Square-well Research Articles

A random batch Ewald method for charged particles in the isothermal-isobaric ensemble.

We develop an accurate, highly efficient, and scalable random batch Ewald (RBE) method to conduct molecular dynamics simulations in the isothermal-isobaric ensemble (the NPT ensemble) for charged particles in a periodic box. After discretizing the Langevin equations of motion derived using suitable Lagrangians, the RBE method builds the mini-batch strategy into the Fourier space in the Ewald summation for the pressure and forces such that the computational cost is reduced to O(N) per time step. We implement the method in the Large-scale Atomic/Molecular Massively Parallel Simulator package and report accurate simulation results for both dynamical quantities and statistics for equilibrium for typical systems including all-atom bulk water and a semi-isotropic membrane system. Numerical simulations on massive supercomputing cluster are also performed to show promising central processing unit efficiency of the RBE.

CFD-DEM Study of Pleated Filter Plugging Process Based on Porous Media Model

The pneumatic conveying process of fine particles through filters was studied by CFD-DEM simulation method. The porous media model and porous structure were used to simulate the airflow state and the blocking effect of fine particles when they flowed through the filter. Under different particle feed rates and initial particle velocities, the effects of the plugging rate and settling velocity in pleated filters were analyzed, and the effect of particle deposition height on fluid zone was studied. The results showed that particles should avoid the feed rate of 250–750 g/s and choose the initial particle velocity of 3–6 m/s to achieve lower plugging rate and faster settling velocity. The position of the filter should avoid the particle inlet to avoid the increase of non-uniformity. Timely cleaning of particles in the filter box can improve the filtering performance.

Simulation Analysis and Test of Pneumatic Distribution Fertilizer Discharge System

Precision fertilizer application technology is necessary to improve the utilization efficiency of fertilizers in agricultural production. Traditional mechanical fertilization systems risk blockages and uneven application when working in multiple crop rows. Pneumatic fertilization systems have improved efficiency and fertilization quality, however, fewer studies have characterized their designs in regards to the motion of the fertilizer particles. Here, we design and evaluate the parameters of the key components of a pneumatic fertilizer discharge system. Numerical simulations were conducted using a coupled EDEM-FLUENT and gas-phase models together with bench tests to examine the effects of inlet wind speed on the efficiency and consistency of the pneumatic fertilization system. The EDEM-FLUENT simulations showed that the number of fertilizer particles in the grid box set by EDEM was 60 particles in the range from t = 0.275 s to t = 0.5 s, and there was no blockage or cut-off in the pipe. The gas-phase simulation showed that there were tiny vortices in the fertilizer conveying pipe, and the maximum flow rate of its backflow was lower than 3.59 m/s, which had little effect on the fertilizer conveyance. The bench test showed that the inlet wind speed was 35–40 m/s, and the fertilization efficiency was 0.29–0.41 kg/s when the maximum variation coefficient of the row discharge consistency of the pneumatic distribution fertilizer discharge system was 5.55%. The coefficient of variation of the average row discharge consistency was 3.93%, and the average fertilizer discharge met the design requirements. Therefore, the pneumatic distribution system achieves stable operation and meets the requirements of fertilization operations.

Particle on a Ring Model for Teaching the Origin of the Aromatic Stabilization Energy and the Hückel and Baird Rules.

Simple mathematical models can serve to reveal the essence of experimental phenomena and scientific concepts. The particle in a box (PIB), for example, is widely used in undergraduate programs to teach the quantum mechanical principles behind the UV–vis spectra of conjugated polyenes and polyynes. In this work, the particle on a ring (POR) and the PIB models are used to elucidate the concept of aromaticity in Introductory Chemistry courses. Thus, we explain the origin of the aromatic stabilization energy, Hückel’s rule, and Baird’s rule. Besides applications, the limitations of the POR and PIB models are also discussed.

Optimizations of multilevel quantum engine with N noninteracting fermions based on Lenoir cycle

We consider optimizations of Lenoir heat engine within a quantum dynamical field consisting of $N$ noninteracting fermions trapped in multilevel infinite potential square-well. Fermions play role as working substance of the engine with each particle nested at different level of energy. We optimized this quantum heat engine model by analysing the physical parameter and deriving the optimum properties of the engine model. The model we investigated consists of one high-energy heat bath and one low-energy sink bath. Heat leakage occurs between these two bathes as expected will degenerate the efficiency of quantum heat engine model. The degeneration increased as we raised the constant parameter of heat leakage. We also obtained loop curves in dimensionless power vs. efficiency of the engine, which efficiency is explicitly affected by heat leakage, but in contrast for the power output. From the curves, we assured that the efficiency of the engine would go back to zero as we raised compression ratio of engine with leakage. Lastly, we checked Clausius relations for each model with various levels of heat leakage. We found that models with leakage have a reversible process on specific compression ratios for each variation of heat leakage. Nevertheless, the compression ratio has limitations because of the $\oint dQ/E>0$ after the reversible point, i.e. violates the Clausius relation.

Particle in a Box: A Basic Paradigm in Quantum Mechanics — Part 2

Particle in a Box: A Basic Paradigm in Quantum Mechanics — Part 2

Excitation spectrum of Bose gases beyond the Gross–Pitaevskii regime

We consider Bose gases of [Formula: see text] particles in a box of volume one, interacting through a repulsive potential with scattering length of order [Formula: see text], for [Formula: see text]. Such regimes interpolate between the Gross–Pitaevskii and thermodynamic limits. Assuming that [Formula: see text] is sufficiently small, we determine the ground state energy and the low-energy excitation spectrum, up to errors vanishing in the limit [Formula: see text].

Self-adjoint extensions for a [formula omitted]-corrected Hamiltonian of a particle on a finite interval

In the present paper we deal with the issue of finding the self-adjoint extensions of a p4-corrected Hamiltonian. The importance of this subject lies on the application of the concepts of quantum mechanics to the minimal-length scale scenario which describes an effective theory of quantum gravity. We work in a finite one dimensional interval and we give the explicit U(4) parametrization that leads to the self-adjoint extensions. Once the parametrization is known, we can choose appropriate U(4) matrices to model physical problems. As examples, we discuss the infinite square-well, periodic conditions, anti-periodic conditions and periodic conditions up to a prescribed phase. We hope that the parametrization we found will contribute to model other interesting physical situations in further works.

Particle in a Box: A Basic Paradigm in Quantum Mechanics — Part 1

Particle in a Box: A Basic Paradigm in Quantum Mechanics — Part 1

Quantum games and interactive tools for quantum technologies outreach and education

We provide an extensive overview of a wide range of quantum games and interactive tools that have been employed by the quantum community in recent years. We present selected tools as described by their developers, including “Hello Quantum, Hello Qiskit, Particle in a Box, Psi and Delta, QPlayLearn, Virtual Lab by Quantum Flytrap, Quantum Odyssey, ScienceAtHome, and the Virtual Quantum Optics Laboratory.” In addition, we present events for quantum game development: hackathons, game jams, and semester projects. Furthermore, we discuss the Quantum Technologies Education for Everyone (QUTE4E) pilot project, which illustrates an effective integration of these interactive tools with quantum outreach and education activities. Finally, we aim at providing guidelines for incorporating quantum games and interactive tools in pedagogic materials to make quantum technologies more accessible for a wider population.

Error measurements for a quantum annealer using the one-dimensional Ising model with twisted boundaries

A finite length ferromagnetic chain with opposite spin polarization imposed at its two ends is one of the simplest frustrated spin models. In the clean classical limit the domain wall inserted on account of the boundary conditions resides with equal probability on any one of the bonds, and the degeneracy is precisely equal to the number of bonds. If quantum mechanics is introduced via a transverse field, the domain wall will behave as a particle in a box, and prefer to be nearer the middle of the chain rather than the ends. A simple characteristic of a real quantum annealer is therefore which of these limits obtains in practice. Here we have used the ferromagnetic chain with antiparallel boundary spins to test a real flux qubit quantum annealer and discover that contrary to both expectations, the domain walls found are non-uniformly distributed on account of effective random longitudinal fields present notwithstanding tuning carried out to zero out such fields when the couplings between qubits are nominally zero. We present a simple derivation of the form of the distribution function for the domain walls, and show also how the effect we have discovered can be used to determine the strength of the effective random fields (noise) characterizing the annealer. The noise measured in this fashion is smaller than what is seen during the single-qubit tuning process, but nonetheless qualitatively affects the outcome of the simulation performed by the annealer.

True gravitational atoms: Spherical cloud of dilatonic black holes

Black hole as elementary particle is a fairly glamorous idea. For ordinary black holes, a surrounding particle inevitably penetrates into the interior of the hole since the center of the hole is an infinite potential well. For the first time we demonstrate that an extreme dilatonic black hole in spherical symmetry perfectly behaves as an atom, in the sense that its surrounding cloud of particles are completely stable. Thus we reach a spherical cloud of dilatonic black hole. We find exact wave functions of the cloud for arbitrary gravitational fine structure constant $\mu M$, and clear the underlying physical nature of the stability. Through careful studies of the exact wave function, we find the spectrum of this system. We discuss the physical meaning of this discovery especially from considerations of entropy, and the resultant possibility to explore quantization of gravitational waves from coming observations.

Solution of Sakata-Taketani Equation via the Caputo and Riemann-Liouville Fractional Derivatives

In this paper, we present the Sakata–Taketani (ST) equation of spin 0 in the framework of fractional derivatives . We propose fractional Lagrangian and Hamiltonian densities. Using the fractional variational principle , the fractional Sakata–Taketani equation and its adjoint are determined. Accordingly, we have solved exactly the following problems: the free Sakata-Taketani equation of spin 0 in both cases (in time fractional and in space-time fractional cases) and the Sakata–Taketani particle of spin 0 in space-time fractional and in a box model (also known as the infinite potential well).

Trapping and binding by dephasing

The binding and trapping of particles usually rely on conservative forces, described by unitary quantum dynamics. We show that both can also arise solely from spatially dependent dephasing, the simplest type of decoherence. This can be based on continuous weak position measurements in only selected regions of space, for which we propose a practical realization. For a single particle, we demonstrate a quantum particle in a box based on dephasing. For two particles, we demonstrate their binding despite repulsive interactions, if their molecular states are dephased at large separations only. Both mechanisms are experimentally accessible, as we show for an example with Rydberg atoms in a cold gas background.

Relativistic spin-0 particle in a box: Bound states, wave packets, and the disappearance of the Klein paradox

The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.

A fluid description based on the Bernoulli equation of the one-body stationary states of quantum mechanics with real valued wavefunctions

A formalism is developed, and applied, that describes a class of one-body quantum mechanical systems as fluids where each stationary state is a steady flow state. The time-independent Schrödinger equation for one-body stationary states with real-valued wavefunctions is shown to be equivalent to a compressible-flow generalization of the Bernoulli equation of fluid dynamics. The mass density, velocity and pressure are taken as functions that are determined by the probability density. The generalized Bernoulli equation describes compressible, irrotational, steady flow with variable mass and a constant specific total energy, i.e, a constant energy per mass for each fluid element. The generalized Bernoulli equation and a generalized continuity equation provide a fluid dynamic interpretation of a class of quantum mechanical stationary states that is an alternative to the unrealistic, static-fluid interpretation provided by the Madelung equations and quantum hydrodynamics. The total kinetic energy from the Bernoulli equation is shown to be equal to the expectation value of the kinetic energy, and the integrand of the expectation value of the kinetic energy is given an interpretation. It is also demonstrated that variable mass is necessary for a satisfactory fluid model of stationary states. However, over all space, the flows conserve mass, because the rate of mass creation from the sources are equal to the rate of mass annihilation from the sinks. The following flows are examined: the ground and first excited-states of a particle in a one-dimensional box, the harmonic oscillator, and the hydrogen s states.

Using quantum mechanics for calculation of different infinite sums

We demonstrate that a certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument p (p > 2) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.

Modeling partition coefficient of liquid droplets impacting on particles by direct numerical simulation

Liquid partition coefficient, defined as the proportion of the liquid volume covered on particle surface to that in bulk phase, is a key parameter in the simulation of liquid-injected fluidized beds. To develop the sub-grid model for liquid partition coefficient as a function of local hydrodynamics, a two-dimensional Volume-of-Fluid (VOF) simulation was conducted to directly resolve droplets impinging, wetting and spreading on particles in a box. The model was first validated by the experiments in literature on the spreading diameter of a droplet impinging on a single particle. Then for a system of multiple droplets-particles impinging, we investigated the effects of Weber number, droplet-to-particle (DTP) ratio and particle volume fraction on several characteristic parameters, e.g., ratio of liquid area adhered to particles to the total liquid area, liquid film thickness, the percentage of wetted particle surface area and liquid holdup. All these parameters decrease with increasing Weber number when We = 10 ∼ 370, and then gradually stabilize. Higher particle volume fraction increases the collisional frequency of droplets and particles, and therefore the ratio of liquid area adhered to particles to the total liquid area. By contrast, DTP is not the dominant factor in liquid partition. Finally a correlation for liquid partition coefficient is proposed as a function of droplet Weber number and particle volume fraction. It may serve as a sub-grid model for CFD-DEM or multifluid models in simulation of large-scale liquid-injected fluidized beds.

Phase-Amplitude Relations for a Particle with a Superposition of Two Energy Levels in a Double Potential Well.

We study the connection between the phase and the amplitude of the wave function and the conditions under which this relationship exists. For this we use the model of particle in a box. We have shown that the amplitude can be calculated from the phase and vice versa if the log analytical uncertainty relations are satisfied.

Soft representation of the square-well and square-shoulder potentials to be used in Brownian and molecular dynamics simulations

The discrete hard-sphere (HS), square-well (SW), and square-shoulder (SS) potentials have become the battle horse of molecular and complex fluids because they contain the basic elements to describe the thermodynamic, structural, and transport properties of both types of fluids. The mathematical simplicity of these discrete potentials allows us to obtain some analytical results despite the nature and complexity of the modeled systems. However, the divergent forces arising at the potential discontinuities may lead to severe issues when discrete potentials are used in computer simulations with uniform time steps. One of the few routes to avoid these technical problems is to replace the discrete potentials with continuous and differentiable forms built under strict physical criteria to capture the correct phenomenology. The match of the second virial coefficient between the discrete and the soft potentials has recently been successfully used to construct a continuous representation that mimics some physical properties of HSs (Báez et al 2018 J. Chem. Phys. 149 164907). In this paper, we report an extension of this idea to construct soft representations of the discrete SW and SS potentials. We assess the accuracy of the resulting soft potential by studying structural and thermodynamic properties of the modeled systems by using extensive Brownian and molecular dynamics computer simulations. Besides, Monte Carlo results for the original discrete potentials are used as benchmark. We have also implemented the discrete interaction models and their soft counterparts within the integral equations theory of liquids, finding that the most widely used approximations predict almost identical results for both potentials.