Showing papers on "Finite potential well published in 2015"
TL;DR: In this paper, the authors propose an alternative formulation in which the potential function does not appear, and obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions.
Abstract: In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrodinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.
56 citations
TL;DR: In this paper, a waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics, based on the well-known resemblance between the guided and radiation modes of waveguide with the bound and scattering states of a quantum well.
Abstract: A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well-known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete because resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.
22 citations
18 Aug 2015
TL;DR: Wertheim's solution of the fundamental statistical mechanical equation given by Percus and Yevick for hard spheres is invoked with a square well attractive part as a perturbation tail to get exact solution of direct correlation function, C ( k ) in momentum space and analytical expressions are obtained for structure factor, S ( k ). These expressions are then used to predict static structure factors for ten liquid metals, leading to fair agreement with experimental data.
Abstract: In the present paper surface tension, Debye temperature, coordination numbers along with microscopic correlations of ten liquid metals are determined using square-well model of correlation functions. Wertheim’s solution of the fundamental statistical mechanical equation given by Percus and Yevick for hard spheres is invoked with a square well attractive part as a perturbation tail to get exact solution of the direct correlation function, C ( k ) in momentum space and the analytical expressions are obtained for structure factor, S ( k ). These expressions are then used to predict static structure factors for ten liquid metals, leading to fair agreement with experimental data. Radial distribution function g ( r ) is obtained by Fourier analysis of computed S ( k ), from which the coordination numbers and the nearest neighbor distances of liquid metals are evaluated. Computed coordination numbers and surface properties of liquid metals using such a simple technique are in good agreement with experimental results.
16 citations
30 Sep 2015
TL;DR: In this paper, a particle that moves under the influence of a cubic potential which is restricted only to the state of the two energy levels is shown as a function of the width ratio between the initial conditions and the farthest wall while expanding.
Abstract: Carnot cycle consists of isothermal and adiabatic processes which are reversible. Using analogy in quantum mechanics, these processes can be well explained by replacing variables in classical process with a quantum system. Quantum system which is shown in this paper is a particle that moves under the influence of a cubic potential which is restricted only to the state of the two energy levels. At the end, the efficiency of the system is shown as a function of the width ratio between the initial conditions and the farthest wall while expanding. Furthermore, the system efficiency will be considered 1D and 2D cases. The providing efficiencies are different due to the influence of the degeneration of energy and the degrees of freedom of the system.
10 citations
TL;DR: In this article, the authors extended the approach to a three-dimensional system of spherical particles interacting by a square-well potential and showed that the mass-action law is equivalent to the complete partition function in the thermodynamic limit.
10 citations
TL;DR: In this article, the most general non-Hermitian Hulthen potential was considered to study the scattering of spinless relativistic particles and conditions for CC, SS and CPA were obtained analytically for this potential.
10 citations
TL;DR: Brownian dynamics is implemented to investigate the static properties of colloidal particles confined anisotropically and interacting via a potential which can be tailored in a repulsive-attractive-respulsive fashion as the interparticle distance increases.
Abstract: We implement Brownian dynamics to investigate the static properties of colloidal particles confined anisotropically and interacting via a potential which can be tailored in a repulsive-attractive-respulsive fashion as the interparticle distance increases. A diverse number of structural phases are self-assembled, which were classified according to two aspects, that is, their macroscopic and microscopic patterns. Concerning the microscopic phases we found the quasicrystalline, triangular, square, and mixed orderings, where this latter is a combination of square and triangular cells in a 3×2 proportion, i.e., the so-called (3(3),4(2)) Archimedian lattice. On the macroscopic level the system could self-organize in a compact or perforated single cluster surrounded or not by fringes. All the structural phases are summarized in detailed phases diagrams, which clearly show that the different phases are extended as the confinement potential becomes more anisotropic.
8 citations
30 Sep 2015
TL;DR: In this paper, the authors generalize the situation to 2D symmetric potential well and show that for the same ratio, 2D potential well is more efficient than 1D potentialWell.
Abstract: Carnot model of heat engine is the most efficient cycle consisting of isothermal and adiabatic processes which are reversible. Although ideal gas usually used as a working fluid in the Carnot engine, Bender used quantum particle confined in 1D potential well as a working fluid. In this paper, by following Bender we generalize the situation to 2D symmetric potential well. The efficiency is express as the ratio of the initial length of the system to the final length of the compressed system. The result then is shown that for the same ratio, 2D potential well is more efficient than 1D potential well.
8 citations
17 Jul 2015
TL;DR: In this article, the exact complex trajectories for the Eckart potential barrier and the soft potential step, plotted in a previous work, show that more trajectories link the left and right regions of the barrier, when the energy is increased.
Abstract: Complex quantum trajectory approach, which arose from a modified de Broglie–Bohm interpretation of quantum mechanics, has attracted much attention in recent years. The exact complex trajectories for the Eckart potential barrier and the soft potential step, plotted in a previous work, show that more trajectories link the left and right regions of the barrier, when the energy is increased. In this paper, we evaluate the reflection probability using a new ansatz based on these observations, as the ratio between the total probabilities of reflected and incident trajectories. While doing this, we also put to test the complex-extended probability density previously postulated for these quantum trajectories. The new ansatz is preferred since the evaluation is solely done with the help of the complex-extended probability density along the imaginary direction and the trajectory pattern itself. The calculations are performed for a rectangular potential barrier, symmetric Eckart and Morse barriers, and a soft potential step. The predictions are in perfect agreement with the standard results for potentials such as the rectangular potential barrier. For the other potentials, there is very good agreement with standard results, but it is exact only for low and high energies. For moderate energies, there are slight deviations. These deviations result from the periodicity of the trajectory pattern along the imaginary axis and have a maximum value only as much as $$0.1\,\%$$
of the standard value. Measurement of such deviation shall provide an opportunity to falsify the ansatz.
7 citations
TL;DR: In this paper, an approximate formula for the energy levels of the bound states of a particle in a finite square well is obtained, without using the Schrodinger equation, and the physics and mathematics involved in this approach are accessible to a gifted high school student.
Abstract: An approximate formula for the energy levels of the bound states of a particle in a finite square well are obtained, without using the Schrodinger equation. The physics and mathematics involved in this approach are accessible to a gifted high school student.
6 citations
Journal Article•
TL;DR: In this paper, it was shown that Laskin"s functions are not amathematical solution to the fractional Schrodinger equation and the equation does not have any nonzero solutions at all in the sense of mathematics.
Abstract: There has been a continuous argument on the correctness of the Laskin"s solution for the infinite square well problem in the fractional quantum mechanics In this paper, we prove that the Laskin"s functions are not amathematical solution to the fractional Schrodinger equation and the equation does not have any nonzero solutions at all in the sense of mathematics As in the standard quantum mechanics, we view the infinite square well problem as the limit of the finite well problem, and define the solution for the infinite square well problem as the limit of the solution for the finite square well problem Using the simple property of the infinite operators, we show that Laskin"s function can be the limit solution for the infinite square well problem The single-sided well problem and the 3 dimensional well problem are included This operator method works for the same problems in the relativistic quantum mechanicsas well
TL;DR: In this article, the effect of a waveguide impurity on an ultracold matter wave-packet at the threshold of delocalization is studied, where the impurity is modeled by a 1D short range square well potential with depth V0 and width 2R0 coupled to a contact impurity at the center of the bend.
Abstract: The behavior of an ultracold matter wave-packet under confinement conditions by an impurity near the delocalization threshold is of fundamental importance in atomic physics, particularly in atom chip technology. We study the effect of a waveguide impurity on an ultracold matter wave-packet at the threshold of delocalization. The impurity is modeled by a 1D short range square well potential with depth V0 and width 2R0 coupled to a contact impurity at the center of the bend. We report the ground state exact solution for the time-independent nonlinear Schrodinger equation that describes a Bose–Einstein condensate at the delocalization threshold, obtaining the density profile and the maximum nonlinear coupling constant, gmax. This allows us to obtain the maximum number of atoms, Nmax, that the defect potential can localize for the ground state. We find that for small impurity reduced size, ξ = V 0 R 0 , the maximum number of particles and nonlinear coupling constant, gmax, that characterize a contact impurity, become constant. For high values of ξ, we recover the Thomas–Fermi result. With the analytic solution, we report the full width at half maximum for the wave-function and density profile finding a large tunneling probability for small confining conditions. Implications of these findings to atom chips are discussed.
TL;DR: In this article, the scattering of spin-1 bosons in a nonminimal vector double-step potential is described in terms of eigenstates of the helicity operator and it is shown that the transmission coefficient is insensitive to the choice of the polarization of the incident beam.
Abstract: The scattering of spin-1 bosons in a nonminimal vector double-step potential is described in terms of eigenstates of the helicity operator and it is shown that the transmission coefficient is insensitive to the choice of the polarization of the incident beam. Poles of the transmission amplitude reveal the existence of a two-fold degenerate spectrum. The results are interpreted in terms of solutions of two coupled effective Schrodinger equations for a finite square well with additional δ-functions situated at the borders.
TL;DR: A system of few colloids confined in a small spherical cavity with event driven molecular dynamics simulations in the canonical ensemble is studied, identifying distinctive qualitative behavior over the thermodynamic plane T - ρ, in a few-particle equivalent to phase diagrams of macroscopic systems.
Abstract: We study a system of few colloids confined in a small spherical cavity with event driven molecular dynamics simulations in the canonical ensemble. The colloidal particles interact through a short range square-well potential that takes into account the basic elements of attraction and excluded-volume repulsion of the interaction among colloids. We analyze the structural and thermodynamic properties of this few-body confined system in the framework of inhomogeneous fluids theory. Pair correlation function and density profile are used to determine the structure and the spatial characteristics of the system. Pressure on the walls, internal energy, and surface quantities such as surface tension and adsorption are also analyzed for a wide range of densities and temperatures. We have characterized systems from 2 to 6 confined particles, identifying distinctive qualitative behavior over the thermodynamic plane T − ρ, in a few-particle equivalent to phase diagrams of macroscopic systems. Applying the extended law of corresponding states, the square well interaction is mapped to the Asakura-Oosawa model for colloid-polymer mixtures. We link explicitly the temperature of the confined square-well fluid to the equivalent packing fraction of polymers in the Asakura-Oosawa model. Using this approach, we study the confined system of few colloids in a colloid-polymer mixture.
TL;DR: In this paper, the transmission and reflection of a quantum particle incident upon a step potential increase, step potential decrease, a square well, and a square barrier were investigated. But the authors focused on the transmission of the particle incident.
Abstract: We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in und...
TL;DR: In this article, the effect of a wave-guide impurity on an ultra-cold matter wavepacket at the threshold of de-localization is modeled by a 1-D square well potential with depth V676 0 and length 2R676 0.
Abstract: The determination of the maximum number of atoms and the density profile of an ultra-cold wave-packet, under confinement conditions by an attractive impurity near the de-localization threshold, have been an open problem in ultra-cold atom physics. In this work, we study the effect of a wave-guide impurity on an ultra-cold matter wave-packet at the threshold of de-localization. The impurity is modeled by a 1-D square well potential with depth V
0 and length 2R
0. Coupling of the square well potential to a contact impurity of strength β at the center is also considered. The time-independent non-linear Schrodinger equation describing a Bose-Einstein condensate at the delocalization threshold is exactly solved. The density profile, maximum non-linear coupling constant, g
max, and maximum number of atoms, N
max, prompt to be localized by the defect potential in the ground and first excited states are also reported. It is shown that g
max and the density profiles become only functions of the reduced impurity size ξ = √V
0
R
0. It is also found that the first excited state at the threshold of de-localization exists only for ξ ≥ π/(2√2), always holding a lower number of atoms than the corresponding ground state for the same reduced impurity size. Also, the addition of a repulsive contact impurity leads to a non-linear coupling constant at the de-localization threshold lower than that of the square well potential. In spite of the non-linear character of the Gross-Pitaevskii equation, it is found that a general scaling-law holds for defects with the same ξ, related with the same g
max, having the same reduced density profile in the quasi-free direction. We report the full width at half maximum for the wave-function and density profile, finding a large spread for small reduced confining conditions. Implications of these results for the determination of the wave-packet properties under confinement in atom chip and Bose-Einstein condensates are presented with the aim to motivate further experimental work.
01 Jan 2015
TL;DR: In this article, the Efimov spectrum for a square well potential as two-body interaction was calculated and the universality of the three-body parameter a0 (−) was investigated.
Abstract: The most important goal of this bachelor thesis is to calculate the Efimov spectrum for a square well potential as two-body interaction and to investigate the universality of the three-body parameter a0 (−) . Since inelastic collisions should be incorporated into Efimov physics, the off-shell two-body T-matrix of the square well potential is calculated first. The obtained T-matrix is used as input for the Skorniakov-Ter-Martirosian equation from which the Efimov bound states can be calculated. The off-shell two-body T-matrix for s-wave scattering depends on the momentum of the incoming wave, the momentum of the scattered wave and on the energy of the two-particle system. It is symmetric under the exchange of the ingoing and outgoing momenta. An import characteristic of the off-shell T-matrix is that at higher order resonances, i.e. K0R = nπ/2 with n = 3, 5, 7, etc., the absolute maximum of the T-matrix does not occur at kR = 0 even if the absolute values of kR and qR are close to zero. In this case the maximum peak occurs at kR = K0R. The Efimov spectrum for a zero-range interaction potential is characterized by a universal scaling behavior of the trimer states. However, finite range effects of the interaction potential can lead to both non-universality in the scaling factor, especially for the lowest energy Efimov states, and a three-body parameter which cuts off the Efimov spectrum from below. The Efimov spectrum is calculated for the resonance condition K0R = π/2. It is shown that the deviation of the first scaling factor (a1 (−) /a0 (−) = 18) from the universal scaling factor 22.7 is probably the result of finite range effects. A surprising result of the Efimov spectrum is the absence of the parameter a0 (+) which means that the energy of the lowest-energy three-body bound state does not converge to the two-body bound state energy. The three-body parameter of the lowest-energy trimer state has also been found for the resonance condition K0R = π/2. It value is a0 (−) = −3R, so it does not equal the universal three-body parameter a0 (−) = −9.8 RvdW which is experimentally observed. The universal value of a0 (−) is more likely to be retrieved for deeper square well potentials. Further research is needed to confirm this hypothesis.
TL;DR: In this paper, the scattering of spin-1 bosons in a nonminimal vector double-step potential is described in terms of eigenstates of the helicity operator and it is shown that the transmission coefficient is insensitive to the choice of the polarization of the incident beam.
Abstract: The scattering of spin-1 bosons in a nonminimal vector double-step potential is described in terms of eigenstates of the helicity operator and it is shown that the transmission coefficient is insensitive to the choice of the polarization of the incident beam. Poles of the transmission amplitude reveal the existence of a two-fold degenerate spectrum. The results are interpreted in terms of solutions of two coupled effective Schrodinger equations for a finite square well with additional $\delta $-functions situated at the borders.
TL;DR: In this paper, a waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics, based on the well known resemblance between the guided and radiation modes of waveguide with the bound and scattering states of a quantum well.
Abstract: A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete since resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.
07 Sep 2015
TL;DR: In this paper, e + C60 and e + A@C60 elastic scattering is scrutinized versus the parameters of Uc(r) in order to identify Uc (r) which is best suited for studying electron-fullerene scattering and how the latter can be controlled by tuning the potential.
Abstract: Modelling of the C60 cage by a square-well potential Uc(r) is a popular approximation. In the literature, some inconsistency is present in choosing the magnitudes of parameters of Uc(r). In the present study, e + C60 and e + A@C60 elastic scattering is scrutinized versus the parameters of Uc(r) in order to identify Uc(r) which is best suited for studying electron-fullerene scattering and how the latter can be controlled by tuning the potential.
TL;DR: In this article, a simple model of the quasi-two-dimensional dipolar Bose gas confined in the one-dimensional square well potential is considered, where all dipoles are oriented along the confining axis.
Abstract: We consider a simple model of the quasi-two-dimensional dipolar Bose gas confined in the one-dimensional square well potential. All dipoles are assumed to be oriented along the confining axis. By means of hydrodynamic approach it is shown that the general structure of the low-lying excitations can be analyzed exactly. We demonstrate that the problem significantly simplifies in the high-density limit for which the density profile in the confined direction as well as the leading-order contribution to the ground-state energy and spectrum of elementary excitations are calculated. The low-temperature result for the damping rate of the phonon mode is also presented.
07 Sep 2015
TL;DR: In this paper, the spatial distribution of electric charges forming a square well potential has been analyzed and two model potentials of that type are proposed and their parameters have been calculated, and it has been shown that the phenomenological potentials simulating the potential of C60 shell belong to a family of potentials with non-flat bottoms.
Abstract: The spatial distribution of electric charges forming a square well potential has been analyzed. It has been shown that two concentric spheres each with a double layer of charges create the potentials of this type. It has been demonstrated that the phenomenological potentials simulating the potential of C60 shell belong to a family of potentials with non-flat bottoms. A C60 shell potential has been calculated under the assumption that it is formed by the averaged charge density of a neutral atom. It has been shown that this potential has a cuspshaped form. Two model potentials of that type are proposed and their parameters have been calculated.
TL;DR: In this paper, the Saxon-Woods potential of a spherical Fermi-Dirac gas is approximated by a truncated Dirac delta function near the center of the gas.
Abstract: Within a special scheme providing significant new results, we show that the (attractive) optical potential due to a spherical (non-relativistic) dilute and degenerate Fermi-Dirac gas becomes a Saxon-Woods potential which, in turn, is approximated by a truncated Dirac delta function near the center of the above spherical gas. This approximation is suitable to investigate phenomena in which fermions are restricted to move in relatively small spatial domains. In relation to this, we determine the corresponding fermion stationary wavefunctions which are found to be proportional to the aforementioned delta function if twice the fermion rest-mass multiplied by the total electron energy is much larger than the square of the reduced Planck constant. If this relationship is not fulfilled, the above fermion system is found to be roughly equivalent to fermions in an infinite one-dimensional potential well. In addition, the force field derived from the optical potential in question is determined as well as the chemical potential of the gas. Finally, application of our formulation to study electron transport in nanostructures is outlined.
Posted Content•
TL;DR: In this article, it was shown that the infinite square well is not the limit of a finite square well and that it needs a more delicate potential to confine particles inside a well.
Abstract: We show that it needs a more delicate potential to confine particles inside a well. The original model containing a vague notation of infinity in the potential energy is ambiguous. Using the Heaviside step function and the Dirac delta-function, we give a precise form for the confining potential. Although such form appears unusual, the ambiguities are resolved. This form also shows that the infinite square well is not the limit of a finite square well.
Posted Content•
04 Apr 2015
TL;DR: In this article, a waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics, based on the well known resemblance between the guided and radiation modes of waveguide with the bound and scattering states of a quantum well.
Abstract: A classical optics waveguide structure is proposed to simulate resonances of short range one-dimensional potentials in quantum mechanics. The analogy is based on the well known resemblance between the guided and radiation modes of a waveguide with the bound and scattering states of a quantum well. As resonances are scattering states that spend some time in the zone of influence of the scatterer, we associate them with the leaky modes of a waveguide, the latter characterized by suffering attenuation in the direction of propagation but increasing exponentially in the transverse directions. The resemblance is complete since resonances (leaky modes) can be interpreted as bound states (guided modes) with definite lifetime (longitudinal shift). As an immediate application we calculate the leaky modes (resonances) associated with a dielectric homogeneous slab (square well potential) and show that these modes are attenuated as they propagate.