Infinite Square Research Articles

Numerical simulation of quantum anharmonic oscillator, embedded within an infinite square well potential, by matrix methods using Gnumeric spreadsheet

The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic () and quartic (λx4) terms is embedded within an infinite square well potential of appropriate width, ‘a’ and its sine eigen functions are used as basis functions ‘N’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘a’ and ‘N’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘λ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.

Coherent states of position-dependent mass trapped in an infinite square well

We develop generalized coherent states based on the Gazeau–Klauder formalism for a particle with position-dependent mass trapped in an infinite square well. We study the quantum statistical properties of these states by means of the Mandel parameter and the second-order correlation function. Our analysis reveals that the constructed coherent states exhibit sub-Poissonian statistics. Moreover, theoretical investigations of wave packet revivals and fractional revivals for the pertaining system have been performed by means of the autocorrelation function and temporal evolution of probability density.

Discrete scattering by two staggered semi-infinite defects: reduction of matrix Wiener–Hopf problem

As an extension of the discrete Sommerfeld problems on lattices, the scattering of a time-harmonic wave is considered on an infinite square lattice when there exists a pair of semi-infinite cracks or rigid constraints. Due to the presence of stagger, also called offset, in the alignment of the defect edges the asymmetry in the problem leads to a matrix Wiener–Hopf kernel that cannot be reduced to scalar Wiener–Hopf in any known way. In the corresponding continuum model the same problem is a well-known formidable one which possesses certain special structure with exponentially growing elements on the diagonal of kernel. From this viewpoint the present paper tackles a discrete analog of the same by reformulating the Wiener–Hopf problem and reducing it to a finite set of linear algebraic equations; the coefficients of which can be found by an application of the scalar Wiener–Hopf factorization. The considered discrete paradigm involving lattice waves is relevant for modern applications of mechanics and physics at small length scales.

Path integral for quantum dynamics with position-dependent mass within the displacement operator approach

In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb 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.

Universal trade-off between proximity and aspect-ratio in optimizing the field enhancement factor of large area field emitters

The apex-field enhancement factor (aFEF) is regarded as a meaningful parameter to characterize field electron emission (FE) devices. If experimentally extracted from orthodox current–voltage characteristics, this parameter reliably quantifies how much the emitter's sharp tip locally magnifies the applied external electrostatic field. Many experimental works have reported FE from arrays of carbon nanotubes with fixed spacing (c) between nearest-neighbor emitters, fixed apex-radii (r), and various height (h) and claimed the existence of an aspect ratio (h/r) at which a local maximum effective aFEF is achieved. Hereafter, it is shown that those results are not consistent with simulations using basic electrostatics for both finite or infinite regular square arrays of emitters. Quite interestingly, our results show that the aFEF (γa), for an emitter in an infinity regular array, simply saturates at γs for h≳0.7c. Additionally, we found a universal behavior in which γa scales as γa∼γs(h/c)0.84, when h≲0.7c, for h/r≳50. These results provide a practical rule for the design of large arrays of field emitters, which can be used to build FE nanoelectromechanical resonators with both mechanical strength and reduced Joule losses.

Nested Square Roots of 2 Revisited

We show how various recently established results on finite and infinite nested square roots of 2 can be derived elegantly via a topological conjugacy linking the tent map and the logistic map. We also prove an additional new and striking consequence of this relationship.

Scattering on a square lattice from a crack with a damage zone.

A semi-infinite crack in an infinite square lattice is subjected to a wave coming from infinity, thereby leading to its scattering by the crack surfaces. A partially damaged zone ahead of the crack tip is modelled by an arbitrarily distributed stiffness of the damaged links. While an open crack, with an atomically sharp crack tip, in the lattice has been solved in closed form with the help of the scalar Wiener–Hopf formulation (Sharma 2015 SIAM J. Appl. Math., 75, 1171–1192 (doi:10.1137/140985093); Sharma 2015 SIAM J. Appl. Math. 75, 1915–1940. (doi:10.1137/15M1010646)), the problem considered here becomes very intricate depending on the nature of the damaged links. For instance, in the case of a partially bridged finite zone it involves a 2 × 2 matrix kernel of formidable class. But using an original technique, the problem, including the general case of arbitrarily damaged links, is reduced to a scalar one with the exception that it involves solving an auxiliary linear system of N × N equations, where N defines the length of the damage zone. The proposed method does allow, effectively, the construction of an exact solution. Numerical examples and the asymptotic approximation of the scattered field far away from the crack tip are also presented.

Finite and infinite nested square roots convergent to unity

We investigate nested square root formulas convergent to \textbf{1} by two methods. In the first method, we derive a general infinitely nested square root formula. In the second method we derive a general finitely nested square root formula.

Quantitative analysis of the applicability limits of the model of a one-dimensional infinite square well

Abstract We discuss the influence of several factors on the deviations from energy spectrum of an infinite square quantum well (QW) for real microscopic systems that can be approximately modelled using particle in a box. We introduce the “blurring” potential in the form of the modified Woods-Saxon potential and solve the corresponding Schrodinger equation. It is found that the increase of the degree of blurring δ of the QW leads to the increase of number of the energy levels inside it and to increase of deviations from the quadratic dependence e (n) (e is the particle energy, n is the energy level number) typical for the infinite square QW, especially, for the energy levels close to the QW “tops”. It is most surprising that for relatively “large” values of δ the difference between the levels energies of such well and the appropriate (with the same n) levels energies of the square QW with the same depth changes sign (from positive to negative) as number n increases. We also conclude that the asymmetry of the QW and non-equality min≠mout (where min and mout are the particle effective mass inside and outside the QW) play a significant role for the relatively “shallow” well near the QW top.

Frustating use of the Laplace transform for the quantum states of a particle in a box

Abstract In a recent paper diffused in the literature, the quantum problem of a particle in a infinite square well potential was claimed to be solved via Laplace transform. Surprisingly, the correct solutions were found for an eigenvalue problem with variable defined on a finite range. As a matter of fact, the authors have misused the Laplace transform of the time-independent Schrodinger equation, and they have also used incorrectly the inversion process. In the present paper it is pointed out carefully and clearly those mistakes that might occur with the use of inappropriate methods for a given eigenvalue problem.

Collectives of automata on infinite grid graph with deterministic vertex labeling

Automata walking on graphs are a mathematical formalization of autonomous mobile agents with limited memory operating in discrete environments. Under this model broad area of studies of the behaviour of automata in finite and infinite labyrinths (a labyrinth is an embedded directed graph of special form) arose and intensively developing. Research in this regard received a wide range of applications, for example, in the problems of image analysis and navigation of mobile robots. Automata operating in labyrinths can distinguish directions, that is, they have a compass. This paper examines vertex labellings of infinite square grid graph thanks to these labellings a finite automaton without a compass can walk along graph in any arbitrary direction. The automaton looking over neighbourhood of the current vertex and may move to some neighbouring vertex selected by its label. We propose a minimal deterministic traversable vertex labelling that satisfies the required property. A labelling is said to be deterministic if all vertices in closed neighbourhood of every vertex have different labels. It is shown that minimal deterministic traversable vertex labelling of square grid graph uses labels of five different types. Minimal deterministic traversable labelling of subgraphs of infinite square grid graph whose degrees are less than four are developed. The key problem for automata and labyrinths is the problem of constructing a finite automaton that traverse a given class of labyrinths. We say that automaton traverse infinite graph if it visits any randomly selected vertex of this graph in a finite time. It is proved that a collective of one automaton and three pebbles can traverse infinite square grid graph with deterministic labelling and any collective with fewer pebbles cannot. We consider pebbles as automata of the simplest form, whose positions are completely determined by the remaining automata of the collective. The results regarding to exploration of an infinite deterministic square grid graph coincide with the results of A.V. Andzhan (Andzans) regarding to traversal of an infinite mosaic labyrinth without holes. It follows from above that infinite grid graph after constructing a minimal traversable deterministic labelling on it and fixing two pairs of opposite directions on it becomes an analogue of an infinite mosaic labyrinth without holes.

Quantum matrix diagonalization visualized

We show how to visualize the process of diagonalizing the Hamiltonian matrix to find the energy eigenvalues and eigenvectors of a generic one-dimensional quantum system. Starting in the familiar sine-wave basis of an embedding infinite square well, we display the Hamiltonian matrix graphically with the basis functions alongside. Each step in the diagonalization process consists of selecting a nonzero off-diagonal matrix element and then rotating the two corresponding basis vectors in their own subspace until this element is zero. We provide mathematica code to display the effects of these rotations on both the matrix and the basis functions. As an electronic supplement, we also provide a javascript web app to interactively carry out this process.

Inferring the smoothness of the autocorrelation function from that of the initial state

We point out that by the ‘smoothness means fast decay’ principle in Fourier analysis, it is possible to infer the smoothness (or nonsmoothness) of the autocorrelation function from a mere glimpse of the initial state. Specifically, for a generic system with smooth eigenstates, the smoother an initial state is, the faster its decomposition coefficients with respect to the eigenstates of the system decay, and in turn the smoother the autocorrelation function is. The idea is illustrated with three increasingly smooth functions in an infinite square well. By using the Mellin transform, we also find that the nonsmoothness or singularity of the initial state affects the short-time behavior of its autocorrelation function. In particular, a sufficiently nonsmooth or singular initial state could decay in a nonquadratic power law, with the exponent continuously tunable. We now understand the periodic cusps of the autocorrelation function in the quench dynamics of a Bloch state, which was observed previously (Zhang and Yang 2016 Europhys. Lett. 114 60001).

The b-coloring of infinite graphs

We consider the b-coloring number of infinite graphs. We prove that the parameters of the infinite square, triangular and hexagonal lattices are 5, 7 and 4 on the plane. We also obtain that the parameters of infinite square lattices and its induction graph of n-dimensional are 2n + 1 and n2 + n + 1.

The ground states of nanomagnetic dipoles placed on an infinite square lattice

• The ground magnetic state of a dipole system has been modeled. • Static Relaxation method, a version of Monte Carlo method at low temperature is used to simulate the system. • The ground state energy per dipole of an infinitive simple square lattice model is calculated. • A dipole system of nanomagnetic particles behaves in a similar manner to paramagnetic material. • Metastable domains of different types of ordering were observed. Nanomagnetic materials are excellent experimental tools for the study of electromagnetic phenomena and are suitable for the manufacture of artificial new materials with unusual physical properties. These properties depend on the magnetic structure of the dipole system in a great degree. In the present study, the emphasis is on the exact magnetic structure of a system with an infinite number of nanomagnetic dipoles arranged on a simple square lattice at zero temperature. Using numerical methods, such as the Static Relaxation and Monte Carlo simulation, we have obtained various magnetic configurations and by analyzing them, we found some exact, degenerated ground states for a system with an infinite number of dipoles, with long-range dipole-dipole interactions between themselves that form a simple two-dimensional square lattice. The degenerated ground states turned out to constitute systems of anti-parallel lines which form a ‘magnetic crystalline’ structure with a zero-magnetization vector. Any equilibrium state of dipole system is degenerated by nature of the dipole-dipole interaction: simultaneous inversion of the direction of each dipole of the system to exact opposite one leaves the system in equilibrium and does not change its energy. The degeneracy of the ground state, in combination with fluctuations, boundary and finite-size effects results in several metastable disordered configurations for a system of finite but large number of dipoles, thus macroscopically a nanomagnetic dipole system has zero-magnetization and behaves in a fashion similar to paramagnetic material.

Optimal strategies to infer the width of an infinite square well by performing measurements on the particle(s) contained in the well

We seek the optimal strategy to infer the width a of an infinite potential well by performing measurements on the particle(s) contained in the well. In particular, we address quantum estimation theory as the proper framework to formulate the problem and to determine the optimal quantum measurement, as well as to evaluate the ultimate bounds to precision. Our results show that in a static framework the best strategy is to measure position on a delocalized particle, corresponding to a width-independent quantum signal-to-noise ratio (QSNR), which increases with delocalisation. Upon considering time-evolution inside the well, we find that QSNR increases with time as t2 (at least for small t). On the other hand, it decreases with a and thus time-evolution is a metrological resource only when the width is not too large compared to the available time evolution. Finally, we consider entangled particles in the well and observe super-additivity of the QSNR: it is the sum of the single-particle QSNRs, plus a positive definite term, which depends on their preparation and may increase with the number of entangled particles. Overall, entanglement represents a resource for the precise characterization of potential wells.

Steady-state kinetic temperature distribution in a two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a point heat source

We consider heat transfer in an infinite two-dimensional square harmonic scalar lattice lying in a viscous environment and subjected to a heat source. The basic equations for the particles of the lattice are stated in the form of a system of stochastic ordinary differential equations. We perform a continualization procedure and derive an infinite system of linear partial differential equations for covariance variables. The most important results of the paper are the deterministic differential–difference equation describing non-stationary heat propagation in the lattice and the analytical formula in the integral form for its steady-state solution describing kinetic temperature distribution caused by a point heat source of a constant intensity. The comparison between numerical solution of stochastic equations and obtained analytical solution demonstrates a very good agreement everywhere except for the main diagonals of the lattice (with respect to the point source position), where the analytical solution is singular.

Tensor Network Annealing Algorithm for Two-Dimensional Thermal States.

Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of two-dimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finite-temperature regime. As a benchmark, we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultracold atoms.

Exotic Patterns of Synchrony in Planar Lattice Networks

Patterns of dynamical synchrony that can occur robustly in networks of coupled dynamical systems are associated with balanced colorings of the nodes of the network. In symmetric networks, the orbits of any group of symmetries automatically determine a balanced orbit coloring. Balanced colorings not of this kind are said to be exotic. Exotic colorings occur in infinite planar lattices, both square and hexagonal, with various short-range couplings. In some cases, a balanced two-coloring remains balanced when colors are swapped along suitable diagonals, giving rise to uncountably many distinct exotic colorings. We explain this phenomenon in terms of iterated orbit colorings, in which the quotient of the lattice by an orbit coloring has extra symmetries, allowing new orbit colorings on the quotient, which then lift back to the lattice. We apply the same construction to several other exotic lattice colorings. Two appendices discuss how to modify the notion of balance for networks with diffusive coupling, and how to formalize the differential equations in infinitely many variables that arise for lattices.