Single-particle Wave Functions Research Articles

Comparative studies on nuclear elastic magnetic form factors between the relativistic and non-relativistic mean-field approaches * *Supported by the National Natural Science Foundation of China (11505292, 11605105, 11822503, 11975167, 12035011), the Shandong Provincial Natural Science Foundation, China (ZR2020MA096), the Fundamental Research Funds for the Central Universities (20CX05013A, 22120210138), and the Graduate Innovative Research Funds of China University of Petroleum (East

It is known that elastic magnetic electron scattering can be used to study the magnetic properties of nuclei and determine the outermost-shell single-particle orbitals. In this study, the magnetic form factors of odd-A nuclei calculated with relativistic and non-relativistic models are systematically compared. We use the relativistic mean-field (RMF) and Skyrme Hartree-Fock (SHF) models to generate single-particle wave functions and calculate the values of selected nuclei under relativistic and non-relativistic frameworks, respectively. Geometric factors are introduced through the spherical limit method to consider the influences of deformation, which improves the agreement between the theoretical results and experimental data. It is shown that both the models have the capability to describe the magnetic form factors in the spherical and deformed cases, and the discrepancies in reflect the differences in the descriptions of the single-particle orbital between the two models.

Effective thermalization of a many-body dynamically localized Bose gas

Dynamical localization is the analog of Anderson localization in momentum space, where the system's energy saturates and the single-particle wave-functions are exponentially localized in momentum space. In the presence of interactions, in the context of a periodically kicked Bose gas, it has been argued that dynamical localization persists. Focusing on the Tonks (strongly interacting) regime, we show that the many-body dynamically localized phase is effectively thermal, a clear deviation from the breaking of ergodicity observed in standard many-body localized systems. We relate the effective temperature to the driving parameters, and thus quantitatively describe the loss of coherence at large distances in this phase. Contrary to the non-interacting case, the momentum distribution decays as a power-law at large momenta, characterized by an effectively thermal Tan's contact. This is a rare example where driving and many-body (dynamical) localization lead to an effectively ergodic state.

Electronic and magnetic properties of many-electron complexes in charged InAsxP1−x quantum dots in InP nanowires

We present here a microscopic theory of electronic complexes in charged $\mathrm{In}{\mathrm{As}}_{x}{\mathrm{P}}_{1\ensuremath{-}x}$ quantum dots in InP nanowires with a hexagonal cross section and determine the potential use of an array of such quantum dots as a synthetic spin chain for the possible construction of a topological qubit. The single-particle energies and wave functions are obtained by diagonalizing a microscopic atomistic tight-binding Hamiltonian of multiple quantum dots in the basis of $s{p}^{3}{d}^{5}{s}^{*}$ local atomic orbitals for a given random distribution of arsenic (As) vs phosphorus (P) atoms. The conduction band electronic states are found grouped into $s, p$, and $d$ quantum dot shells. For a double dot, the electronic shells can be understood in terms of interdot tunneling despite the random distribution of As atoms in each quantum dot. The single- and double-dot structures were charged with a finite number of electrons. The many-body Hamiltonian including Coulomb electron-electron interactions was constructed using single atomistic particle states and then diagonalized in the space of many-electron configurations. For a single dot filled with ${N}_{e}=1--7$ electrons, the ground state of a half-filled $p$-shell configuration with ${N}_{e}=4$ was found with total electronic spin $S=1$. The low-energy spectrum obtained using exact diagonalization of a Hamiltonian of a charged double dot filled with ${N}_{e}=8$ electrons, i.e., half-filled $p$ shells in each dot, was successfully fitted to the Hubbard-Kanamori and antiferromagnetic Heisenberg spin-1 Hamiltonians. The atomistic simulation confirmed the potential of InAsP/InP quantum dots in a nanowire for the design of synthetic spin chains.

Role of triaxiality in deformed halo nuclei

It is known that nuclear deformation plays an important role in inducing the halo structure in neutron-rich nuclei by mixing several angular momentum components. While previous theoretical studies on this problem in the literature assume axially symmetric deformation, we here consider non-axially symmetric deformations. With triaxial deformation, the $\Omega$ quantum number is admixed in a single-particle wave function, where $\Omega$ is the projection of the single-particle angular momentum on the symmetric axis, and the halo structure may arise even when it is absent with the axially symmetric deformation. In this way, the area of halo nuclei may be extended when triaxial deformation is considered. We demonstrate this idea using a deformed Woods-Saxon potential for nuclei with neutron number N=13 and 43.

Tensor force and deformation in even-even nuclei

The variational principle is used to build a model which describes open shell nuclei with ground state deformations. Hartree-Fock equations are solved by using single particle wave functions whose radial parts depend on the projection of the angular momentum on the quantization axis. Pairing effects are taken into account by solving Bardeen-Cooper-Schrieffer equations in each step of the minimisation procedure. The Gogny D1S finite-range interaction and an extension of it that includes tensor terms are consistently used in both parts of our calculations. The model is applied to study a set of isotopes with 34 protons and of isotones with 34 neutrons. Total energies, density distributions, their radii and single particle energies are analysed and the results of our calculations are compared with the available experimental data. We focused our attention on the effects of the deformation and of the tensor force on these observables. Our model describes open shell nuclei from a peculiar perspective and opens the possibility of future theoretical developments.

Ground and excited state characteristics of the nuclei with A = 6

The binding energy, the root-mean-square radius, the magnetic dipole moment, the electric quadrupole moment, and the moment of inertia of the nucleus 6Li are calculated by applying different models. The translation invariant shell model is applied to calculate the binding energy, the root-mean-square radius, and the magnetic dipole moment by using two- and three-body interactions. Also, the spectra of the nuclei with A = 6 are calculated by using the translation-invariant shell model. Moreover, the ft-value of the allowed transition: 6He(Jπ=0+;T=1)β- → 6Li(Jπ=1+;T'=1) is also calculated. Furthermore, the concept of the single-particle Schrodinger fluid for axially symmetric deformed nuclei is applied to calculate the moment of inertia of 6Li. Also, we calculated the magnetic dipole moment and the electric quadrupole moment of the nucleus 6Li in this case of axially symmetric shape. Moreover, the nuclear superfluidity model is applied to calculate the moment of inertia of 6Li, based on a single-particle deformed anisotropic oscillator potential added to it a spin-orbit term and a term proportional to the square of the orbital angular momentum, as usual in this case. The single-particle wave functions obtained in this case are used to calculate the magnetic dipole moment and the electric quadrupole moment of 6Li.

Inelastic Form Factor Calculations for 46,48,50Ti Isotopes Using Tassie Model

Inelastic form factors of electrical transition have been calculated for 46,48,50Ti isotopes using the Tassie model. The form factors have been calculated for different exciting energies. The harmonic oscillator (HO) wave function has been used as a single-particle wave function. The model space has been considered as 1f7/2, 2p3/2, 2p1/2, and 2f5/2. Gx1 has been used as effective interaction in all calculations. In all calculations, the effective charge has been considered as 1.5e for proton and 0.5e for neutron. All obtained results have been compared with data from an experiment. The calculations show the Tassie model gives a good description of longitudinal form factors of 46,48,50Ti isotopes in E(2+) transitions as compared with experimental data, especially in the region below 2 fm−1 of momentum transfer, but in the E(4+), the theoretical results deviated slightly from experimental data especially in the region greater than 1.5 fm−1 of momentum transfer.

Studying Inelastic Electron Scattering of Magnetic Form Factors M1 for 48Ca Nucleus Using Skyrme Interaction

The M1 inelastic scattering form factors from 48Ca nucleus have been investigated through nuclear shell theory. The nuclear effective two body interaction fpbm of Richter has been utilized to generate the two body wave vectors with Harmonic oscillator wave function as a single particle wave function in Fp shell model space. Core polarization effect has been used to include the discarded space (core+ higher configuration) via model space with realistic interaction of Skyrme Sly5 to couple the model space active particles with the pair (particle-hole) excited in energy of 2ћω. Calculated results has been compared to available experimental data.

Probing nuclear forces beyond the nuclear drip line: the cases of $$^{16}$$F and $$^{15}$$F

The unbound proton-rich nuclei $^{16}$F and $^{15}$F are investigated experimentally and theoretically. Several experiments using the resonant elastic scattering method were performed at GANIL with radioactive beams to determine the properties of the low lying states of these nuclei. Strong asymmetry between $^{16}$F-$^{16}$N and $^{15}$F-$^{15}$C mirror nuclei is observed. The strength of the $nucleon-nucleon$ effective interaction involving the loosely bound proton in the $s_{1/2}$ orbit is significantly modified with respect to their mirror nuclei $^{16}$N and $^{15}$C. The reduction of the effective interaction is estimated by calculating the interaction energies with a schematic zero-range force. It is found that, after correcting for the effects due to changes in the radial distribution of the single-particle wave functions, the mirror symmetry of the $n-p$ interaction is preserved between $^{16}$F and $^{16}$N, while a difference of 63\% is measured between the $p-p$ versus $n-n$ interactions in the second excited state of $^{15}$F and $^{15}$C nuclei. Several explanations are proposed.

Kernels for non interacting fermions via a Green’s function approach with applications to step potentials

The quantum correlations of N non interacting spinless fermions in their ground state can be expressed in terms of a two-point function called the kernel. Here we develop a general and compact method for computing the kernel in a general trapping potential in terms of the Green’s function for the corresponding single particle Schrödinger equation. For smooth potentials in one dimension the method allows a simple alternative derivation of the local density approximation for the density and of the sine kernel in the bulk part of the trap in the large N limit. It also recovers the density and the kernel of the so-called Airy gas at the edge. This method allows the analysis of the quantum correlations in the ground state when the potential has a singular part with a fast variation in space. For the square step barrier of height V 0 in one dimension, we derive explicit expressions for the density and for the kernel. For large Fermi energy μ > V 0 it describes the interpolation between two regions of different densities in a Fermi gas, each described by a different sine kernel. Of particular interest is the critical point of the square well potential when μ = V 0. In this critical case, while there is a macroscopic number of fermions in the lower part of the step potential, there is only a finite O(1) number of fermions on the shoulder, and moreover this number is independent of μ. In particular, the density exhibits an algebraic decay ∼1/x 2, where x is the distance from the jump. Furthermore, we show that the critical behavior around μ = V 0 exhibits universality with respect to the shape of the barrier. This is established (i) by an exact solution for a smooth barrier (the Woods–Saxon potential) and (ii) by establishing a general relation between the large distance behavior of the kernel and the scattering amplitudes of the single-particle wave-function.

Dynamical transition orbitals: A particle-hole description in real-time TDDFT dynamics.

We expand the concept of natural transition orbitals in the context of real-time time-dependent density functional theory (RT-TDDFT) and show its application in practical calculations. Kohn-Sham single-particle wavefunctions are propagated in RT-TDDFT simulation, and physical properties remain invariant under their unitary transformation. In this work, we exploit this gauge freedom and expand the concept of natural transition orbitals, which is widely used in linear-response TDDFT, for obtaining a particle-hole description in RT-TDDFT simulation. While linear-response TDDFT is widely used to study electronic excitation, RT-TDDFT can be employed more generally to simulate non-equilibrium electron dynamics. Studying electron dynamics in terms of dynamic transitions of particle-hole pairs is, however, not straightforward in the RT-TDDFT simulation. By constructing natural transition orbitals through projecting time-dependent Kohn-Sham wave functions onto occupied/unoccupied eigenstate subspaces, we show that linear combinations of a pair of the resulting hole/particle orbitals form a new gauge, which we refer to as dynamical transition orbitals. We demonstrate the utility of this framework to analyze RT-TDDFT simulations of optical excitation and electronic stopping dynamics in the particle-hole description.

Role of continuum in nuclear direct reactions with one-neutron halo nuclei: A one-dimensional model

We study the evolution of a single-particle wave function during the collision of a one dimensional potential well by another well, which can be regarded as a simple model for the problem of the scattering of a one-neutron halo nucleus by another nucleus. This constitutes an effective three-body problem, whose solution in three dimensions can be extremely complicated, particularly when breakup and rearrangement channels are to be considered. Our one-dimensional model provides the essential three-body nature of this problem, and allows for a much simpler application and assessment of different methods of solution. To simplify further the problem, we assume that the potential well representing the projectile moves according to a predetermined classical trajectory, although the internal motion of the "valence" particle is treated fully quantum-mechanically. This corresponds to a semiclassical approach of the scattering problem. Different approaches are investigated to understand the dynamics involving one-body halo-like systems: the "exact" time-dependent solution of the Schr\"odinger equation is compared to a numerical continuum-discretized coupled-channels (CC) calculation presenting various model cases including different reaction channels. This framework allows us to discuss the reaction mechanism and the role of continuum, whose inclusion in the CC calculation results to be crucial to reproduce the "exact" solution, even when the initial and final states are well bound. We also link each dynamical situation with analogous problem solved in a three dimensional (3D) CC framework, discussing the main challenges experienced in the usual 3D models.

About the Nature of the Quantum System—An Examination of the Random Discontinuous Motion Interpretation of the Quantum Mechanics

What is the quantum system? Consider the wave-function of the electron—what we call “single particle wave-function”—and assume that it contains N wave-packets. If we pass all the wave-packets through an electric field, all are deflected, as if each one of them contains an electron. However, if we bring any two wave-packets to travel close to one another, they don’t repel one another, as if at least one of them contains no charge. In trying to solve the measurement problem of the quantum mechanics (QM), different interpretations were proposed, each one coming with a particular ontology. However, only one interpretation paid explicit attention to the contradiction mentioned above. This interpretation was proposed by S. Gao who named it “random discontinuous motion” (RDM), because it assumes the existence of a particle that jumps from place to place at random. The particle carries all the physical properties of the respective type of particle, mass, charge, magnetic momentum, etc. It jumps under the control of an “instantaneous condition” about which Gao did not give details so far. Along with presenting problems of the QM that this interpretation solves, this text reveals difficulties vis-a-vis entanglements and the special relativity.

Root Mean Square Radius of 6Li Nucleus using Cluster Model Wavefunction

The calculation of root mean square nuclear charge radius is one of the most important nuclear parameters regarding the size and structure of the nucleus. In this paper calculations for root mean square (r.m.s.) radius of the ground state of 6Li nucleus using high energy electron scattering is presented. The initial work involves writing first the cluster model wavefunction employing the resonating group method, generator coordinate method and complex generator coordinate technique. The wavefunction is written with definite parity, spin, total angular momentum and relative motion between the alpha cluster and deuteron cluster and the center-of-mass of the two clusters. The application of complex generator coordinate technique transforms the cluster model wavefunction into antisymmetrized products of single particle wavefunction written in terms of single particle co-ordinates, the centerof-mass coordinates, parameter coordinates and generator coordinates. The width parameters of alpha and deuteron clusters are adjusted to obtain predictions close to experimental values.

CLUSTER SHELL MODEL WAVE FUNCTION: STRUCTURE OF 6LI NUCLEUS

In this paper cluster model wave function for 6Li using Shell Model with definite parity and angular momentum is written along with cluster co-ordinates, which are relative to the center-of-mass of various clusters and involve with parameters. These parameters can be adjusted to some extent to obtain predictions close to experimental properties. The cluster model wave function is written along with resonating group method (RGM) and the Complex Generator Coordinate Technique (CGCT). The Complex Generator Coordinate Technique allows the transformation of the cluster model wave function written in terms of cluster co-ordinates into anti-symmetrized product of single particle wave function. This wave function is written in terms of single particle co-ordinates, the center-of-mass co-ordinates, parameter coordinates and generator coordinates.

Single- and two-photon interference within the simulated photon wave function in coordinate representation

The main provisions of the construction of the single-particle wave function of a photon in the coordinate representation are expounded and its application to the description of single-photon interference phenomena is discussed. The connection between the concepts of electromagnetic field strengths used in quantum electrodynamics and the characteristics of wave packets in photon quantum mechanics is substantiated. For illustration, we consider the relationship between the structure of the photon wave function in coordinate representation and the polarization of photon in the formalism of quantum transition amplitudes. The meaning of the concept of linear polarization of a photon in the state of a wave packet is explained. The wave function is constructed in the coordinate representation of the state of two entangled photons at the exit from a nonlinear crystal by modeling the polarization vectors and Gaussian momentum distribution for each of the photons.

Probing the Planck scale: the modification of the time evolution operator due to the quantum structure of spacetime

The propagator which evolves the wave-function in non-relativistic quantum mechanics, can be expressed as a matrix element of a time evolution operator: i.e. GNR(x) = 〈x2|UNR(t)|x1〉 in terms of the orthonormal eigenkets |x〉 of the position operator. In quantum field theory, it is not possible to define a conceptually useful single-particle position operator or its eigenkets. It is also not possible to interpret the relativistic (Feynman) propagator GR(x) as evolving any kind of single-particle wave-functions. In spite of all these, it is indeed possible to express the propagator of a free spinless particle, in quantum field theory, as a matrix element 〈x2|UR(t)|x1〉 for a suitably defined time evolution operator and (non-orthonormal) kets |x〉 labeled by spatial coordinates. At mesoscopic scales, which are close but not too close to Planck scale, one can incorporate quantum gravitational corrections to the propagator by introducing a zero-point-length. It turns out that even this quantum-gravity-corrected propagator can be expressed as a matrix element 〈x2|UQG(t)|x1〉. I describe these results and explore several consequences. It turns out that the evolution operator UQG(t) becomes non-unitary for sub-Planckian time intervals while remaining unitary for time interval is larger than Planck time. The results can be generalized to any ultrastatic curved spacetime.

Spectroscopic factors, overlaps, and isospin symmetry from an R -matrix point of view

Background: Spectroscopic factors, overlaps, and isospin symmetry are often used in conjunction with single-particle wave functions for the phenomenological analysis of nuclear structure and reactions. Many differing prescriptions for connecting these quantities to physically relevant asymptotic normalization constants or widths are available in the literature, but their relationship and degree of validity are not always clear. Purpose: This paper derives relationships among the above quantities of interest using well-defined methodology and starting assumptions. Method: $R$-matrix theory is used as the primary tool to interoperate between the quantities of interest to this work. Particular attention is paid to effects arising from beyond the nuclear surface, where isospin symmetry is strongly violated. Results: Relationships among the quantities of interest are derived. Example applications of these methods to mirror levels in nucleon+${}^{12}{\rm C}$, nucleon+${}^{16}{\rm O}$, and nucleon+${}^{26}{\rm Al}$ are presented. A new approach to multi-level mirror symmetry is derived and applied to the first three $2^+$ states of ${}^{18}{\rm O}$ and ${}^{18}{\rm Ne}$. Conclusions: The relationship between the quantities of interest is clarified and certain procedures are recommended. It is found that the asymptotic normalization constant of the second $2^+$ state in ${}^{18}{\rm Ne}$ deduced from the mirror state in ${}^{18}{\rm O}$ is significantly larger than found in previous work. This finding has the effect of increasing the ${}^{17}{\rm F}(p,\gamma){}^{18}{\rm Ne}$ reaction rate in novae.

Robustness and independence of the eigenstates with respect to the boundary conditions across a delocalization-localization phase transition

We focus on the many-body eigenstates across a localization-delocalization phase transition. To characterize the robustness of the eigenstates, we introduce the eigenstate overlaps $\mathcal{O}$ with respect to the different boundary conditions. In the ergodic phase, the average of eigenstate overlaps $\bar{\mathcal{O}}$ is exponential decay with the increase of the system size indicating the fragility of its eigenstates, and this can be considered as an eigenstate-version butterfly effect of the chaotic systems. For localized systems, $\bar{\mathcal{O}}$ is almost size-independent showing the strong robustness of the eigenstates and the broken of eigenstate thermalization hypothesis. In addition, we find that the response of eigenstates to the change of boundary conditions in many-body localized systems is identified with the single-particle wave functions in Anderson localized systems. This indicates that the eigenstates of the many-body localized systems, as the many-body wave functions, may be independent of each other. We demonstrate that this is consistent with the existence of a large number of quasilocal integrals of motion in the many-body localized phase. Our results provide a new method to study localized and delocalized systems from the perspective of eigenstates.

Tunable Flux Vortices in Two-Dimensional Dirac Superconductors.

The nontrivial geometry encoded in the quantum mechanical wave function has important consequences for both noninteracting and interacting systems. Yet, our understanding of the relationship between geometrical effects in noninteracting systems and their interacting counterparts is far from complete. Here, we demonstrate how the single-particle Berry curvature associated with the normal phase in two dimensions modifies the fluxoid quantization of a Bardeen-Cooper-Schrieffer superconductor. A discussion of the experimental scenarios where this anomalous quantization is expected is provided. Our work demonstrates the importance of variational Ansätze in making a clear connection between the Berry phases of single-particle and many-body wave functions.