Square-well Research Articles

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.

Modeling mechanical equilibration processes of closed quantum systems: A case study.

We model the dynamics of a closed quantum system brought out of mechanical equilibrium, undergoing a nondriven, spontaneous, thermodynamic transformation. In particular, we consider a quantum particle in a box with a moving and insulating wall, subjected to a constant external pressure. Under the assumption that the wall undergoes classical dynamics, we obtain a system of differential equationsthat describes the evolution of the quantum system and the motion of the wall. We study the dynamics of such a system and the thermodynamics of the process of compression and expansion of the box. Our approach is able to capture several properties of the thermodynamic transformations considered and goes beyond a description in terms of an ad hoc time-dependent Hamiltonian, considering instead the mutual interactions between the dynamics of the quantum system and the parameters of its Hamiltonian.

Quantum Chirikov criterion: Two particles in a box as a toy model for a quantum gas

We consider a toy model for emergence of chaos in a quantum many-body short-range-interacting system: two one-dimensional hard-core particles in a box, with a small mass defect as a perturbation over an integrable system, the latter represented by two equal mass particles. To that system, we apply a quantum generalization of Chirikov's criterion for the onset of chaos, i.e. the criterion of overlapping resonances. There, classical nonlinear resonances translate almost automatically to the quantum language. Quantum mechanics intervenes at a later stage: the resonances occupying less than one Hamiltonian eigenstate are excluded from the chaos criterion. Resonances appear as contiguous patches of low purity unperturbed eigenstates, separated by the groups of undestroyed states-the quantum analogues of the classical KAM tori.

The Particle in a Box Warrants an Examination

The particle in a box is a simple model that has a classical Hamiltonian H = p2 (using 2m = 1), with a limited coordinate space, -b q b, where 0 b < ∞. Using canonical quantization, this example has been fully studied thanks to its simplicity, and it is a common example for beginners to understand. Despite its repeated analysis, there is a feature that puts the past results into question. In addition to pointing out the quantization issue, the procedures of affine quantization can lead to a proper quantization that necessarily points toward more complicated eigenfunctions and eigenvalues, which deserve to be solved.

Stationary structures of spin-orbit coupled polariton condensates in Bessel lattices

&lt;sec&gt;Bessel optical lattice yields a non-spatially periodic column-symmetric optical lattice potential field, which has the characteristics of both infinite deep potential well and the ring-shaped potential well. A deep potential is formed in the center of the 0-order Bessel optical lattice. In the non-zero-order Beseel optical lattice, a ring-shaped shallow potential well with a central barrier can be formed. Exciton-polariton is a semi-light and semi-matter quasi-particle, which can achieve the Bose-Einstein condensate phase transition even at room temperature to form a polariton condensate. In addition, the polariton condensate is likely to realize sufficiently strong spin-orbit coupling due to the cavity-induced TE-TM splitting of the polariton energy levels. The polariton condensate can be realized at room temperature, and there can be spin-orbit coupling in it, which provides a new platform for the studying of quantum physics. &lt;/sec&gt;&lt;sec&gt;In this paper, the Bessel optical lattice is introduced into a polariton condensate. The stationary state structure of spinor two-component polariton condensate with spin-orbit coupling is investigated. By solving the Gross-Pitaevskii equation, we first give a stationary state structures of the polariton condensate both in the laboratory coordinate frame and in the rotating coordinate frame. Owing to the introduction of the Bessel optical lattice, the stationary state structures of polariton condensate are diverse. We dispaly the stationary state structures of the basic Gaussian solitons and multipole solitons in the central deep potential well in the laboratory coordinate frame, and the ring solitons and multipole solitons in the central shallow potential well. We also dispaly the vortex ring soliton that exists in the rotating coordinate frame, and the stationary state structure of the component separation caused by the spin-orbit interaction. We analyze not only the influences of the spin-orbit coupling on the stationary state structures in the two coordinate frames, but also the stability of the multipole solitons in the rotating coordinate frame. It is found that the multipole solitons formed in the ring-shaped shallow potential well have better stability than in the central deep potential well, and they can maintain the relative structure and spatial distribution for a long time in the rotation process. In the rotating coordinate frame, even if the two-component separation conditions are not satisfied, the introduction of spin-orbit coupling can cause the two components to separate.&lt;/sec&gt;

About Degeneration of Landau’s Levels

Through the non separable solution of the eigenvalue problem associated to the problem of a charged particle in a flat box and a constant transversal magnetic field, with Landau and symmetric gauges, it is found that the Landau’s levels are numerably degenerated in both cases. A mathematical proposition is proven to carry out this statement.

Formation of the quasi-planar B56 boron cluster: topological path from B12 and disk aromaticity.

Formation and stability of the B56 boron cluster were investigated using a topological approach and the disk aromaticity model. An extensive global energy minimum search for the B56 system which was carried out by means of the Mexican Enhanced Genetic Algorithm (MEGA) in conjunction with density functional theory computations, confirms a quasi-planar structure as its energetically most stable isomer. Such a structural motif is derived by applying a topological leapfrog operation to a B12 form. Its high thermodynamic stability can be explained by the disk aromaticity model in which the delocalization of its π orbitals can be assigned to the levels of a particle in a circular box with the [(1σ)2 (1π)4 (1δ)4 (1φ)4 (2σ)2 (1γ)4 (2π)4 (2δ)4 (1η)4 (2φ)4 (1θ)2] electronic configuration. This π delocalization is confirmed by other delocalization indices. While the B56 has a similar electron delocalization to that of the quasi-planar B50, they have opposite magnetic ring current properties because of the symmetry selection rules of their HOMO-LUMO electronic transitions. The π delocalization in the boron clusters is larger at long distances as compared to carbon clusters at similar sizes, but such a trend is reversed at shorter distances.

Physics Simulation with Excel Spreadsheet

this study present several examples of using Excel spreadsheets to study and simulate physical phenomena, namely projectile motion, falling motion in The air, The wave function of particles in a one-dimensional box, and nuclear radioactive decay. Spreadsheets are used to analyze the motion of projectiles with and without resistance, falling objects with different terminal velocities, graphs of wave functions and probabilities with different values of n, i.e. n=1 to n=4, and nuclear decay with different probability values of P. The use of excel spreadsheets has proven to be very useful in the physics learning process, namely the rapid response to changes in data and graphs, according to the values given

Detailed fluctuation theorem bound for apparent violations of the second law.

The second law of thermodynamics is a statement about the statistics of the entropy production, 〈Σ〉≥0. For small systems, it is known that the entropy production is a random variable and negative values (Σ<0) might be observed in some experiments. This situation is sometimes called apparent violation of the second law. In this sense, how often is the second law violated? For a given average 〈Σ〉, we show that the strong detailed fluctuation theorem implies a lower tight bound for the apparent violations of the second law. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a bosonic mode in the weak-coupling approximation, a levitated nanoparticle, and a classical particle in a box.

Wave functions for high-symmetry, thin microstrip antennas, and two-dimensional quantum boxes

For a spinless quantum particle in a one-dimensional box or an electromagnetic wave in a one-dimensional cavity, the respective Dirichlet and Neumann boundary conditions both lead to non-degenerate wave functions. However, in two spatial dimensions, the symmetry of the box or microstrip antenna is an important feature that has often been overlooked in the literature. In the high-symmetry cases of a disk, square, or equilateral triangle, the wave functions for each of those two boundary conditions are grouped into two distinct classes, which are one- and two-dimensional representations of the respective point groups, $C_{\infty v}$, $C_{4v}$, and $C_{3v}$. Here we present visualizations of representative wave functions for both boundary conditions and both one- and two-dimensional representations of those point groups. For the one-dimensional representations, color contour plots of the wave functions are presented. For the two-dimensional representations, the infinite degeneracies are presented as common nodal points and/or lines, the patterns of which are invariant under all operations of the respective point group. The wave functions with the Neumann boundary conditions have important consequences for the coherent terahertz emission from the intrinsic Josephson junctions in the high-temperature superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$: the enhancement of the output power from electromagnetic cavity resonances is only strong for wave functions that are not degenerate.

The investigation of Carnot engine in the presence of deformed formalism

In this paper, after introducing a kind of [Formula: see text]-deformation in quantum mechanics, first [Formula: see text]-deformed form of Schrödinger equation for a single particle in a box is derived. Then, the energy eigenvalues and wave function in Schrödinger equation are studied. Also, we discuss the Carnot cycle by using of the energy eigenvalues. We obtain the thermodynamic properties such as force, heat transferred, work done and efficiency in the cycle. Finally, all results have satisfied what we had expected before.

Influence of atomic-scale structure on the transport, ordering, and thermodynamics of partially ordered Cu-In alloys

We demonstrate the atomic-level structural properties of liquid Cu-In alloy as a function of In concentration using Square-Well (SW) model potential function under random phase approximation. It was found that the concentration-dependent structure factor S(k) and its Fourier transform, radial distribution function g(r) are in good agreement with the results obtained by the X-ray diffraction technique. Transport coefficients like diffusion coefficient and shear viscosity of liquid Cu-In alloys were estimated through computed structural functions and SW potential function. Interdiffusion coefficients in the melts were computed by solving the well-known Einstein's equation for Brownian particles with SW pair potential under linear trajectory principal. The composition-dependent shear viscosity is calculated using computed diffusion data in modified Stokes-Einstein (SE) relation. Specifically, we demonstrate a breakdown of SE relation at low In concentration in liquid Cu-In alloy. Using model calculations, we investigate the thermodynamic properties of alloy such as enthalpy of mixing, Gibbs free energy of mixing, and entropy of mixing. The concentration-concentration fluctuation at the zero momentum vector SCC(0)shows that there is a strong chemical bond between hetero-atoms in the Cu-rich region of liquid Cu-In alloys. It is worth mentioning here that the computed values of SCC(0), without using any adjusting or experimental parameter are in excellent agreement with experimental values measured by activity data.

Alternative momentum concept for a quantum mechanical particle in a box

For a particle in a box, the operator p̂=−i∂x is not self-adjoint. We provide an alternative construction of a momentum operator p̂R+ip̂I, which has two self-adjoint components p̂R and p̂I. This leads to a description of momentum measurements performed on a particle that is strictly limited to the interior of a box of size L, which yields quantized momentum values πn/L with n∈Z.Received 22 December 2020Revised 14 July 2021Accepted 15 September 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.L042008Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasNonrelativistic wave equationQuantum formalismQuantum measurementsGeneral Physics

Effects of chiral MIT boundary conditions for a Dirac particle in a box

Abstract We investigate the effects of the chiral MIT boundary conditions for a Dirac particle in a 1D box. We show how the boundary condition affects the discrete momentum, energy level, transition frequency, and spin state in a box. The effects of the chiral MIT boundary conditions on the probability and scalar densities are also demonstrated. The results show that an asymmetric distribution appears in the box depending on the parameters of the spin orientation and chiral angle.

Quantum states of hydrogen cations confined into spherical and nanotube-like potential wells: , (C 2h and C 2v ) and (D 3h )

In this work, the potential energy surfaces (PESs) of hydrogen-based cations and placed inside infinite potential wells, in the shape of an infinite cylinder and a sphere, are investigated. The calculations are performed using the diffusion Monte Carlo (DMC) technique. Different symmetries are studied, some of them for the first time. The PESs trends are rationalized. The Raman shift of mode of spherically confined is also calculated.

Electric field modified quantum entropic measures of particle in a spherical box

Electric field modified quantum entropic measures of particle in a spherical box

Adiabatic theorem for closed quantum systems initialized at finite temperature

The evolution of a driven quantum system is said to be adiabatic whenever the state of the system stays close to an instantaneous eigenstate of its time-dependent Hamiltonian. The celebrated quantum adiabatic theorem ensures that such pure state adiabaticity can be maintained with arbitrary accuracy, provided one chooses a small enough driving rate. Here, we extend the notion of quantum adiabaticity to closed quantum systems initially prepared at finite temperature. In this case adiabaticity implies that the (mixed) state of the system stays close to a quasi-Gibbs state diagonal in the basis of the instantaneous eigenstates of the Hamiltonian. We prove a sufficient condition for the finite temperature adiabaticity. Remarkably, it turns out that the finite temperature adiabaticity can be more robust than the pure state adiabaticity with respect to increasing the system size. This can be the case for one-body systems with large Hilbert spaces, such as a particle in a large box, as well as for certain many-body systems. In particular, we present an example of a driven many-body system where, in the thermodynamic limit, the finite temperature adiabaticity is maintained, while the pure state adiabaticity breaks down. On the other hand, for generic many-body systems the scaling of the finite temperature adiabatic condition with the system size is exponential, analogously to pure state adiabatic conditions.

Quantum dynamics of a nucleon in the Fermi accelerator

The quantum dynamics of a particle in a one-dimensional box with an oscillating wall (the Fermi accelerator) is investigated. The model is applied to the motion of a single nucleon in the mean-field potential of a heavy atomic nucleus whose surface vibrates. By directly solving the time-dependent Schrödinger equation, both the state of the particle and its mean-energy are studied. The effects of the frequency of the wall oscillation on the nucleon’s energy are addressed. Its energy oscillates in phase with the moving wall for all frequencies, showing no chaotic behaviour. There is a large initial peak of the nucleon’s energy as the particle adjusts to the sudden change in the size of the box and a varying relaxation time as it plateaus towards lower energy and a partial equilibrium. Small oscillations in energy continue, since there cannot be a true equilibrium while the wall is moving. The quantum coherence between the different parts of the nucleon’s wave-function in real space is very much preserved. This research lays the foundation for future investigations into quantum tunnelling in the Fermi accelerator.

Considerations about the incompleteness of the Ehrenfest’s theorem in quantum mechanics

We describe a study motivated by our interest to examine the incompleteness of the Ehrenfest’s theorem in quantum mechanics and to resolve a doubt regarding whether or not the hermiticity of the Hamiltonian operator is sufficient to justify a simplification of the expression of the macroscopic-observable time derivative that promotes the one usually found in quantum-mechanics textbooks. The study develops by considering the simple quantum system ‘particle in one-dimensional box’. We propose theoretical arguments to support the incompleteness of the Ehrenfest’s theorem in the formulation he gave, in agreement with similar findings already published by a few authors, and corroborate them with the numerical example of an electric charge in an electrostatic field. The contents of this study should be useful to Bachelor and Master students; the style of the discussions is tailored to stimulate, we hope, the student’s ability for independent thinking.

Fast driven quantum systems: interaction picture and boundary conditions

The work continues systematic applications of the method developed by Di Martino and Facchi (2015) for obtaining, for systems with variable boundary conditions, an equivalent Schrödinger equation with constant boundary conditions. From the resulting equation with rapidly varying terms, an effective equation is then derived for the averaged (slowly varying) wave functions. In this work, we obtain this effective equation for rapidly driven systems in the interaction picture (in first order in reverse frequency of external driving). This is the most suitable picture for systems with external driving. As an example, an extended two-state fast driven system is considered. As a second example, we consider the one-dimensional motion of a particle in a potential box with high-frequency oscillations of its width. Oscillations of width lead to additional terms in the effective equation proportional not only to the second derivative of the potential, but to the first derivative as well. In the case of width oscillations, compared to the oscillations of the box as a whole, the changes are even more significant. In our examples, phenomena such as slow averaged oscillations, resonance, the appearance of volcano potentials are observed, as is often the case in driven systems.