusion theory of particles in a medium, as simple as it is, we describe that quantum mechanics is just an elegant and subtle equation to describe the probability of all the trajectories that a particle can take to propagate in time by a predictive wavefunction. Therefore, information on the position of particles through time in quantum theory is embedded in the wavefunction, which predicts the evolution of an ensemble of individual Brownian particles.

Programming Relativity As the Mathematics of Perspective in a Planck Unit Simulation Hypothesis

The Simulation Hypothesis proposes that all of reality is in fact an artificial simulation, analogous to a computer simulation. Outlined here is a method for programming relativistic mass, space and time at the Planck level as applicable for use in Planck Universe-as-a-Simulation Hypothesis. For the virtual universe the model uses a 4-axis hyper-sphere that expands in incremental steps (the simulation clock-rate). Virtual particles that oscillate between an electric wave-state and a mass point-state are mapped within this hyper-sphere, the oscillation driven by this expansion. Particles are assigned an N-S axis which determines the direction in which they are pulled along by the expansion, thus an independent particle motion may be dispensed with. Only in the mass point-state do particles have fixed hyper-sphere co-ordinates. The rate of expansion translates to the speed of light and so in terms of the hyper-sphere co-ordinates all particles (and objects) travel at the speed of light, time (as the clock-rate) and velocity (as the rate of expansion) are therefore constant, however photons, as the means of information exchange, are restricted to lateral movement across the hyper-sphere thus giving the appearance of a 3-D space. Lorentz formulas are used to translate between this 3-D space and the hyper-sphere co-ordinates, relativity resembling the mathematics of perspective.

α decay preformation probabilities across the N = 126 shell closure based on the single particle energy spectra

We put forward a concise approach to calculate the α decay preformation probabilities by defining the microscopic valence nucleon and hole numbers based on the single particle energy spectra, which is obtained within the relativistic Hartree–Bogoliubov model. The residual interactions including neutron–neutron, proton–proton pairing and proton–neutron correlations could trigger the collective motions among the valence nucleons and holes, along with the independent particle motions under the mean field assumption. Therefore the more the number of valence nucleons and holes, the stronger the collective motions such as α particle clustering. We assume that the valence orbits locate around the Fermi surface obeying a Fermi–Dirac distribution, i.e. nucleons (holes) occupying the orbits with much lower (higher) energies away from the Fermi surface can not be considered as the valence nucleons (holes). The α decay preformation probabilities Pα are calculated within the Np Nn scheme in terms of the valence proton–neutron interaction. As an example, in the case of even–even polonium, radon, radium and thorium isotopes the extracted α decay preformation probabilities , which are obtained within the generalized liquid drop model, can be well reproduced by employing two different formulas to take into account the shell effect at N = 126 shell closure.

Unified description of structure and reactions: implementing the nuclear field theory program

The modern theory of the atomic nucleus results from the merging of the liquid drop model of Niels Bohr and Fritz Kalckar, and of the shell model of Marie Goeppert Meyer and Hans Jensen. The first model contributed the concepts of collective excitations. The second, those of independent-particle motion. The unification of these apparently contradictory views in terms of the particle-vibration and particle-rotation couplings carried out by Aage Bohr and Ben Mottelson has allowed for an ever more complete, accurate and detailed description of nuclear structure. Nuclear field theory (NFT), developed by the Copenhagen–Buenos Aires collaboration, provided a powerful quantal embodiment of this unification. Reactions are not only at the basis of quantum mechanics (statistical interpretation, Max Born), but also the specific tools to probe the atomic nucleus. It is then natural that NFT is being extended to deal with processes which involve the continuum in an intrinsic fashion, so as to be able to treat them on an equal footing with those associated with bound states (structure). As a result, spectroscopic studies of transfer to continuum states could eventually make use of the NFT rules, properly extended to take care of recoil effects. In the present contribution we review the implementation of the NFT program of structure and reactions, setting special emphasis on open problems and outstanding predictions.

New charge radius relations for atomic nuclei

We show that the charge radii of neighboring atomic nuclei, independent of atomic number and charge, follow remarkably very simple relations, despite the fact that atomic nuclei are complex finite many-body systems governed by the laws of quantum mechanics. These relations can be understood within the picture of independent-particle motion and by assuming neighboring nuclei having similar pattern in the charge density distribution. A root-mean-square (rms) deviation of 0.0078 fm is obtained between the predictions in these relations and the experimental values, i.e., a comparable precision as modern experimental techniques. Such high accuracy relations are very useful to check the consistence of nuclear charge radius surface and moreover to predict unknown nuclear charge radii, while large deviations from experimental data is seen to reveal the appearance of nuclear shape transition or coexsitence.

Some Aspects of Transfer Reactions in Light and Heavy Ion Collisions

Transfer reactions played a key role in unraveling many properties of the nucleus, in particular they have been essential to define the relevance of the independent particle motion and to define the properties of two-particle correlations. In view of the now available radioactive beams, it is important to summarize some aspects of the transfer processes that will be relevant to define the properties of nuclei close to the neutron and proton drip lines.

Aage Niels Bohr

Aage Niels Bohr

Covariant theory of particle-vibrational coupling and its effect on the single-particle spectrum

The relativistic mean field (RMF) approach describing the motion of independent particles in effective meson fields is extended by a microscopic theory of particle vibrational coupling. It leads to an energy dependence of the relativistic mass operator in the Dyson equation for the single-particle propagator. This equation is solved in the shell-model of Dirac states. As a result of the dynamics of particle-vibrational coupling we observe a noticeable increase of the level density near the Fermi surface. The shifts of the single-particle levels in the odd nuclei surrounding $^{208}\mathrm{Pb}$ and the corresponding distributions of the single-particle strength are discussed and compared with experimental data.

Generation of low-density plasma by two-step photoionization of barium and its evolution in electric field

Two-color, two-step resonant photoionization has been used to produce low-density (∼106 – 107cm−3) barium photoplasma in an atomic beam. The two-photon induced specific absorption coefficient at 355nm for the photoionization process has been measured. The motion of the finite size photoplasma bounded by vacuum is experimentally studied in a static electric field in parallel-plate electrode configuration. It is observed that at sufficiently high electric fields all electrons leave the plasma and the evolution of the left over ion bunch is understood on the basis of independent particle motion.

Center-of-Mass Energy Removal in Relativistic Three-Quark Models of the Proton

A variant of a squared three-body Dirac equation is used to determine center-of-mass energy effects in independent particle motion approximations for three quarks in the nucleon. A scalar linear flux tube potential is used to confine the quarks. The relativistic nearly massless three-quark system, in the rest frame where the total momentum is zero, has a squared energy that is 3/5 the value compared to when the quarks are assumed to move independently. This is smaller than the 2/3 energy ratio determined using the non-relativistic harmonic oscillator model. This analytic model has one parameter, the flux tube constant. Choosing the flux tube constant to reproduce the proton rest energy, results in the analytic wave function well reproducing the proton axial charge and rms charge radius. The proton magnetic moment predicted is 2.235, lower than experiment.

Collective and independent-particle motion in two-electron artificial atoms

Investigations of the exactly solvable excitation spectra of two-electron quantum dots with a parabolic confinement, for different values of the parameter R(W) expressing the relative magnitudes of the interelectron repulsion and the zero-point kinetic energy, reveal for large R(W) a rovibrational spectrum associated with a linear trimeric rigid molecule composed of the two electrons and the infinitely heavy confining dot. This spectrum transforms to that of a "floppy" molecule for smaller R(W). The conditional probability distribution calculated for the exact two-electron wave functions allows identification of the rovibrational excitations as rotations and stretching/bending vibrations.

Nonadiabatic dynamics via the classical limit Schrödinger equation

The coupled Schrödinger equations that describe nonadiabatic dynamics are recast using the Bohm formulation of quantum mechanics. The resulting coupled Bohm equations are solved numerically for two scattering models, giving results that are essentially identical to wave-packet solution of the original coupled Schrödinger equations. The classical limit of the set of coupled Bohm equations is then described, producing a mixed quantum-classical theory incorporating classical-like motion on each potential energy surface accompanied by quantum transitions between the quantum states. Numerical tests of the mixed quantum-classical method are in excellent agreement with the accurate full-quantum results for the model problems. The method is contrasted with the related surface-hopping approach. It is shown that computing the dynamics of a distribution of classical particles is more consistent and more accurate than computing the motion of independent point particles as with surface hopping.

Why are nuclei described by independent particle motion ?

Why are nuclei described by independent particle motion ?

Independent particle motion and correlations in fermion systems

The independent-particle model explains many features of atomic nuclei and other fermion systems. The low-energy states of nearly closed-shell systems can be interpreted as having quasiparticles in single-particle orbitals. The difference between physical particles and quasiparticles results from the effects of correlations in the system. In this Colloquium the authors consider the consequences of these correlations. They discuss in particular, mainly for the case of nuclei, the quasihole strength $z$ (spectroscopic factor) that gives the probability of the quasiparticle's being a physical particle. Results from both theory and experiment indicate that $z\ensuremath{\sim}0.65$ and imply that only $\ensuremath{\sim}2/3$ of the time a nucleon acts as an independent particle bound in an average potential. The fraction of $\ensuremath{\sim}1/3$ of correlated nucleons is larger than believed in the past.

Diffusion approximation for billiards with totally accommodating scatterers

We study the 2D motion of independent point particles colliding with a periodic array of circular obstacles. The interaction between the particles and the obstacles is described by a total accommodation reflection law. Assuming that the array of scatterers has finite horizon, the density of particles is approximated by the solution of a diffusion equation in the long-time and large-scale regime. The proof relies on a multiscale asymptotics and gives the order of approximation.

On the nature of nuclear dissipation, as a hallmark for collective dynamics at finite excitation

We study slow collective motion of isoscalar type at finite excitation. The collective variable is parameterized as a shape degree of freedom and the mean field is approximated by a deformed shell model potential. We concentrate on situations of slow motion, as guaranteed, for instance, by the presence of a strong friction force, which allows us to apply linear response theory. The prediction for nuclear dissipation of some models of internal motion are contrasted. They encompass such opposing cases as that of pure independent particle motion and the one of "collisional dominance". For the former the wall formula appears as the macroscopic limit, which is here simulated through Strutinsky smoothing procedures. It is argued that this limit hardly applies to the actual nuclear situation. The reason is found in large collisional damping present for nucleonic dynamics at finite temperature $T$. The level structure of the mean field as well as the $T$-dependence of collisional damping determine the $T$-dependence of friction. Two contributions are isolated, one coming from real transitions, the other being associated to what for infinite matter is called the "heat pole". The importance of the latter depends strongly on the level spectrum of internal motion, and thus is very different for "adiabatic" and "diabatic" situations, both belonging to different degrees of "ergodicity".

Shell model and high-spin and deformation

Taking deformation of the average potential as the key to understanding nuclear collective phenomena, some aspects of the current status and perspectives of spheroidal, triaxial, octupole, and superdeformed nuclear shapes at low- and high-spins are reviewed with emphasis on the shell structure associated with the independent particle-motion in appropriate potentials.

Functional integral approach to the Lipkin model.

A quantum-mechanical formulation involving both collective and independent-particle motions in many-fermion systems is proposed by using the path-integral technique. A semiclassical method of evaluating the functional integral over both fields is described. As an illustration, the Lipkin model is utilized.

An Extension of Time-Dependent Hatree-Fock Theory Including Grassmann Variables. I: --Basic Formulation in the Framework of Dirac's Canonical Theory with Constraints--

A microscopic theory of collective and independent-particle motions in many-fermion system is developed in the framework of Dirac's canonical theory with constraints. The basic idea is an extension of the conventional time-dependent Hartree-Fock theory based on the use of the Grassmann variables and, in certain sense, it supplements incomplete parts of the paper presented by the present authors. Canonicity condition to certify the canonical property of the variables, equation to determine collective submanifold and constraints to govern the collective and the Grassmann variables are given.

Transition from Collective to Independent-Particle Motion Within the Valence Shell of the Second- and Third-Row Atoms

Construction explicite d'une constante de mouvement exacte pour un hamiltonien des couches de valence de ces atomes. Analyse des sommes des operations a 1 et a 2 electrons, et de leurs contributions relatives