Square-well Research Articles

Numerical analysis and performance optimization of a spiral fertilizer distributor in side deep fertilization of a paddy field

To improve the quality of side deep fertilization in paddy fields and study the dynamic change in fertilizer particles in fertilizer boxes, a model connecting the mechanical components and fertilizer based on the discrete element method was established, and a numerical simulation analysis of the spiral fertilizer distributor was conducted with EDEM software. The primary factors affecting the uniformity and stability of the fertilizer were studied. Using an orthogonal experimental design and numerical simulation technology, the speed, lead and diameter of a steel wire were selected as experimental factors, the mean fertilization value and the fertilization coefficient of the variation were selected as evaluating indicators, a mathematical model between the factors and indicators was established by virtual orthogonal testing, and the numerical processing was optimized with Design-Expert 6.0.10 software. The results showed that the primary and secondary factors affecting the comprehensive performance occurred in the following sequence: steel wire speed > steel wire diameter > steel wire lead. When the speed, lead, and diameter of the steel wire were 15 r/min, 65 mm and 7 mm, respectively, the operation performance was superior. High-speed video camera analysis and bench tests showed that the results were consistent with the simulation optimization. The mean value and the coefficient of variation of fertilization were 9.39 g and 3.71%, respectively, which satisfy the agronomic requirements for paddy field side deep fertilization.

The time independent fractional Schrödinger equation with position-dependent mass

In this paper, we present a fractional form of Schrödinger equation using the Kallil’s derivative method. Then we investigate the capability of this equation to obtain the wave functions and its energy levels for a particle trapped in infinite potential well (IPW) for the range1<α≤2. We also, present the fractional form of the Schrödinger equation for a particle with position-dependent mass (PDM) and by using it, we obtain wave functions and its energy levels in range 0 <α≤1 and in different amounts of η and μ in which μ + η = 1/2. It seems that our method is very useful to solve the problems in PDM case.

On the virial theorem for a particle in a box: Accounting for Cauchy's boundary condition

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.

Shannon entropy and Fisher information for endohedral confined one- and two-electron atoms

In this work, we study the Shannon entropy and Fisher information of the H, He, Li+, and Be2+ atoms confined by a C60 cavity. The fullerene cavity is described by three model potentials: the square well (SW), Woods-Saxon (WS), and the Gaussian model (GM). We solved the time-independent Schrödinger and Hartree-Fock equations by means of a numerical method. We find that the C60 endohedral cavity modifies the ground- and excited-states of the hydrogen atom where the largest effect is given for the 2s state leading to an increase of ∼2.5% of the Shannon entropy. The Fisher information results for the excited-states are highly sensitive to the presence of the C60 cavity where the largest increase is reported for the 5s state with 50%. For the two-electron atoms, we find slight changes on the ground-state properties where the larger increase is for the Shannon entropy and Fisher information of the He atom in the ground-state. Our findings show that the GM potential leads to the largest change in electronic properties due to the overlap with the wave-functions. In the absence of the C60 cavity, our findings are in excellent agreement with other theoretical results.

Bound States of Spherically Symmetric Potentials

We solve the time-independent Schrodinger equation for spherically symmetric potentials. First, we consider simple cases of a particle on a ring and a particle on a sphere to illustrate the degeneracy arising due to symmetry. We then consider three different spherically symmetric potentials: (i) spherical well potential, (ii) isotropic three-dimensional harmonic oscillator, and (iii) spherically confined isotropic three-dimensional harmonic oscillator. Our discussion mainly focuses on the energy levels of the bound states and the associated degeneracies. Finally, we calculate the heat capacity of endohedral fullerenes using two simple models—particle in a spherical box and confined harmonic oscillator.

Dimensional analysis and the correspondence between classical and quantum uncertainty

Heisenberg’s uncertainty principle is often cited as an example of a ‘purely quantum’ relation with no analogue in the classical limit where ℏ → 0. However, this formulation of the classical limit is problematic for many reasons, one of which is dimensional analysis. Since ℏ is a dimensionful constant, we may always work in natural units in which ℏ = 1. Dimensional analysis teaches us that all physical laws can be expressed purely in terms of dimensionless quantities. This indicates that the existence of a dimensionally consistent constraint on ΔxΔp requires the existence of a dimensionful parameter with units of action, and that any definition of the classical limit must be formulated in terms of dimensionless quantities (such as quantum numbers). Therefore, bounds on classical uncertainty (formulated in terms of statistical ensembles) can only be written in terms of dimensionful scales of the system under consideration, and can be readily compared to their quantum counterparts after being non-dimensionalized. We compare the uncertainty of certain coupled classical systems and their quantum counterparts (such as harmonic oscillators and particles in a box), and show that they converge in the classical limit. We find that since these systems feature additional dimensionful scales, the uncertainty bounds are dependent on multiple dimensionless parameters, in accordance with dimensional considerations.

On 3D and 1D Weyl particles in a 1D box

We construct the most general families of self-adjoint boundary conditions for three (equivalent) Weyl Hamiltonian operators, each describing a three-dimensional Weyl particle in a one-dimensional box situated along a Cartesian axis. These results are essentially obtained by using the most general family of self-adjoint boundary conditions for a Dirac Hamiltonian operator that describes a one-dimensional Dirac particle in a box, in the Weyl representation, and by applying simple changes of representation to this operator. Likewise, we present the most general family of self-adjoint boundary conditions for a Weyl Hamiltonian operator that describes a one-dimensional Weyl particle in a one-dimensional box. We also obtain and discuss throughout the article distinct results related to the Weyl equations in (3+1) and (1+1) dimensions, in addition to their respective wave functions, and present certain key results related to representations for the Dirac equation in (1+1) dimensions.

The quantum character of buckling instabilities in thin rods

Here the buckling of inextensible rods due to axial body forces is mapped to 1D, nonrelativistic, time-independent quantum mechanics. Focusing on the pedagogical case of rods confined to 2D, three simple and physically realizable applications of the mapping are given in detail; the quantum counterparts of these are particle in a box, particle in a delta-function well, and particle in a triangular well. A fourth application examines the buckling counterpart of a quantum many-body problem (in the Hartree approximation). Through a fifth application, given in the form of an exercise, the reader can explore the surprising consequences of adding a second transverse dimension to the rod buckling problem and imposing periodic boundary conditions.

Entropy production of an active particle in a box

A run-and-tumble particle in a one-dimensional box (infinite potential well) is studied. The steady state is analytically solved and analyzed, revealing the emergent length scale of the boundary layer where particles accumulate near the walls. The mesoscopic steady state entropy production rate of the system is derived from coupled Fokker-Planck equations with a linear reaction term, resulting in an exact analytic expression. The entropy production density is shown to peak at the walls. Additionally, the derivative of the entropy production rate peaks at a system size proportional to the length scale of the accumulation boundary layer, suggesting that the behavior of the entropy production rate and its derivatives as a function of the control parameter may signify a qualitative behavior change in the physics of active systems, such as phase transitions.

Maple’s Quantum Chemistry Package in the Chemistry Classroom

An introduction to the Quantum Chemistry Package (QCP), implemented in the computer algebra system Maple, is presented. The QCP combines sophisticated electronic structure methods and Maple's easy-to-use graphical interface to enable computation and visualization of the electronic energies and properties of molecules. Here we describe how the QCP can be used in the chemistry classroom using lessons provided within the package. In particular, the calculation and visualization of molecular orbitals of hydrogen fluoride, the application of the particle in a box to conjugated dyes, the use of geometry optimization and normal mode analysis for hypochlorous acid, and the thermodynamics of combustion of methane.

The Quijote Simulations

The Quijote simulations are a set of 44,100 full N-body simulations spanning more than 7,000 cosmological models in the $\{\Omega_{\rm m}, \Omega_{\rm b}, h, n_s, \sigma_8, M_\nu, w \}$ hyperplane. At a single redshift the simulations contain more than 8.5 trillions of particles over a combined volume of 44,100 $(h^{-1}{\rm Gpc})^3$; each simulation follow the evolution of $256^3$, $512^3$ or $1024^3$ particles in a box of $1~h^{-1}{\rm Gpc}$ length. Billions of dark matter halos and cosmic voids have been identified in the simulations, whose runs required more than 35 million core hours. The Quijote simulations have been designed for two main purposes: 1) to quantify the information content on cosmological observables, and 2) to provide enough data to train machine learning algorithms. In this paper we describe the simulations and show a few of their applications. We also release the Petabyte of data generated, comprising hundreds of thousands of simulation snapshots at multiple redshifts, halo and void catalogs, together with millions of summary statistics such as power spectra, bispectra, correlation functions, marked power spectra, and estimated probability density functions.

κ-Deformed quantum and classical mechanics for a system with position-dependent effective mass

We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews–Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.

Position-dependent mass charged particles in magnetic and Aharonov–Bohm flux fields: separability, exact and conditionally exact solvability

Using cylindrical coordinates, we consider position-dependent mass (PDM) charged particles moving under the influence of magnetic, Aharonov–Bohm flux, and a pseudoharmonic or a generalized Killingbeck-type potential fields. We implement the PDM minimal coupling recipe (Mustafa in J Phys A Math Theor 52:148001, 2019), along with the PDM-momentum operator (Mustafa and Algadhi in Eur Phys J Plus 134:228, 2019), and report separability under radial cylindrical and azimuthal symmetrization settings. For the radial Schrödinger part, we transform it into a radial one-dimensional Schrödinger-type and use two PDM settings, $$g\left( \rho \right) =\eta \rho ^{2}$$ and $$g\left( \rho \right) =\eta /\rho ^{2}$$ , to report on the exact solvability of PDM charged particles moving in three fields: magnetic, Aharonov–Bohm flux, and pseudoharmonic potential fields. Next, we consider the radial Schrödinger part as is and use the biconfluent Heun differential forms for two PDM settings, $$g\left( \rho \right) =\lambda \rho $$ and $$g\left( \rho \right) =\lambda /\rho ^{2}$$ , to report on the conditionally exact solvability of our PDM charged particles moving in three fields: magnetic, Aharonov–Bohm flux, and generalized Killingbeck potential fields. Yet, we report the spectral signatures of the one-dimensional z-dependent Schrödinger part on the overall eigenvalues and eigenfunctions, for all examples, using two z-dependent potential models (infinite potential well and Morse-type potentials).

Energy and stability of negatively charged trion in cylindrical quantum dot under temperature effect

The uncorrelated energy of negatively charged trion confined in a semiconductor cylindrical quantum dot (QD) under temperature effect is theoretically investigated using a variational procedure within the effective mass approximation. We have used a wave function satisfying for the strong confinement regime and also presenting well results for the weak confinement regimes. The trion confinement is described by a finite and infinite square potential wells under the temperature effect. Our findings show that the uncorrelated energy of the negatively charged trion is more important for the small size of the QD. We have found a strong dependence of negatively charged trion energy on the confinement potential. Besides, our numerical results have shown that the effect of temperature leads to an increase in uncorrelated energy. The temperature has a significant effect on the stability of the three non-bound charge carriers.

Separation and alignment of chiral active particles in a rotational magnetic field.

We propose a method for the chiral separation and alignment of active paramagnetic particles in a two-dimensional square box with periodic boundary conditions. In a rotational magnetic field, the dynamic behavior of magnetized particles is strongly determined by the competition between the magnetic interaction and differing chirality. By suitably tailoring the parameters, active particles with different chirality can be aggregated into different clusters and separated. However, when either the magnetic interaction or chirality difference is dominant, the particles are prone to mixing. In addition, the external rotational magnetic field plays a decisive role in aligning particles. The numerical results show that there exists an optimal strength and rotation frequency of the magnetic field, as well as a rotational diffusion coefficient, self-propulsion velocity, and packing fraction, at which the separation coefficient takes its maximal value. The proposed method can be exploited to separate naturally occurring chiral active particles.

Withdrawn: Spatio-temporal of Dirac and Klein Gordon particles in a one-dimensional box

Abstract It has come to the attention of the Editors of Chinese Physics B that the article entitled 'Spatio-temporal of Dirac and Klein Gordon particles in a one-dimensional box' should not have been accepted for publication because of its substantial replication of a published article. Consequently, this paper has been withdrawn from publication.

Thickness dependence of infrared lattice absorption and excitonic absorption in ZnO layers on Si and SiO2 grown by atomic layer deposition

Using spectroscopic ellipsometry from the midinfrared (0.03 eV) to the deep ultraviolet (6.5 eV), the authors determined the thickness dependence of the dielectric function for ZnO thin layers (5–50 nm) on Si and quartz in comparison to bulk ZnO. They observed a small blueshift of the band gap (∼80 meV) in thin ZnO layers due to quantum confinement, which is consistent with a simple effective mass theory in an infinite potential well. There is a drastic reduction in the excitonic effects near the bandgap, especially for thin ZnO on Si, which not only affects the excitonic absorption peak but also lowers the high-frequency dielectric constant by up to 40%. No significant change of the phonon parameters (except an increased broadening) in thin ZnO layers was found.

An infinite square-well potential as a limiting case of a square-well potential in a minimal-length scenario

One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same as one found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement with those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution operator.

The effect of fractional calculus on the formation of quantum‐mechanical operators

In this paper, the deformation of the ordinary quantum mechanics is formulated based on the idea of conformable fractional calculus. Some properties of fractional calculus and fractional elementary functions are investigated. The fractional wave equation in 1 + 1 dimension and fractional version of the Lorentz transformation are discussed. Finally, the fractional quantum mechanics is formulated; infinite potential well problem, density of states for the ideal gas, and quantum harmonic oscillator problem are discussed.

MS-Excel Spreadsheet Applications in Introductory UnderGraduate Physics-A Review

The role of information technology has been quite instrumental in bringing improvisations in the the field of pedagogy. The institutions are shifting to ICT based techniques, because teaching science subjects with ICT based simulations has partially replaced the requirements of laboratories along with enhancement of conceptual understanding of the students. Presently, many science teachers prefer to use self made spreadsheet computer simulations and virtual laboratories. For example, the most common MS-Excel spreadsheet can be programmed to imitate the movement of many physical systems such as a motion of projectile, trajectory of charged particle in electric and magnetic fields; and motion of particle in a box. Excel spreadsheet is frequently available on desktop computers, and because of their massive data manipulation functions and powerful plotting facilities they are finding increasing use in physics teaching. This review paper highlights the role of MS-Excel spreadsheet in simulating many real world problems in the teaching of Physics.