Search Results (875)

This study investigates the electronic and nonlinear optical properties of coupled GaN/AlN quantum dots using a numerical approach based on coordinate transformation combined with the finite difference method (FDM). The Schrödinger equation is solved to determine the electronic energy levels and wave functions of the system, which are subsequently used to evaluate the nonlinear optical rectification (NOR) response. Since numerical simulations become computationally expensive for large quantum dot systems, several regression-based models, including Polynomial Regression, Ridge Regression, LASSO, and Elastic Net, are trained on high-fidelity numerical data. These models learn the relationship between structural parameters and the resulting electronic and optical properties, enabling fast and reliable predictions for larger quantum dot configurations. The predictive performance of the ML models is assessed by comparing their results with the numerical simulations, showing excellent agreement while significantly reducing computational effort. The proposed hybrid physics–machine learning framework therefore provides an efficient and reliable approach for predicting the electronic and nonlinear optical behavior of coupled GaN/AlN quantum dots. Full article

The critical binding of quantum states in Screened Coulomb Potentials such as Yukawa/Debye, Hulthén, and ECSC (Exponential Cosine Screened Coulomb) potentials is of perennial interest and relevance in many fields of science, ranging from nuclear and particle physics; plasma physics, astrophysics, cosmology, and nuclear fusion; physical chemistry, condensed matter, and materials physics; to synthetic nanostructures and nanophotonics. The purpose of this paper is to heuristically explore two related mysteries, one new, the other more than 50 years old. The solutions to these mysteries have implications for a much broader class of potentials, those addressed by Klaus and Simon. In our recent paper we presented numerical calculations using the Phase Method (PM), which is accurate to 60 digits and to screening lengths $\mathcal{D}\le {10}^3$ au and $l=0$–20 of the critical binding parameters for these potentials and, for Yukawa and ECSC, $l=0$–12 to $\mathcal{D}\le {10}^5$ au, at 30 digits. In doing so, we discovered an anomalous period-40 sawtooth structure in the critical parameters of the ECSC potential that is not observed for the Yukawa potential. In this second paper, we quantitatively explain the origin and periodicity of this newly discovered structure. To do so, we use two complementary approaches: a “neoclassical” (NC) variant of conventional semiclassical phase-space quantization and the PM for very precise fully quantum calculations. The observed period-40 sawtooth structure is quantitatively explained in terms of a novel “tick-tock” mechanism. The periodicity is calculated in terms of the ratio of phase-space integrals for the primary and secondary potential wells. A quartic double-well potential is used as a simple model to further illustrate the tick-tock mechanism. Using the NC method, an approximate expression is derived to predict the locations of tick-tock glitches from higher-order wells; it is confirmed by a PM calculation up to $\mathcal{D}\le {10}^6$ au. The second mystery is a strangely linear dependence of the total number of bound states vs. screening length for both the Yukawa and ECSC potentials. Using the PM, we confirm and extend these empirical relations. We show, using the PM, that an approximate trivariate linear relation between the square root of the critical screening length $\sqrt{D_c}$, state number n, and angular momentum l applies to these potentials. This, plus a geometrical state accumulation argument, solve the second mystery. We show these properties derive from the scaling relation between screening length and coupling constant and, as such, are predicted to be applicable to the whole class of potentials. These results are expected to be of both theoretical interest and experimental relevance when interpreting spectra or calculating thermal properties. The significance of these results, and the applicability of these methods and conclusions to a vast array of related potentials, is briefly discussed. Full article

This paper focuses on the construction of exact solutions for the derivative nonlinear Schrödinger equation. By reformulating the generalized Darboux transformation, we provide a simplified representation for n-fold solutions. We systematically classify the physical structures of the second-, third-, and fourth-order solutions, identifying various patterns ranging from multi-soliton interactions to breather-type and periodic waves. In particular, we clarify how the reductions of spectral parameters determine localization, oscillation, and coalescence effects, and we discuss the physical implications of the resulting soliton, breather, bound-state, and periodic structures. These results provide a comprehensive dynamical classification, deepening the understanding of nonlinear wave propagation governed by the derivative nonlinear Schrödinger equation. Full article

We present a theoretical study focused on the photoelectron spectrum of near-infrared (NIR) laser-driven ionization of hydrogen atoms by attosecond pulse trains composed of several HHs of the former. We analyze the effects of increasing the intensity of the NIR probe laser to account for the interference of multiple quantum pathways arising from mainbands formed in ionization by the attosecond pulse train within the strong-field approximation (SFA) beyond the commonly used first-order perturbative (in the NIR laser intensity) reconstruction of attosecond beating by interference of two-photon transitions (RABBIT). The structure of the energy bands formed in the photoelectron spectrum is governed by quantum interferences of the photoelectron wave packet released within one optical cycle of the NIR probe laser field—intracycle interference—and by the number of active high harmonic components, leading to higher-order Fourier contributions as a function of the NIR–XUV relative phase delay. We show that Fourier terms can be interpreted in terms of well-defined semiclassical trajectories. Our results demonstrate a significant departure from the standard two-path quantum-interference RABBIT picture, showing that both the phase-dependent oscillations of mainbands and sidebands and the extracted phase delays depend strongly on the probing laser intensity. The predictions of the SFA reveal that the above-threshold ionization bands exhibit systematic splitting and oscillation patterns as a function of the NIR intensity. SFA predictions are compared with results obtained within ab initio solutions of the time-dependent Schrödinger equation (TDSE), showing an excellent agreement, which evidences the minor effect of the Coulomb potential of the remaining ion on the escaping photoelectron for high energy above-threshold ionization. The precise study of the SFA reference phases is essential for the determination of the effect of the Coulomb potential on the escaping photoelectron for what these findings provide new insights into attosecond chronoscopy in the strong-field regime. Full article

In this work, we investigate the spectral, information-theoretic, and thermodynamic properties of a fractal Schrödinger system with position-dependent mass subject to an effective semiconductor-like confinement. We employ a fractal momentum operator and a Von Roos Hamiltonian with BenDaniel–Duke ordering to obtain exact analytical solutions for the energy spectrum and wave functions. The interplay between the fractal parameter $\alpha$, the effective lattice scale $l_0$, and the harmonic confinement strength $\omega$ is explored. We perform a comprehensive analysis of the Shannon entropy, Fisher information, and Fisher–Shannon complexity in both coordinate and momentum spaces. Our results demonstrate that these parameters directly control the localization–delocalization transition and the informational architecture of the quantum states, while satisfying the entropic and Fisher uncertainty relations. Furthermore, we derive the exact partition function and the corresponding thermodynamic properties (free energy, internal energy, entropy, and specific heat) of the system. The analytical framework presented offers valuable insights into the spectral, information-theoretic, and thermodynamic behavior of quantum systems in fractal semiconductor-like environments. Full article

We address the paradoxical transformation of a classical-mechanical particle motion when the space and time scales of observation pass below the uncertainty principle threshold. This is analyzed in the language of classical statistical mechanics, considering specifically many-particle systems inhomogeneous along one spatial direction. We employ the density functional formalism in its square-gradient form and find: (i) The macroscopic solution is analogous to the classical trajectory of a particle under a potential of force given by (minus) the free energy density. Whereas, (ii) fluctuations around the solution in (i) are equal to the quantum-mechanical wave functions of a particle under a potential given by the curvature of the free energy density. We illustrate this situation with three textbook examples: A particle in a box, the harmonic oscillator, and the hydrogen atom. We show that their time-independent Schrödinger equation wave functions describe, respectively, the fluctuations of a fluid interface, of critical point fluctuations, and of a confined ideal gas. At large scales, sharp probability distributions make fluctuations irrelevant; the vanishing of the first variation yields the macroscopically observable statistical-mechanical non-uniformity, equivalent to the classical particle trajectory. But at sufficiently small scales, with necessarily very few particles, distributions appear much wider, fluctuations dominate, and one obtains the Schrödinger equation (for the microscopic potential). Full article

In this work, we report an exactly solvable quantum model featuring a spin-dependent Coulomb interaction, described by the spin vector potential $\overrightarrow{\mathcal{A}}=k(\overrightarrow{r}\times \overrightarrow{S})/r^2$ together with a Coulomb-type scalar potential $\phi =\kappa /r$. The model is governed by the Schrödinger-type Hamiltonian ${\mathcal{H}}_{\mathrm{S}}={\overrightarrow{\Pi }}^2/\left(2M\right)+q\phi$ in nonrelativistic quantum mechanics and by the Dirac-type Hamiltonian ${\mathcal{H}}_{\mathrm{D}}=c\overrightarrow{\alpha }·\overrightarrow{\Pi }+\beta M{c}^2+q\phi$ in relativistic quantum mechanics, where $\overrightarrow{\Pi }=\overrightarrow{p}-(q/c)\overrightarrow{\mathcal{A}}$ is the canonical momentum. We demonstrate two main results: (i) Just as the Coulomb-type scalar potential ${\mathcal{S}}_{\mathrm{Maxwell}}=\{\overrightarrow{\mathcal{A}}=0,\phi =\kappa /r\}$ is a local exact solution of Maxwell’s equations on $r\ne 0$, the gauge potential ${\mathcal{S}}_{\mathrm{YM}}=\{\overrightarrow{\mathcal{A}}=k(\overrightarrow{r}\times \overrightarrow{S})/r^2,\phi =\kappa /r\}$ constitutes a local exact solution of the Yang–Mills equations on the punctured region $r\ne 0$. (ii) Both Hamiltonians ${\mathcal{H}}_{\mathrm{S}}$ and ${\mathcal{H}}_{\mathrm{D}}$ can be solved exactly in the presence of this spin-dependent Coulomb interaction. The resulting energy spectra are derived, and they naturally reduce to those of the ordinary hydrogen atom when the spin-dependent terms are neglected. Finally, we clarify the quantization conditions and the fixed-background interpretation of the model. Full article

The built-in electric field induced by polarization in ZnO/Mg_0.2Zn_0.8O quantum wells can be screened to modulate the conduction-band potential profile and intersubband energy levels. To optimize the screening of the built-in electric field, we analyze the influence of an external electric field, temperature, and modulation doping. The position of the doped layer is varied within the heterostructure to improve field compensation, providing additional control over electron localization and intersubband energy separation. In this work, within the effective mass approximation and by self-consistently solving the Poisson and Schrödinger equations using the finite-difference method, we calculate the electronic structure and nonlinear optical response of an n-type doped ZnO/Mg_0.2Zn_0.8O quantum well heterostructure. Our results indicate a strong dependence of the confinement potential on the applied external electric field and the electrostatic potential arising from the doped layer. We demonstrate electronic Raman gain values on the order of ${10}^3$–${10}^4$ cm⁻¹ for specific values of field strength, temperature, and doped-layer position. This approach enables fine-tuning of the nonlinear optical response, which is crucial for the development of ZnO-based optoelectronic devices. Full article

We propose an exploratory framework for a Bohmian model of quantum matter propagating on a classical curved spacetime background. The gravitational sector is governed by classical Einstein field equations throughout; no quantisation of spacetime is attempted. The wave function emerges as the scalar contraction $\mathrm{\Psi }={\psi }_{\nu }{\psi }^{\nu }\in \mathbb{C}$ of a complex-valued tensorial field ${\psi }^{\mu }$, encoding quantum dynamics in a geometric object. The wave tensor interacts with spacetime via the stress–energy tensor $T_{\mu \nu }$, mediated by a real scalar field a of dimension volume, so that $a{T}_{\mu \nu }{\psi }^{\mu }{\psi }^{\nu }$ yields the correct potential energy. We derive a covariant Adapted Schrödinger Equation as the unique minimal covariant lift of the standard equation, justify it from four guiding principles, and verify three internal consistency checks. Under seven explicit approximations the framework reproduces the Schrödinger equation with Coulomb potential for the hydrogen atom. We also derive a dynamical equation for ${\psi }^{\mu }$ that entails the Adapted Schrödinger Equation by contraction. Two open problems are then resolved. First, a complete Lagrangian formulation is provided: a real-valued action for $\mathrm{\Psi }$ yields the Adapted Schrödinger Equation via the Euler–Lagrange equations; a separate action for ${\psi }^{\mu }$, extended by a non-polynomial term, yields the full dynamical equation variationally. Second, two experimental predictions are derived. Expanding to first post-Newtonian order, the perturbation Hamiltonian has coefficients $(3, 1)$ on the kinetic and potential operators; via the virial theorem these produce a coordinate-time blueshift, which after photon propagation yields the universal Einstein gravitational redshift $\delta \nu /\nu =\mathrm{\Phi }/c^2$, confirming consistency with the equivalence principle. The same kinetic coefficient independently predicts that free quantum wave packets spread more slowly by the fractional amount $3\left|\mathrm{\Phi }\right|/c^2$, a correction absent in standard non-relativistic quantum mechanics. Full article

In this study, we utilize the ${\varphi }^6$-model expansion method to derive a diverse set of Jacobi elliptic function solutions for the conformable resonant Nonlinear Schrödinger Equation (NLSE) with parabolic law nonlinearity. As the modulus of the Jacobi elliptic functions approaches 1 and 0, the solutions transform into hyperbolic and trigonometric functions, respectively. This methodology yields various exact traveling wave solutions, including kink solitons, singular solitons, periodic solutions, and singular periodic solutions. Notably, this work represents the first investigation into identifying Jacobi elliptic function solutions for the conformable resonant NLSE. These results enhance the understanding of the nonlinear dynamical properties intrinsic to the NLSE. We use graphical illustrations to highlight the dynamical features of the solutions. Moreover, our approach showcases versatility in addressing other nonlinear partial differential equations, offering insights applicable to nonlinear optics, fluid dynamics, and quantum physics. Full article

This paper presents a four-component integrable extension of the derivative nonlinear Schrödinger (DNLS) soliton hierarchy, namely, the Kaup–Newell hierarchy of soliton equations. Motivated by a general extension idea for the Kaup–Newell spectral matrix, we propose a specially constructed 4th-order matrix-valued eigenvalue problem involving four potentials and derive the corresponding integrable Hamiltonian hierarchy via the Lax pair framework. A recursion operator and a bi-Hamiltonian structure are established to demonstrate the Liouville integrability of the resulting hierarchy. As an illustrative example, we derive an integrable system of four DNLS equations, each containing two linear dispersion terms, which differs from standard integrable systems. Full article

Low-temperature photoluminescence (PL) has been studied under hydrostatic pressure and varying excitation powers in three samples of single In_0.17Ga_0.83N quantum wells with different widths: 2.6 nm, 5.2 nm, and 10.4 nm. Transitions involving ground states were strong in the 2.6 nm well, weak in the 5.2 nm well, and absent in the 10.4 nm well. Pressure coefficients of PL lines have been used to estimate the electric field in the wells. In the widest well, the field seems to be fully screened (at high excitation powers). Simulations involving Poisson and Schrödinger equations allowed us to identify the experimental PL lines. Pressure evolution of the PL spectra agreed with the simulation. We present diagrams showing the dependence of the field in the well on pressure and on carrier concentration. In wide wells, these diagrams illustrate the transition from a 2D-like system to a 3D-like system. Full article

In this paper, we investigate the long-term asymptotic behavior of solutions to the Cauchy problem for the n-component coupled higher-order nonlinear Schrödinger (nC-HNLS) equation on a line with decaying initial data. Through the application of nonlinear steepest descent methods to an associated $(n+1)\times (n+1)$ matrix Riemann–Hilbert (RH) problem, we find that, within the sector defined by $\left|\frac{x}{t}-\frac{1}{3\epsilon }\right|t^{\frac{2}{3}}⩽C$, where $C>0$ is a constant, the asymptotics can be characterized in relation to the solution to a coupled modified Painlevé II equation. This relationship is connected to a corresponding $(n+1)\times (n+1)$ matrix RH problem. Full article

In this article, the different types of self-coagulation discovered in the fluid simulations of inductively coupled plasma (abbreviated as ICP) at both the electronegative and electropositive cases are presented. Among these, the electronegative plasma sources include Ar/O₂, Ar/Cl₂, and Ar/SF₆, and the electropositive plasma source is the inertial argon plasma itself. The fluid simulation versions are not the same. Concretely, the Comsol software version 5.4 is used to simulate the Ar/O₂, Ar/Cl₂, Ar/SF₆, and the pure argon ICPs, and the self-written code of the fluid model is used to simulate the pure argon ICP as well, but in a different framework of fluid design. The types of self-coagulation refined from these fluid simulations are the physically ambi-polar self-coagulation of ions, the chemically ambi-polar self-coagulation of ions, the mono-polar self-coagulation of electrons, and the non-polar self-coagulation of argon metastable atoms. These self-coagulations are based on mass and founded through the Comsol fluid simulations, and moreover, the self-coagulation of thermal energy of electrons is founded through the self-written fluid code simulation. Based on the self-coagulations of mass and energy, together with the accompanying discharge hierarchy, we hypothesize (1) the correlation of ambi-polar self-coagulation and diffusion, (2) the mean of using the Schrodinger equation to describe the quasi-particle of anions given by self-coagulation in a certain potential barrier, (3) the analogy of the $\beta$ and ${\beta }^{+}$ decay and the asymmetry given by two types of ICP source simulation, (4) the picture of spin orientations of neutrino and anti-neutrino, and (5) the model for photon sustainment. The self-coagulation behavior is seen to be general and the interdisciplinary works of plasma physics with quantum mechanics, particle physics, nuclear physics, and optics are helpful for us to better understand the mass and energy general dynamics. Full article

An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms. Full article