Delta Potential Research Articles

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

This study employs the use of the fourth order Numerov scheme to determine the eigenstates and eigenvalues of particles, electrons in particular, in single and double delta function potentials. For the single delta potential, it is found that the eigenstates could only be attained by using specific potential depths. The depth of the delta potential well has a value that varies depending on the delta strength. These depths are used for each well on the double delta function potential and the eigenvalues are determined. There are two bound states found in the computation, one with a symmetric eigenstate and another one which is antisymmetric. Keywords—Double Delta Potential, Eigenstates, Eigenvalue, Numerov Method, Single Delta Potential

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

Eigenvalues of Particle Bound in Single and Double Delta Function Potentials through Numerical Analysis

THE EFFECT OF EXTREME CONFINEMENT ON THE NONLINEAR-OPTICAL RESPONSE OF QUANTUM WIRES

This work focuses on understanding the nonlinear-optical response of a 1-D quantum wire embedded in 2-D space when quantum-size effects in the transverse direction are minimized using an extremely weighted delta function potential. Our aim is to establish the fundamental basis for understanding the effect of geometry on the nonlinear-optical response of quantum loops that are formed into a network of quantum wires. It is shown that in the limit of full confinement, the sum rules are obeyed when the transverse infinite-energy continuum states are included. While the continuum states associated with the transverse wavefunction do not contribute to the nonlinear optical response, they are essential to preserving the validity of the sum rules. This work is a building block for future studies of nonlinear-optical enhancement of quantum graphs (which include loops and bent wires) based on their geometry. These properties are important in quantum mechanical modeling of any response function of quantum-confined systems, including the nonlinear-optical response of any system in which there is confinement in at least one dimension, such as nanowires, which provide confinement in two dimensions.

Application of Quantum-Behaved Particle Swarm Optimization in Engineering Constrained Optimization Problems

To overcome the shortage of standard Particle Swarm Optimization(SPSO) on premature convergence, Quantum-behaved Particle Swarm Optimization (QPSO) is presented to solve engineering constrained optimization problem. QPSO algorithm is a novel PSO algorithm model in terms of quantum mechanics. The model is based on Delta potential, and we think the particle has the behavior of quanta. Because the particle doesn’t have a certain trajectory, it has more randomicity than the particle which has fixed path in PSO, thus the QPSO more easily escapes from local optima, and has more capability to seek the global optimal solution. In the period of iterative optimization, outside point method is used to deal with those particles that violate the constraints. Furthermore, compared with other intelligent algorithms, the QPSO is verified by two instances of engineering constrained optimization, experimental results indicate that the algorithm performs better in terms of accuracy and robustness.

Efficient thermodynamic description of multicomponent one-dimensional Bose gases

We present a new method of obtaining nonlinear integral equations characterizing the thermodynamics of one-dimensional multi-component gases interacting via a delta-function potential. In the case of the repulsive two-component Bose gas we obtain a simple system of two NLIE allowing for an efficient numerical implementation in contrast with the infinite number of coupled equations obtained by employing the Thermodynamic Bethe Ansatz. Our technique makes use of the Quantum Transfer Matrix and the fact that in a certain continuum limit multi-component gases can be obtained from appropriate anisotropic spin chains.

Bound States in Delta Function Potentials

We present a systematic treatment of the bound state structure of a short-range attractive interatomic potential in one, two, and three dimensions as its range approaches zero. This allows the evaluation of the utility of delta function potentials in the modeling of few-body systems such as nuclei, atoms, and clusters. The relation to scattering by delta function potentials is also discussed.

Neutron bound states in a supersymmetric double Dirac delta electrostatic potential

AbstractWe study neutron bound states in a particular (1+1)‐supersymmetric configuration. A simple physical setup is chosen so that the effective interaction becomes a particular kind of double Dirac delta potential. In this configuration it is shown that the neutron distinguishes between positive and negative electric charge as a consequence of a purely quantum mechanical effect. The theoretical background is presented and a possible application to a real‐life physical system is discussed.

Decay modes of two repulsively interacting bosons

We study the decay of two repulsively interacting bosons tunnelling through a delta potential barrier by a direct numerical solution of the time-dependent Schrödinger equation. The solutions are analysed according to the regions of particle presence: both particles inside the trap (in–in), one particle in and one particle out (in–out) and both particles outside (out–out). It is shown that the in–in probability is dominated by the exponential decay, and its decay rate is predicted very well from outgoing boundary conditions. Up to a certain range of interaction strength, the decay of in–out probability is dominated by the single-particle decay mode. The decay mechanisms are adequately described by simple models.

Acceleration of trapped particles and beams

The dynamics of a quantum particle bound by an accelerating delta-functional potential is investigated. Three cases are considered, using the reference frame moving along with the δ-function, in which the acceleration is converted into the additional linear potential. (i) A stationary regime, which corresponds to a resonance state, with a minimum degree of delocalization, supported by the accelerating potential trap. (ii) A pulling scenario: an initially bound particle follows the accelerating delta-functional trap, within a finite time. (iii) The pushing scenario: the particle, which was initially localized to the right of the repulsive delta-function, is shoved to the right by the accelerating potential. For the two latter scenarios, the lifetime of the trapped particle and the largest velocity to which it can be accelerated while staying trapped are found. Analytical approximations are developed for the cases of small and large accelerations in the pulling regime and also for a small acceleration in the stationary situation, and in the regime of pushing. The same regimes may be realized by Airy-like planar optical beams guided by a narrow bending potential channel or crest. Physical estimates are given for an atom steered by a stylus of a scanning tunnelling microscope, and for the optical beam guided by a bending stripe.

Solution of the quantum harmonic oscillator plus a delta-function potential at the origin: the oddness of its even-parity solutions

We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of the solution of the harmonic oscillator problem.

Effect of Coulombic friction on spatial displacement statistics

The phenomenon of Coulombic friction enters the stochastic description of dry friction between two solids and the statistic characterization of vibrating granular media. Here we analyze the corresponding Fokker-Planck equation including both velocity and spatial components, exhibiting a formal connection to a quantum mechanical harmonic oscillator in the presence of a delta potential. Numerical solutions for the resulting spatial displacement statistics show a crossover from exponential to Gaussian displacement statistics. We identify a transient intermediate regime that exhibits multiscaling properties arising from the contribution of Coulombic friction. The possible role of these effects during observations in diffusion experiments is briefly discussed.

TRANSMISSION PROPERTIES OF THE ONE-DIMENSIONAL ARRAY OF DELTA POTENTIALS

The problem of one-dimensional quantum wire along which a moving particle interacts with a linear array of N delta-function potentials is studied. Using a quantum waveguide approach, the transfer matrix is calculated to obtain the transmission probability of the particle. Results for arbitrary N and for specific regular arrays are presented. Some particular symmetries and invariances of the delta-function potential array for the N = 2 case are analyzed in detail. It is shown that perfect transmission can take place in a variety of situations.

The conductance of the asymmetric Aharonov–Bohm ring with a delta-potential barrier in the presence of Rashba spin–orbit interaction

This Letter deals with the effect of a delta-potential barrier on the one-dimensional and asymmetric Aharonov–Bohm ring with Rashba spin–orbit interaction. The analytic expressions of the transmission amplitude and conductance are derived by applying the transfer matrix method in the one-electron scattering formalism. Resorting to an adequate tuning of the Aharonov–Bohm magnetic flux and Rashba coupling strength, a control of the conductance can be done by varying the strength and the location of the barrier. This may be useful in the design of further mesoscopic spin filters by using suitable asymmetry configurations.

Tunneling in fractional quantum mechanics

We study tunneling through delta and double delta potentials in fractional quantum mechanics. After solving the fractional Schrödinger equation for these potentials, we calculate the corresponding reflection and transmission coefficients. These coefficients have a very interesting behavior. In particular, we can have zero energy tunneling when the order of the Riesz fractional derivative is different from 2. For both potentials, the zero energy limit of the transmission coefficient is given by , where α is the order of the derivative (1 < α ⩽ 2).

Stability and control of a confined 1D quantum system with time-dependent delta potentials

The evolution problem for a quantum particle confined in a 1D box and interacting with one fixed point through a time-dependent point interaction is considered. Under suitable assumptions of regularity for the time profile of the Hamiltonian, we prove the existence of strict solutions to the corresponding Schrödinger equation. The result is used to discuss the stability and steady-state local controllability of the wavefunction when the strength of the interaction is used as a control parameter.

Spin-dependent transport and spin-switching effect in graphene with magnetoelectric modulations

We investigate spin transport through magnetic potential structures in single graphene layer with the consideration of Zeeman effect. We find that the magnetic step leads to the spin-dependent transport, and the spin-polarized conductance has the oscillatory behavior as a function of fermion energy. Furthermore, the electrostatic delta-potential added to the magnetic step causes dramatic changes in the spin transport. The spin polarization can be reversed by the application of the electrostatic delta potential, leading to so called spin-switching effect. We expect that the spin-switching effect in this study gives rise to great convenience for experimental investigations on graphene spintronics.

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

We consider the one-dimensional focusing nonlinear Schr\odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski.

Schrödinger Equations for Delta Potential Barrier in Quantum Phase Space

The collision with the delta potential barrier and the bound state in the delta potential well for a particle are both typical problems in quantum mechanics. We solve strictly the eigenequations of a particle in the delta potential fields in phase space in order to offer another soluble example for physical systems. We define the Schrödinger equations of the delta potential fields in the phase space.

Coherent Tunneling Through Quantum Wire Tailored by Gaussian Profile

In this paper, we propose a novel structure of quantum waveguide. In this structure we tailored the quantum wire by Gaussian Profile. Thus, the Dirac-Delta function potentials are weighted according to Gaussian distribution function. We studied the electronic transmission properties through this tailored quantum waveguide structure. We have assumed that single free-electron channel is incident on the structure and the scattering of electrons is solely from the geometric nature of the problem. We have used the transfer matrix method to study the electron transmission. Coherent Tunneling is achieved through this structure, which is well-defined allowed conduction bands. The electronic conductance spectrum depends on the number of the Dirac delta function potential in the quantum wire. When the number of Dirac delta function potentials in the structure and their strengths are increased, both well defined conductance bands and sharper and narrower forbidden bands are formed. This novel structure has a good defect tolerance. The structure tolerates strength defect and tolerates position defect for the central Dirac delta function in the Gaussian distribution.

Stability and control of a confined 1D quantum system with time-dependent delta potentials

Stability and control of a confined 1D quantum system with time-dependent delta potentials