Equation In Momentum Space Research Articles

Virtual states in the coupled-channel problems with an improved complex scaling method

We improve the complex scaling method (CSM) to obtain virtual states, which is challenging in the conventional CSM. Our approach solves the Schrödinger equation in momentum space as an eigenvalue problem by choosing flexible contours. It proves to be highly effective in identifying the poles across different Riemann sheets in multichannel scatterings. This approach is more straightforward and efficient than employing the Lippmann-Schwinger equation and using a root-finding algorithm to search for the zeros of the Fredholm determinant. This advancement significantly extends the capabilities of the CSM in accurately characterizing the virtual states in quantum systems. Published by the American Physical Society 2024

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Effect of pulse duration on the above-threshold ionization of a hydrogen atom irradiated by a 400 nm intense laser.

The photoelectron emission spectra generated by the interaction between ultrashort intense laser pulses and atoms can reveal the ultrafast dynamics of electrons. By using the numerical solution of the time-dependent Schrödinger equation in momentum space, the photoelectron emission spectra of atoms irradiated by 400 nm intense lasers with different durations of the pulse has been investigated. In the photoelectron emission spectrum, in addition to the above-threshold ionization peaks due to ionization interference in multiple cycles and the sideband peaks mainly due to the interference of ionized electrons at different moments along the rising edge of the laser pulse envelope, additional peaks of photoelectron emission whose intensity appears to oscillate with the increasing duration of the laser pulse can also be observed. Based on strong-field approximation and the population's analysis of the bound state, it is found that these photoelectron peaks originate from the ionization of the excited state and the oscillations of these peaks are due to the superposition of their peak energy positions with the sideband energy positions. Furthermore, it is demonstrated that the energy positions of the maximum intensity of the photoelectron emission spectra move towards the higher energy end as the duration of the driving laser pulse extends. This phenomenon can be attributed to the fact that the main moment of ionization of atoms changes with the increasing duration of the driving laser pulse, thus allowing the real-time ionization of atoms to be probed using photoelectron emission spectra.

Trion clustering structure and binding energy in two-dimensional semiconductor materials: Faddeev equations approach

In this work, we develop the basic formalism to study trions in semiconductor layered materials using the Faddeev equations in momentum space for three different particles lying in two dimensions. We solve the trion Faddeev coupled integral equations for both short-range one-term separable Yamaguchi potential and Rytova-Keldysh (RK) interaction applied to the MoS$_2$ layer. We devise two distinct regularization methods to overcome the challenge posed by the repulsive electron-electron RK potential in the numerical solution of the Faddeev equations in momentum space. The first method regulates the repulsive interaction in the infrared region, while the second regulates it in the ultraviolet region. By extrapolating the trion energy to the situation without screening, the two methods gave consistent results for the MoS$_2$ layer with a trion binding energy of $-49.5(1)$~meV for the exciton energy of $-753.3$~meV. We analyzed the trion structure for the RK and Yamaguchi potentials in detail, showing their overall similarities and the dominant cluster structure, where the strongly bound exciton is weakly bound to an electron. We found that this property is manifested in the dominance of two of the Faddeev components over the one where the hole is a spectator of the interacting electron pair.

An analytical solution of Balitsky–Kovchegov equation using homotopy perturbation method

An approximate analytical solution of the Balitsky–Kovchegov (BK) equation using the homotopy perturbation method (HPM) is suggested in this work. We carried out our work in perturbative Quantum Chromodynamics (QCD) (pQCD) dipole picture of deep inelastic scattering (DIS). The BK equation in momentum space with some change of variables and truncation of the Balitsky–Fadin–Kuraev–Lipatov (BFKL) kernel can be reduced to Fisher–Kolmogorov–Petrovsky–Piscounov (FKPP) equation. The observed geometric scaling phenomena are similar to the travelling wave solution of the FKPP equation. We solved the BK equation using the HPM. The obtained solution in this work also suggests the travelling wave nature of the measured scattering amplitude [Formula: see text] plotted at various rapidities. We also extracted the saturation momentum, [Formula: see text], from the obtained solution and plotted it against different rapidities. The result obtained in this work can be helpful for various phenomenological studies in high-density QCD.

First-Principles Fermi Acceleration in Magnetized Turbulence.

This Letter provides a concrete implementation of Fermi's model of particle acceleration in magnetohydrodynamic (MHD) turbulence, connecting the rate of energization to the gradients of the velocity of magnetic field lines, which it characterizes within a multifractal picture of turbulence intermittency. It then derives a transport equation in momentum space for the distribution function. This description is shown to be substantiated by a large-scale numerical simulation of strong MHD turbulence. The present general framework can be used to model particle acceleration in a variety of environments.

Influence of multiphoton resonance excitation on the above-threshold ionization of a hydrogen atom.

The photo-electron emission of a hydrogen atom irradiated by an ultraviolet laser pulse is investigated by numerically solving the time-dependent Schrödinger equation in momentum space. A subpeak structure with high intensity is observed in the photo-electron emission spectrum, and the peak of the enhanced structure shifts to a higher energy as the laser intensity increases. Through the strong-field approximation and the analysis of the population of the bound state , it is found that this subpeak structure is generated from the interference between the ionized electrons from the ground state and the ionized electrons from the 2p state after the resonant transition from the ground state to the 2p state. Analyzing the change rule of the photo-electron emission spectrum can further deepen the understanding of the energy change of the dressed bound state for an atom irradiated by an intense laser pulse.

Solving time-dependent Schrödinger equation in momentum space with application to strong-field problems

We present a numerical method to solve the time-dependent Schrödinger equation (TDSE) in momentum representation (p-space). We show that the method is practically useful and easier than the coordinate space (r-space) method when continuous states are involved. For a single-active electron (SAE) atom, the numerically complete eigenset can be accurately constructed in p-space by quadrature method which bypasses the singularities in the Coulombic kernel. Although there is an ingenious Lande subtraction for dealing with the singularity but is not straightforward. We formulate the time marching algorithms for an SAE atom in linearly polarized (LP) laser pulse and in circularly polarized (CP) pulse, respectively. We show calibrations to literature results to justify the formulations. Argon in a resonant and a nonresonant CP pulse are investigated and show distinct properties from the case of tunneling regime. Together with the currently available powerful graphics processing unit (GPU) for massively parallel computing, the p-space method could provide a useful alternative tool for some problems such as atoms in intense light pulses.

Deuteron-deuteron elastic and three- and four-body breakup scattering using the Faddeev-Yakubovskii equations

The deuteron-deuteron elastic and three- and four-body breakup scattering cross section have been calculated using the Faddeev-Yakubovskii (FY) chain-of-partition momentum-space equations. In this calculation the initial two-cluster potential is split into separable and nonseparable components, and the effective potential is reduced to elastic two-body and three- and four-body breakup open channels and a closed-channel many-body contribution. The closed-channel contribution is determined by minimizing a variational bound. The Coulomb interaction was included by expanding the initial and final Coulomb states in a Coulomb-Sturmian basis. The three sets of chain-of-partition integral equations were solved for the elastic and three- and four-body breakup scattering amplitudes. The calculations were performed for the $S=2\mathrm{spin}/L=0$ angular-momentum state. The elastic and double-breakup calculations were performed for energies up to $E=5.48\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$, while the single-breakup calculations were performed for energies up to $E=4.17\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. In the case of elastic scattering, the calculated scattering length of ${}^{5}{a}_{dd}=7.8\ifmmode\pm\else\textpm\fi{}0.3$ fm is in good agreement with a FY cluster reduction calculation. The calculated phase shift is smaller than that predicted by the resonating group model and this difference is believed to be due to the differences in the potential and calculation methods. The breakup cross sections were calculated as a function of initial deuteron momentum and fragmented-deuteron momentum. The $d+d\ensuremath{\rightarrow}d+n+p$ cross sections were compared with neutron yield measurements and, while the measurements also included the $Lg0$ components, the general features were consistent. Estimates of the calculational uncertainties/bias are provided.

The Staggered Fermion for the Gross-Neveu Model at Non-Zero Temperature and Density

The 2 + 1d Gross-Neveu model with finite density and finite temperature is studied by the staggered fermion discretization. The kinetic part of this staggered fermion in momentum space is used to build the relation between the staggered fermion and Wilson-like fermion. In the large Nf limit (the number Nf of staggered fermion flavors), the chiral condensate and fermion density are solved from the gap equation in momentum space, and thus the phase diagram of fermion coupling, temperature and chemical potential is obtained. Moreover, an analytic formula for the inverse of the staggered fermion matrix is given explicitly, which can be calculated easily by parallelization. The generalization to the 1 + 1d and 3 + 1d cases is also considered.

Momentum space spinning correlators and higher spin equations in three dimensions

In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator overline{psi}psi is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of overline{psi}psi operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.

The infinite square well potential in momentum space

We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law potential in the limit that the power approaches infinity.

Coulomb bare interactions in inhomogeneous 4-layer graphene structures

We investigate Coulomb bare interactions in 4-layer graphene systems, taking into account the inhomogeneity of dielectric environment. By solving Poisson equation in momentum space, 16 elements of Coulomb potential tensor have been formed with analytical expressions. The formulae illustrate that Coulomb bare interactions in inhomogeneous 4-layer graphene system differ noticeably from those in homogeneous one. Nevertheless, both intra- and interlayer Coulomb potentials become independent with dielectric constants of contacting media, but they are decided by those of dielectric layer surrounding the system in long wavelength approximation. Besides, numerical results demonstrate that Coulomb bare interactions decrease considerably with the increase in carrier density in graphene sheets as well as their separated distance. In all investigations, the inhomogeneity of dielectric background should not be neglected for improvement in the model.

Renormalization and scale evolution of the soft-quark soft function

Soft functions defined in terms of matrix elements of soft fields dressed by Wilson lines are central components of factorization theorems for cross sections and decay rates in collider and heavy-quark physics. While in many cases the relevant soft functions are defined in terms of gluon operators, at subleading order in power counting soft functions containing quark fields appear. We present a detailed discussion of the properties of the soft-quark soft function consisting of a quark propagator dressed by two finite-length Wilson lines connecting at one point. This function enters in the factorization theorem for the Higgs-boson decay amplitude of the h → γγ process mediated by light-quark loops. We perform the renormalization of this soft function at one-loop order, present a conjecture for its two-loop anomalous dimension and discuss solutions to its renormalization-group evolution equation in momentum space, in Laplace space and in the “diagonal space”, where the evolution is strictly local in the momentum variable.

Generalized Uncertainty Principle in Cosmology with Supersymmetry Quantum Mechanics

In this paper, we exactly solve the Wheeler-DeWitt equation in the presence of a conformally coupled scalar field and in the context of the generalized uncertainty principle (GUP). The GUP-corrected Wheeler—De Witt equation in momentum space leads us to introduce factorization method. This method helps us to obtain the exact solution for the corresponding system. So, here we factorize the second-order equation in terms of first-order operators. These first-order operators help us to arrange partner potential and superpotential. Also, we achieve the general quantum stats and energy spectrum for GUP with conformally coupled scalar field system. Also, we show that the stability of a system with the energy spectrum with some conditions.

First-principles Demonstration of Diffusive-advective Particle Acceleration in Kinetic Simulations of Relativistic Plasma Turbulence

Abstract Nonthermal relativistic plasmas are ubiquitous in astrophysical systems like pulsar wind nebulae and active galactic nuclei, as inferred from their emission spectra. The underlying nonthermal particle acceleration (NTPA) processes have traditionally been modeled with a Fokker–Planck (FP) diffusion-advection equation in momentum space. In this Letter, we directly test the FP framework in ab initio kinetic simulations of driven magnetized turbulence in relativistic pair plasma. By statistically analyzing the motion of tracked particles, we demonstrate the diffusive nature of NTPA and measure the FP energy diffusion (D) and advection (A) coefficients as functions of particle energy . We find that scales as in the high-energy nonthermal tail, in line with second-order Fermi acceleration theory, but has a much weaker scaling at lower energies. We also find that A is not negligible and reduces NTPA by tending to pull particles toward the peak of the particle energy distribution. This study provides strong support for the FP picture of turbulent NTPA, thereby enhancing our understanding of space and astrophysical plasmas.

Trion formation and unconventional superconductivity in a three-dimensional model with short-range attraction

A three-fermion problem in a three-dimensional lattice with anisotropic hopping is solved by discretizing the Schrödinger equation in momentum space. Interparticle interaction comprises on-site Hubbard repulsion and in-plane nearest-neighbor attraction. By comparing the energy of three-fermion bound clusters (trions) with the energy of one pair plus one free particle, a trion formation threshold is accurately determined, and the region of pair stability is mapped out. It is found that the “close-packed” density of fermion pairs, which is associated with a maximum pair condensation temperature in this model, is the highest in a strongly anisotropic case. It is also argued that pair superconductivity with the highest critical temperature is always close to trion formation, which makes the system prone to phase separation and local charge ordering.

Microscopic Calculation of the Optical Model Potential from One Boson Exchange Poten­tials

A new method for calculating the optical model potential from One Boson Exchange Potentials (OBEPs) is developed. The G-matrix is calculated by solving the Bethe-Goldstone equation in momentum space. Using vector brackets these G-matrix elements can be transformed from the center of mass representation into the laboratory system. This allows the evaluation of the (r-matrix interaction between nucleons in bound states and those in a plane-wave state. The lowest order contribution to the real part of the potential comes from the Hartree-Fock term, while the lowest order contribution to the imaginary part comes from the two-particle-one-hole (2p1h) diagram. Calculations for 16O and 40Ca have been carried out. Local approximations are obtained by describing the results for the central part in terms of a Woods-Saxon potential and those for the spin-orbit part in terms of the corresponding derivatives. The dependence of these potentials on energy and angular momentum is discussed. The parameters for these local approximations are in good agreement with empirical fits.

Deuteron- α scattering: Separable versus nonseparable Faddeev approach

{\bf Background} Deuteron induced reactions are widely used to probe nuclear structure and astrophysical information. Those (d,p) reactions may be viewed as three-body reactions and described with Faddeev techniques. {\bf Purpose} Faddeev-AGS equations in momentum space have a long tradition of utilizing separable interactions in order to arrive at sets of coupled integral equations in one variable. However, it needs to be demonstrated that observables calculated based on separable interactions agree exactly with those based on nonseparable forces. {\bf Methods} Momentum space AGS equations are solved with separable and nonseparable forces as coupled integral equations. {\bf Results} Deuteron-alpha scattering is calculated via momentum space AGS equations using the CD-Bonn neutron-proton force and a Woods-Saxon type neutron(proton)-$^4$He force, for which the Pauli-forbidden S-wave bound state is projected out. Elastic as well as breakup observables are calculated and compared to results in which the interactions in the two-body sub-systems are represented by separable interactions derived in the Ernst-Shakin-Thaler (EST) framework. {\bf Conclusions} We find that the calculations based on the separable representation of the interactions and the original interactions give results that are in excellent agreement. Specifically, integrated cross sections and angular distributions for elastic scattering agree within $\approx$ 1\%, which is well below typical experimental errors. In addition, the five-fold differential cross sections corresponding to breakup of the deuteron agree extremely well.

One-body Langevin dynamics in heavy-ion collisions at intermediate energies

We present a new framework to treat the dissipation and fluctuation dynamics associated with nucleon-nucleon scattering in heavy-ion collisions. The two-body collision processes are effectively described in terms of the diffusion of nucleons in viscous nuclear media, governed by a set of Langevin equations in momentum space. The new framework combined with the usual mean field dynamics can be used to simulate heavy-ion collisions at intermediate energies. As a proof of principle, we simulate Au + Au reactions and obtain results consistent with other existing codes under the same constrained conditions. We also study the formation of fragments in Sn + Sn reactions at 50 MeV/nucleon, and results are discussed and compared with two other models commonly employed for collisions.