Three-dimensional Wave Function Research Articles

Quantum evolution of quarkonia with correlated and uncorrelated noise

In the quark-gluon plasma (QGP), it is well known that the evolution of quarkonia is affected by the screening of the interaction between the quark and the anti-quark. In addition, exchange of energy and color with the surrounding medium can be included via the incorporation of noise terms in the evolution Hamiltonian. For noise correlated locally in time, these dynamics have been studied in a simple setting by Kajimoto et al.[PHYS. REV. D 97, 014003(2018)]. We extend this calculation by considering non-Abelian dynamics for a three-dimensional wavefunction. We also propose a modification of the noise correlation, allowing it to have a finite correlation in time with the motivation to include long-lived gluonic correlations. We find that in both cases the results differ significantly from solutions of rate equations.

Simple and Multi-layered Quantum dot in various structures

Abstract: Quantum dots have interesting optical properties. They absorb incoming light of one color and emit out light of a completely different color. This research paper discloses eigen states of a simple and multilayer quantum dot in various structures for cuboid, cylinder, dome, cone, and pyramid, and its three-dimensional wave function, energy states, light and dark transitions (X-polarized), light and dark transitions (Y-polarized), light and dark transitions (Zpolarized), light and dark transitions (phi = 0 and theta= 45), absorption (phi = 0 and theta = 45), absorption sweep of angle theta, and integrated absorption are plotted and the observations of high peak values are noted and documented.

Ab initio modelling of molecular hydrogen rotation in the outside of carbon nanotubes

ABSTRACTA first-principle modelling of hydrogen molecular rotation in the outside of carbon nanotubes is presented. Density functional theory (DFT)-based symmetry-adapted perturbation theory (SAPT) is first applied to analyse the influence of the rotation on the dispersion and dispersionless H-nanotube interaction for both sub- and nanometer-sized tubes. An adsorbate three-dimensional wave-function treatment is then applied to calculate the molecular energy levels of the rotating hydrogen molecule. As a key difference with the H located inside the tubes, the SAPT-based analysis indicates a marked influence of a nanotube curvature-induced dipole on the angular-dependent balance of exchange-repulsion, electrostatic, and dispersion contributions for narrow nanotubes. As a result, the landscape of molecular energy levels depends strongly on the diameter of the porous material. In addition, an effective one-dimensional model is proposed to account for the nuclear motion, reproducing full-dimensional approach within less than 1%.

Classical and Quantum Modeling of Li and Na Diffusion in FePO4

Lithium diffusion in olivine phosphates has been widely studied both experimentally and theoretically. However, nuclear quantum effects (NQEs) of the Li ions have not been accounted for in theoretical studies thus far. In the current work, we compared Li and Na diffusion in Li0.25FePO4 and Na0.25FePO4 by computing density functional theory based classical diffusion barriers in conjunction with NQEs for the Li and Na ions. The NQEs are computed using a novel three-dimensional wave function method based on a path integral formulation. The calculations of both the potential and free energy diffusion barriers suggest that Li diffusion is faster than Na diffusion, in agreement with recent experiments. The NQEs for lithium ions in Li0.25FePO4 are higher than those for sodium ions in Na0.25FePO4. Although the contribution of NQEs to the computed Li and Na ion diffusion rates is rather small, the quantum behavior of the Li ions is unusual. Indeed, we observe a reduction in the computed diffusion rate for Li ions due to quantization. We ascribe this effect to the ability of FePO4 to tightly bind the Li ions in the transient tetrahedral transition state, which reduces the classical diffusion barrier but also enhances quantum confinement.

Electronic properties of corrugated graphene: the Heisenberg principle and wormholegeometry in the solid state

Adopting a purely two-dimensional relativistic equation for graphene’s carriers contradictsthe Heisenberg uncertainty principle since it requires setting the off-the-surface coordinateof a three-dimensional wavefunction to zero. Here we present a theoretical framework fordescribing graphene’s massless relativistic carriers in accordance with this mostfundamental of all quantum principles. A gradual confining procedure is used to restrict thedynamics onto a surface and normal to the surface parts, and in the process the embeddingof this surface into the three-dimensional world is accounted for. As a result an invariantgeometric potential arises in the surface part which scales linearly with the meancurvature and shifts the Fermi energy of the material proportional to bending. Straininduced modification of the electronic properties or ‘straintronics’ is clearly animportant field of study in graphene. This opens an avenue to producing electronicdevices: micro- and nano-electromechanical systems (MEMS and NEMS), where theelectronic properties are controlled by geometric means and no additional alteration ofgraphene is necessary. The appearance of this geometric potential also providesus with clues as to how quantum dynamics looks in the curved space–time ofgeneral relativity. In this context we explore a two-dimensional cross-section of thewormhole geometry, realized with graphene as a solid state thought experiment.

Single and double ionization inC6++Hecollisions

Single- and double-ionization processes in ${\mathrm{C}}^{6+}$ collisions with the He atom at an incident energy of $100$ MeV/amu are studied by direct solution of the time-dependent Schr\"odinger equation. A time-dependent close-coupling method based on an expansion of a one-electron three-dimensional wave function in the field of He${}^{+}$ is used to calculate single-ionization cross sections. A time-dependent close-coupling method based on an expansion of a two-electron six-dimensional wave function in the field of He${}^{2+}$ is used to calculate single- and double-ionization cross sections. Electron energy and angle differential cross sections for single ionization are presented for various projectile impact parameters. For relatively large impact parameters, the differential cross sections are in qualitative agreement with ion-atom experiments. Electron energy and angle differential cross sections for double ionization are also presented for a relatively large projectile impact parameter.

Comment on “Exact three-dimensional wave function and the on-shelltmatrix for the sharply cut-off Coulomb potential: Failure of the standard renormalization factor”

The solutions analytically derived by Gloeckle et al. [Phys. Rev. C79, 044003 (2009)] for the three-dimensional wave function and on-shell t matrix in the case of scattering on a sharply cut-off Coulomb potential appear to be fallacious. And their renormalization factor lacks mathematical grounds.

Approximate Three-Dimensional Wave Function and the T-Matrix for the Sharply Cut Off Coulomb Potential

For a sharply cut-off Coulomb potential we derive analytically the asymptotic form of the three- dimensional wave function and the related scattering amplitude. We show a failure of the standard renormal- ization factor which is believed to be generally valid for any type of screening. We obtain also the asymptotic form of the corresponding three-dimensional half-shell t-matrix. Our results are fully supported by the numerical solutions of the three-dimensional Lippmann-Schwinger equation.

Exact three-dimensional wave function and the on-shelltmatrix for the sharply cut-off Coulomb potential: Failure of the standard renormalization factor

The three-dimensional wave function for a sharply cut-off Coulomb potential is analytically derived. The asymptotic form of the related scattering amplitude reveals a failure of the standard renormalization factor which is believed to be generally valid for any type of screening.

Quantum filtering of electron emission from ultrasharp tips

This paper derives an exact fully three-dimensional wave-function solution and presents representative calculations for the problem of an electron confined in an atomically sharp three-dimensional tip. The presented solution has important implications in technologies including self-collimated electron sources, ultrafine electron-beam lithography, electron holography, and cold cathode radio-frequency generation. The tip is modeled in the prolate-spheroidal coordinate system because the principal characteristics of an ultrasharp tip, including cone angle and tip radius, are naturally represented. This quantum-mechanical analysis corroborates several experimental observations on atomically sharp tips, including unexpectedly narrow energy bandwidth for the emitted electrons and pronounced collimation, or self-focusing, of the emitted electron beam. The three-dimensional analysis developed in this paper indicates that these experimental phenomena are attributable to a ``quantum filtering'' effect that arises in the geometric constriction characteristic of ultrasharp vacuum field emitters. The results of this paper suggest that the free-electron supply function, routinely invoked in classical field-emission analysis, may be an inaccurate approximation for the electronic states of ultrasharp tips having quantum-filtering characteristics. Surface plots of the electron wave-function density for various total electron energies and tip geometries demonstrate the tip selectivity with respect to electron states.

Adiabatic Approximation of the Schrödinger--Poisson System with a Partial Confinement

Asymptotic quantum transport models of a two-dimensional electron gas are presented. The starting point is a singular perturbation of the three-dimensional Schrodinger--Poisson system. The small parameter $\eps$ is the scaled width of the electron gas and appears as the lengthscale on which a one-dimensional confining potential varies. The rigorous $\epsilon\rightarrow 0$ limit is performed by projecting the three-dimensional wavefunction on the eigenfunctions corresponding to the confining potential. This leads to a two-dimensional Schrodinger--Poisson system with a modified Poisson equation keeping track of the third dimension. This limit model is proven to be a first-order approximation of the initial model. An intermediate model, called the ``2.5D adiabatic model' is then introduced. It shares the same structure as the limit model but is shown to be a second-order approximation of the three-dimensional model.

Hamiltonians and Green's functions which interpolate between two and three dimensions

I propose to use Hamiltonians with both two-dimensional and three-dimensional kinetic terms for the description of two-dimensional systems in physics. As a model system the evolution of three-dimensional wavefunctions in the presence of an infinitely thin layer is studied. The model predicts distance laws for correlation functions which interpolate between two-dimensional and three-dimensional behavior. It also predicts that in certain cases transmission probabilities through thin layers should depend not only on the transverse, but also on the longitudinal momentum of the infalling particles. The model also yields a static potential which interpolates between the two-dimensional logarithmic potential at small distances and the three-dimensional 1/r-potential at large distances.

Quantum mechanical methods for enzyme kinetics.

This review discusses methods for the incorporation of quantum mechanical effects into enzyme kinetics simulations in which the enzyme is an explicit part of the model. We emphasize three aspects: (a) use of quantum mechanical electronic structure methods such as molecular orbital theory and density functional theory, usually in conjunction with molecular mechanics; (b) treating vibrational motions quantum mechanically, either in an instantaneous harmonic approximation, or by path integrals, or by a three-dimensional wave function coupled to classical nuclear motion; (c) incorporation of multidimensional tunneling approximations into reaction rate calculations.

Exact wave functions from classical orbits: The isotropic harmonic oscillator and semiclassical applications

The wave function for a state of definite angular momentum of the three-dimensional isotropic harmonic oscillator is expressed exactly in terms of the corresponding classical trajectories. In particular, the three-dimensional wave function as well as the radial wave function and the spherical harmonics are obtained as integrals over quantities determined entirely by the classical motion. The expression for the harmonic-oscillator radial wave function is shown also to yield the exact radial wave function for the free particle. The expressions are cast in forms suitable for use as uniform semiclassical approximations for wave functions of other systems. Numerical examples confirm that such wave functions obey boundary conditions appropriate for spherical coordinates and that they are free of caustic singularities. The wave functions obtained by this technique can be quite accurate even for low-energy states where semiclassical approximations are expected to be poor.

A New Perspective on Isomerization Dynamics Illustrated by HCN → HNC

We discuss the role of delocalized states in isomerization dynamics and illustrate the ideas by considering isomerization in HCN. We analyze results from an earlier study, where HCN was treated as a semirigid bender, and isomerization was induced by collision with an Ar atom. That analysis is applied here to three-dimensional HCN, using exact three-dimensional wave functions, obtained using a slight modification of a previous global potential based on high-quality ab initio calculations.

Energy levels of charged particles confined in a multiply connected structure in a magnetic field

We have calculated the wave functions and energy levels of a two-dimensional mesoscopic electron system having four-fold symmetry and multiply connected topology, in both zero and finite magnetic fields (B fields). The electron distribution can be controlled in the multiply connected structure by changing the peripheral channel thickness. The energy levels oscillate as a function of the B field, with pronounced gaps, and the electronic states are classified into four kinds of eigenstates under the 90° rotation operation. The three-dimensional wave functions of cylindrical and helical structures having this cross section are constructed.

Quantum-molecular-dynamics simulations of liquid metals and highly degenerate plasmas

A numerical scheme based on time-dependent density-functional theory, for the simulation of systems of classical ions and quantum mechanical electrons, has been developed with the aim of modeling simple metals or highly degenerate plasmas from first principles. The electrons are represented by a set of three-dimensional wave functions obeying one-particle Schrödinger equations, subject to an effective potential. A pseudospectral method is used for the solution of the Schrödinger equations, coupled to a classical molecular-dynamics simulation for the ions. The scheme yields accurate predictions for the radial distribution functions and the ionic diffusion coefficients in liquid sodium, and is shown to correctly account for shielding due to the fully responsive electron background. Preliminary simulation results for bulk hydrogen under astrophysical conditions are also presented, including the successful modeling of chemical bonding in the isolated hydrogen molecule.

QED test of a Bethe-Salpeter solution method.

A method is described of solving Bethe-Salpeter equations for spin-\textonehalf{} particles using a standard quasipotential approximation with a correction series. A three-dimensional, 16-component bound-state equation is obtained whose form is different from the Breit equation. The equation has two equivalent first-order forms which show each particle obeying a Dirac equation in the presence of the other. The formalism also reconstitutes the four-dimensional bound-state vertex function from the three-dimensional wave function. The method is tested on positronium and the hydrogen atom, using a single-photon-exchange kernel in the Coulomb gauge. In the bound-state equation, in addition to the Coulomb potential and Breit interaction, the formalism gives a subtracted box potential $(\frac{1}{2E})\frac{{\ensuremath{\alpha}}^{2}}{{r}^{2}}$ where $E$ is the mass of the bound state. With this term included, the energy levels (fine structure and hyperfine structure) are correct to order ${\ensuremath{\alpha}}^{4}$. The reconstituted bound-state vertex function, when substituted in Feynman triangle diagrams, gives the lowest-order atomic dipole transition amplitudes correctly.

Three-dimensional, spherically symmetric, saturating model of an N-boson condensate.

The variational Hartree method is applied to a system of N bosons interacting via Skyrme-type attractive and repulsive forces. When the system condenses into its ground state, the wave function for a single boson, phi, may be described by means of the nonlinear, time-independent Schroedinger equation, del/sup 2/phi = epsilon/sub 1/phi-Achemically bondphichemically bond/sup 2/phi+Bchemically bondphichemically bond/sup 4/phi. In one dimension, this equation has a single, analytic, bound state solution and exhibits the property of saturation. In the physically more interesting case of three dimensions with spherical symmetry, no analytic solutions are known, so that a numerical solution must be resorted to, and this shows that saturation is again obtained. However, an analytic approximation to the three-dimensional wave function, which is very accurate for values of epsilon/sub 1/B/A/sup 2/>0.1, is deduced and studied. Approximate analytic expressions for the Hartree potential energy, kinetic energy, and mean square radius of the system are thereby derived, and applications to finite nuclei and infinite nuclear matter are considered.

Caustic singularities in the wave function of the relative motion of heavy ions

The properties of the three-dimensional wave function for the elastic channel of the relative motion of heavy ions at short distances are investigated. For a near-barrier energy, the sharp increase in the amplitude of the wave function in the neighborhood of the cusp that is formed leads to spatial localization of inelastic processes characterized by long-range form factors.