Infinite Square Research Articles

The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

The Gross–Pitaevskii Equation for an Infinite Square Well with a Delta-Function Barrier

Optimal Local Identifying and Local Locating-dominating Codes

We introduce two new classes of covering codes in graphs for every positive integer r. These new codes are called local r-identifying and local r-locating-dominating codes and they are derived from r-identifying and r-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small n optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid and the king grid. We prove that seven out of eight of our constructions have optimal densities.

Effects of Impurities and Deformations on Electronic Effective Mass in Quantum Revival Time within the Infinite Square Well

Effects of Impurities and Deformations on Electronic Effective Mass in Quantum Revival Time within the Infinite Square Well

In situ cell-scale probing nucleation, growth and evolution of CsPbBr3 nanowires by optical absorption spectroscopy

The growth kinetics of colloidal lead halide perovskite nanomaterials are an integral part of their applications, remains poorly understood due to complex nucleation processes and lack of in situ size monitoring method. Here we demonstrated that absorption spectra can be used to observe in situ growth processes of ultrathin CsPbBr3 nanowires in solution with reference to the effective mass infinite deep square potential well model. By means of this method, we have found that the ultrathin nanowires, fabricated by hot injection method, were firstly formed within one minute. Subsequently, they merge with each other into a thicker structure with increasing reaction time. We revealed that the nucleation, growth, and merging of the CsPbBr3 nanowires are determined by the acid concentration and ligand chain length. At lower acidity, the critical nucleation size of the nanowire is smaller, while the shorter the ligand chain length, the faster the merging among the nanowires. Moreover, the merging mode between nanowires changed with their nucleation size. This growth kinetics of CsPbBr3 nanowires provides a reference for optimizing the synthesis conditions to obtain the one-dimensional CsPbBr3 with desired size, thus enabling accurate control of the nanowire shape.

Harnessing the Matrix Method with SMath to Explore the Finite Square Well Potential

This innovative research makes use of a matrix-method driven numerical voyage into the foundation of the time-independent Schr?dinger equation for a one-dimensional finite square well potential to unlock the secrets of the quantum realm. This is made possible by the unmatched power of the open-source computational juggernaut, SMath Studio. The work unfolds the quantum tapestry by cleverly placing the finite square well inside the infinitely large potential of an infinite square well, using an infinite number of basis vectors to invoke wavefunctions from the very fabric of the universe. This scholarly masterwork not only clarifies the elusive low-lying bound states of the finite square well potential but also acts as a quantum lighthouse, illuminating the complex quantum landscape for undergraduate students with unmatched clarity and understanding.

Matrix Mechanics of a Particle in a One-Dimensional Infinite Square Well

We solve the infinite-potential well problem using the method of Heisenberg’s matrix mechanics. In addition to being of educational value, the matrix mechanics allows us to deal with various unphysical issues caused by this potential in a seemingly unproblematic fashion. We also show how to treat many particles within this representation.

Axially symmetric quadrupole-octupole incorporating sextic potential

We present an extended application of the analytic quadrupole octupole axially symmetric model, originally employed to study the octupole deformation and vibrations in light actinides using an infinite well potential (IW). In this work, we extend the model's applicability to a broader range of nuclei exhibiting octupole deformation by incorporating a sextic potential instead of the Davidson potential. Similarly to conventional models, such as AQOA-IW (for infinite square potential) and AQOA-D (for the Davidson potential), our proposed model is referred to as AQOA-S. By employing the sextic potential, phenomenologically represented as v(β˜)=a1β˜2+a2β˜4+a3β˜6, we can derive analytical expressions for the energy spectra and transition rates (B(E1), B(E2), B(E3)). The energy spectra of the model are essentially governed by two critical parameters: ϕ0, indicating the balance between octupole and quadrupole strain, and α, a key factor in adjusting the shape and behavior of the spectra through the sextic potential. In terms of applications, the study encompasses five isotopes, namely 222−226Ra and 224,226Th. Significantly, our model demonstrates remarkable agreement with the corresponding experimental data, particularly for the recently determined B(EL) transition rates of 224Ra, surpassing the performance of the model that employs the Davidson potential. The stability of the octupole deformation in 224Ra adds particular significance to these findings.

Transition Energy For a Polar Quantum Disc with Conical Disclination in Parabolic Confining Electric Potential

The Transition energy of an electron for a polar quantum disc with conical disclination is investigated theoretically. For charge carrier confinement, we consider the infinite polar square well potential (IPSW), and parabolic potential (PP). The disclination in the system is characterized by the kink parameter k. The energy levels of the system were calculated using the Schrödinger equation with the effective mass approximation. Our study reveals that the transition energy decreases as the kink parameter k increases.

Quantum infinite square well as a limiting case of higher order anharmonic oscillator

We present a new type of continuous anharmonic potential and analyze its higher order asymptotic limit. The potential display infinite square well-like behavior when anharmonic order N goes to infinity. Consequently, wave functions and eigenvalues of ground and first excited states of anharmonic oscillators also asymptotically approach to those of infinite square well.

The geodesic mutual visibility problem: Oblivious robots on grids and trees

The Mutual Visibility is a well-known problem in the context of mobile robots. For a set of n robots disposed in the Euclidean plane, it asks for moving the robots without collisions so as to achieve a placement ensuring that no three robots are collinear. For robots moving on graphs, we consider the Geodesic Mutual Visibility (GMV) problem. Robots move along the edges of the graph, without collisions, so as to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means that there is a shortest path (i.e., a “geodesic”) between each pair of robots along which no other robots reside. We study this problem in the context of trees and (finite or infinite) square grids, for robots operating under the standard Look–Compute–Move model. In both scenarios, we provide resolution algorithms along with formal correctness proofs, highlighting the most relevant peculiarities arising within the different contexts, while optimizing the time complexity.

Quantum evolution with random phase scattering

We consider the quantum evolution of a fermion–hole pair in a d-dimensional gas of non-interacting fermions in the presence of random phase scattering. This system is mapped onto an effective Ising model, which enables us to show rigorously that the probability of recombining the fermion and the hole decays exponentially with the distance of their initial spatial separation. In contrast, without random phase scattering the recombination probability decays like a power law, which is reflected by an infinite mean square displacement. The effective Ising model is studied within a saddle point approximation and yields a finite mean square displacement that depends on the evolution time and on the spectral properties of the deterministic part of the evolution operator.

Exact Treatment of the Infinite Square Well in One Dimension with λδ^' (x) Potential

Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge

Probing DDM and ML quantum concepts in shape phase transitions of γ-unstable nuclei

In a recent paper (S. Ait El Korchi et al. (2020) [32]), we explored, inside the context of Critical Point Symmetries (CPSs) X(3) and Z(4), a correlation between two exceedingly known quantum concepts, the Minimal Length (ML) and the Deformation-Dependent Mass (DDM), that are commonly applied in various areas of physics. Such a correlation has been strongly identified in transition nuclei by calculating some physical observables of that quantum system, like as energy spectra, moments of inertia and transition probabilities. In this paper we extend that study to E(5) dynamical symmetry corresponding to the shape phase transition U(5)↔O(6). The experimental realization of the models was found to occur in some nuclei, using the existing phenomenological potentials: Infinite Square Well, Davidson and Kratzer, whose models fits provide the best agreement. Importantly the calculations performed in this work using these potentials, corroborate the fact that the revealed correlation between both quantum concepts is not destructively affected by the presence of other model's parameters and hence its existence is independent of the form or type of the used potential. Undoubtedly, the present work will open the way for more investigations of this correlation in the limits of other critical points symmetries in nuclear shape phase transitions which play today a major role in nuclear structure research from theoretical as well as experimental point of view.

Cation Charge as a Tool to Change Dimensionality in Organic–Inorganic Hybrids Based on Copper Thiocyanate Templated by 1,4-Diazabicyclo[2.2.2]octane

The first three compounds based on a {copper-thiocyanate-dabco} combination, namely, (Hdabco)[Cu2(NCS)3] (1), (H2dabco)[Cu(NCS)3] (2), and [Cu(Hdabco)2(NCS)4]∙2dmso (3), where dabco = 1,4-diazabicyclo[2.2.2]octane were synthesized and characterized by single-crystal XRD, elemental analysis, Raman, and partial IR spectroscopy. In copper(I) derivatives, the influence of the charge of the organic cation on the dimensionality of the crystal structure is observed. Thus, in the case of 1, monoprotonated Hdabco+ cations provide the template for the formation of a polymeric anionic 3D framework {[Cu2(NCS)3]-}n, while in the case of 2, diprotonated H2dabco2+ cations together with discrete [Cu(SCN)3]2- anions generate a simple ionic 0D structure with an island-like crystal lattice. The anionic {[Cu2(SCN)3]-}n framework has infinite square channels of 10 × 10 Å size running along the 001 crystallographic direction. In 3, both the Hdabco+ and thiocyanato units behave as terminal monodentate ligands attached to copper(II) centers via N-donor atoms, forming neutral molecular complexes with an elongated (4+2) octahedral environment. The crystallization molecules of dmso are hydrogen bonded to the protonated parts of the coordinated dabco molecules. A series of by-products Cu(SCN)2(dmso)2 (4), (Hdabco)SCN (5), (H2dabco)(SCN)2 (6), and (H2dabco)(SCN)2∙H2O (7) were identified and characterized.

Perfect Colorings of the Infinite Square Grid: Coverings and Twin Colors

A perfect coloring (equivalent concepts are equitable partition and partition design) of a graph $G$ is a function $f$ from the set of vertices onto some finite set (of colors) such that every node of color $i$ has exactly $S(i,j)$ neighbors of color $j$, where $S(i,j)$ are constants, forming the matrix $S$ called quotient. If $S$ is an adjacency matrix of some simple graph $T$ on the set of colors, then $f$ is called a covering of the target graph $T$ by the cover graph $G$. We characterize all coverings by the infinite square grid, proving that every such coloring is either orbit (that is, corresponds to the orbit partition under the action of some group of graph automorphisms) or has twin colors (that is, two colors such that unifying them keeps the coloring perfect). The case of twin colors is separately classified.

Influence of four confinement phonon models on the electron self-energy in a finite-square potential quantum-well

In this article, we have used the variational method and four different confinement longitudinal optical (LO) phonon models to calculate the self-energy of the electron–confined LO-phonon interaction in a finite square well potential as a function of the width of the well, and the main results are compared with each other. This potential of the quantum well is well suited for the above purpose as it is more realistic than the infinite square potential quantum well (QW). Based on the calculated results, we show good agreement between the confined optical phonon models in two-dimensional structures. The results obtained show that the Huang-Zhu, the Fuchs-Kliewer, and the Knipp-Reinecke models are suitable for studies of the electron–LO-phonon (e–LO-p) interaction in QWs in which the use of the Fuchs-Kliewer model is the easiest because its theoretical model is very simple but effective. The result also shows that the Ridley model is not suitable for theoretical studies of the e–LO-p interaction in QWs because the electromagnetic boundary condition is not satisfied by this model.

Penentuan Density Matrix Sistem Kuantum Satu Partikel Dimensi Satu dengan Metode Discretized Path Integral (DPI)

A mixed quantum system can be represented by a density matrix. By knowing the density matrix of the system, other thermodynamic properties of the system such as Helmholtz free energy and entropy can be determined. Computation of the density matrix of a one-dimensional single-particle quantum system using the discretized path integral(DPI) method is presented. The results of the DPI method were validated using the finite difference time domain (FDTD) method for a particle in an infinite square well and harmonic oscillator potentials. The results of density matrix, Helmholtz’s free energy and entropy have shown that the DPI method produces correct numerical values compared to the FDTD results. Then, the DPI method is used in the double-well potential by giving variations in barrier potential and temperature of the system. The results show that the particle will be tunneling when the barrier potential energy is more than the kinetic energy of the particle. The particle also can tunnel easily if the system has the higher temperature.

Dimer–Dimer Correlations at the Rough–Smooth Boundary

Three phases of macroscopic domains have been seen for large but finite periodic dimer models; these are known as the frozen, rough and smooth phases. The transition region between the frozen and rough region has received a lot of attention for the last twenty years and recently work has been underway to understand the rough–smooth transition region in the case of the two-periodic Aztec diamond. We compute uniform asymptotics for dimer–dimer correlations of the two-periodic Aztec diamond when the dimers lie in the rough–smooth transition region. These asymptotics rely on a formula found in Chhita and Johansson (Adv Math 294:37–149, 2016) for the inverse Kasteleyn matrix, they also apply to the infinite square grid dimer model with a variable weighting which interpolates between the rough and smooth phase (Kenyon et al. Ann Math (2) 163(3):1019–1056, 2006). In particular, we find that distant dimers initially decay exponentially when the magnetic coordinates are very close to the bounded complementary component of the associated amoebae, they then transition to a power law decay once far enough apart.

Collective excitations in even-even nuclei with a stepped infinite square well potential

A solution is worked out for a γ-unstable Bohr Hamiltonian with an infinite square well potential having an adjustable step. The evolution of the model’s spectral characteristics are investigated with varying height and width of the additional step, revealing shape coexisting an mixing features. The description of the dynamical shape phase transition in the collective bands of the Te even-even nuclei is offered as a numerical application.

Thermodynamics of a particle in square well potential

We have presented the thermodynamic properties of a particle moving freely both in an infinite and finite square well in contact with a heat reservoir at constant temperature. In both cases, we have computed the temperature dependence of the partition function, entropy, average energy and specific heat of the particle in an exact way. The exact results for various thermodynamic parameters deviate considerably from its known theoretical values and reveal the importance of such calculations. The effect of the width of the finite well on the different thermodynamic quantities have also been examined critically.