Piecewise Constant Potentials Research Articles

A sum-over-paths approach to one-dimensional time-independent quantum systems

We present an alternative treatment for simple time-independent quantum systems in one dimension, which can be used in the context of an elementary introduction to quantum physics using the Feynman approach. The method is based on representation of the energy-dependent propagator (or Green function) as a sum of complex amplitudes over all possible paths, classical and non-classical, at fixed energy. We treat both confined and open systems with piecewise-constant potentials, obtaining exact results. We introduce an approximation scheme to extend the method to smooth potentials, recovering the Van Vleck-Gutzwiller propagator. Finally, we discuss the educational application of the method.

The similarity of attractive and repulsive forces on a lattice

On a lattice, as the momentum space is compact, the kinetic energy is bounded not only from below but also from above. It is shown that this somehow removes the distinction between repulsive and attractive forces. In particular, it is seen that a region with attractive force would appear forbidden for states with energies higher than a certain value, while repulsive forces could develop bound-states. An explicit transformation is introduced which transforms the spectrum of a system corresponding to a repulsive force, to that of a similar system corresponding to an attractive force. Explicit numerical examples are presented for discrete energies of bound-states of a particle experiencing repulsive force by a piecewise constant potential. Finally, the parameters of a specific one-dimensional (1D) translationally invariant system on continuum are tuned so that the energy of the system resembles the kinetic energy of a system on a 1D lattice. In particular, the parameters are tuned so that while the width of the first energy band and its position are kept finite, the gap between the first energy band and the next energy band goes to infinity, so that effectively only the first energy band is relevant.

Band structure analysis of an analytically solvable Hill equation with continuous potential

This paper concerns analytically solvable cases of Hill’s equation containing a continuously differentiable periodic potential. We outline a procedure for constructing the Floquet–Bloch fundamental system, and analyze the band structure of the system. The similarities to, and differences from, the cases of a piecewise constant periodic potential and the Mathieu potential, are illuminated.

The combined viscous semi-classical limit for a quantum hydrodynamic system with barrier potential

We investigate the viscous model of quantum hydrodynamics, which describes the charge transport in a certain semiconductor. Quantum mechanical effects lead to third order derivatives, turning the stationary system into an elliptic system of mixed order in the sense of Douglis–Nirenberg. In the case most relevant to applications, the semiconductor device features a piecewise constant barrier potential. In the case of thermodynamic equilibrium, we obtain asymptotic expansions of interfacial layers of the particle density in neighbourhoods of the jump points of this barrier potential, and we present rigorous proofs of uniform estimates of the remainder terms in these asymptotic expansions.

A renormalized potential-following propagation algorithm for solving the coupled-channels equations.

We derive a general renormalized potential-following propagation method that efficiently solves the coupled-channels equations. The step size is variable, the method is compatible with reactive boundary conditions, and the algorithm may be combined with other renormalized algorithms, such as renormalized Numerov. We diagonalize the coupling matrix and consider piece-wise constant and linear reference potentials. The constant reference potential algorithm is very simple to implement, yet for multichannel problems almost as accurate as the linear reference potential method. The applicability of the proposed algorithms to realistic problems is demonstrated for cold collisions of NH radicals. The renormalized approach has the advantage of producing wave functions in a straightforward way, which is illustrated for a shape resonance in NH-NH collisions. These scattering wave functions can be used to study ultracold photoassociation and near-threshold photodissociation.

Relativistic Corrections to the Theory of Nanostructures

On the example of a simple exactly solved model of the potential well formed by three piecewise-constant potentials, it is demonstrated that spin splitting of electronic levels (the Rashba effect) exists only for an approximate description with the Pauli equation. An exact problem solution based on the Dirac equation demonstrates that such splitting of levels in the potential well is absent. This result is obtained both analytically and numerically. It is demonstrated that this is valid for any systems with piecewise-constant potentials forming the potential well.

Saturation of black hole lasers in Bose-Einstein condensates

To obtain the end-point evolution of the so-called black hole laser instability, we study the set of stationary solutions of the Gross-Pitaevskii equation for piecewise constant potentials which admit a homogeneous solution with a supersonic flow in the central region between two discontinuities. When the distance between them is larger than a critical value, we recover that the homogeneous solution is unstable, and we identify the lowest energy state. We show that it can be viewed as determining the saturated value of the first (node-less) complex frequency mode which drives the instability. We also classify the set of stationary solutions and establish their relation both with the set of complex frequency modes and with known soliton solutions. Finally, we adopt a procedure \`a la Pitaevski-Baym-Pethick to construct the effective functional which governs the transition from the homogeneous to non-homogeneous solutions.

A study of electron transport through a multiple quantum well structure in the presence of an electric field, using the relativistic transfer matrix

We propose a computational model for obtaining the transmission coefficient of a relativistic electron transported through a semiconductor-based multiple quantum well structure, in the presence of an external, constant electric field. For this purpose, a relativistic version of the transfer matrix formalism for piecewise constant potentials, based on the Dirac equation, is used. The effects of the electric field and geometry on the relativistic transmission coefficient are investigated. The results obtained in the relativistic framework are compared with the non-relativistic ones. This model could be used in problems with polarization and point-dependent effective mass, with possible applications to the field of optoelectronic devices.

Comparing the relativistic and non-relativistic transmission coefficients of the electron transport through a quantum heterostructure

Abstract The Letter deals with the transport of relativistic electron through a quantum heterostructure in the presence of a constant bias voltage. The transmission coefficient associated to the relativistic moving through the considered system is computed using the transfer matrix formalism for piecewise constant potentials. The results are compared to those which correspond to the non-relativistic case, in which the Schrodinger equation is used. The comparison is performed for different values of the system length and the effect of the bias voltage is also studied.

Structure of discrete-potential fluids interacting via two piece-wise constant potentials with a hard-core

The structure of fluids whose molecules interact via potentials with a hard-core plus two piece-wise constant sections of different widths and heights has been studied by the perturbation theory based on the power series of an inverse temperature. The perturbation theory based on the reference system which incorporates both the repulsive and attractive potentials predicts accurate radial distribution function and direct correlation function of the discrete-potential fluids for the whole density regions. In particular, it is the most successful for the square-shoulder plus square-shoulder fluid and shifted square-shoulder fluid with a purely repulsive potential rather than the square-well plus square-well fluids with an attractive potential. It is found that the present theory predicts a more accurate structure than the rational-function approximation and conventional integral equations such as Percus–Yevick and hypernetted-chain theories in most of the cases.

Influence of Rashba spin-orbit interaction on the transmission and the dwell time of the electron tunneling through a multiple quantum well structure

In this paper we propose a model for computing the transmission coefficient and the dwell time of the electron tunneling through a multiple quantum well structure with Rashba spin-orbit interaction and with an external and constant electric field, using the transfer matrix formalism with piecewise constant potentials. The results indicate a difference between the transmission coefficients and the dwell times for spin-up and spin-down orientations. This difference can be modified by varying the electric field and the number of the barriers and wells, this opening the way to create a spin filtering in the time domain.

Structural properties of fluids interacting via piece-wise constant potentials with a hard core

The structural properties of fluids whose molecules interact via potentials with a hard core plus two piece-wise constant sections of different widths and heights are presented. These follow from the more general development previously introduced for potentials with a hard core plus n piece-wise constant sections [A. Santos, S. B. Yuste, and M. Lopez de Haro, Condens. Matter Phys. 15, 23602 (2012)] in which use was made of a semi-analytic rational-function approximation method. The results of illustrative cases comprising eight different combinations of wells and shoulders are compared both with simulation data and with those that follow from the numerical solution of the Percus-Yevick and hypernetted-chain integral equations. It is found that the rational-function approximation generally predicts a more accurate radial distribution function than the Percus-Yevick theory and is comparable or even superior to the hypernetted-chain theory. This superiority over both integral equation theories is lost, however, at high densities, especially as the widths of the wells and/or the barriers increase.

On the Spectrum of the Dirac Operator and the Existence of Discrete Eigenvalues for the Defocusing Nonlinear Schrödinger Equation

We revisit the scattering problem for the defocusing nonlinear Schrödinger equation with constant, nonzero boundary conditions at infinity, i.e., the eigenvalue problem for the Dirac operator with nonzero rest mass. By considering a specific kind of piecewise constant potentials we address and clarify two issues, concerning: (i) the (non)existence of an area theorem relating the presence/absence of discrete eigenvalues to an appropriate measure of the initial condition; and (ii) the existence of a contribution to the asymptotic phase difference of the potential from the continuous spectrum.

Quantum temporal probabilities in tunneling systems

We study the temporal aspects of quantum tunneling as manifested in time-of-arrival experiments in which the detected particle tunnels through a potential barrier. In particular, we present a general method for constructing temporal probabilities in tunneling systems that (i) defines ‘classical’ time observables for quantum systems and (ii) applies to relativistic particles interacting through quantum fields. We show that the relevant probabilities are defined in terms of specific correlation functions of the quantum field associated with tunneling particles. We construct a probability distribution with respect to the time of particle detection that contains all information about the temporal aspects of the tunneling process. In specific cases, this probability distribution leads to the definition of a delay time that, for parity-symmetric potentials, reduces to the phase time of Bohm and Wigner. We apply our results to piecewise constant potentials, by deriving the appropriate junction conditions on the points of discontinuity. For the double square potential, in particular, we demonstrate the existence of (at least) two physically relevant time parameters, the delay time and a decay rate that describes the escape of particles trapped in the inter-barrier region. Finally, we propose a resolution to the paradox of apparent superluminal velocities for tunneling particles. We demonstrate that the idea of faster-than-light speeds in tunneling follows from an inadmissible use of classical reasoning in the description of quantum systems.

Interfaces supporting surface gap soliton ground states in the 1D nonlinear Schrödinger equation

We consider the problem of verifying the existence of H1 ground states of the 1D nonlinear Schrödinger equation for an interface of two periodic structures: −u″+V(x)u−λu=Γ(x)|u|p−1uon R with V(x)=V1(x),Γ(x)=Γ1(x) for x≥0 and V(x)=V2(x),Γ(x)=Γ2(x) for x<0. Here V1,V2,Γ1,Γ2 are periodic, λ<minσ(−d2dx2+V), and p>1. The article [T. Dohnal, M. Plum, W. Reichel, Surface gap soliton ground states for the nonlinear Schrödinger equation, Comm. Math. Phys. 308 (2011) 511–542] provides for the 1D case an existence criterion in the form of an integral inequality involving the linear potentials V1,V2 and the Bloch waves of the operators −d2dx2+V1,2−λ. We choose here the classes of piecewise constant and piecewise linear potentials V1,2 and check this criterion for a set of parameter values. For the piecewise constant case the Bloch waves are calculated explicitly and for the piecewise linear case verified enclosures of the Bloch waves are computed numerically. The integrals in the criterion are evaluated via interval arithmetic, so rigorous existence statements are produced. Examples of interfaces supporting ground states are reported including ones for which ground state existence follows for all periodic Γ1,2 with ess supΓ1,2>0.

Local symmetries in one-dimensional quantum scattering

We introduce the concept of parity symmetry in restricted spatial domains -- local parity -- and explore its impact on the stationary transport properties of generic, one-dimensional aperiodic potentials of compact support. It is shown that, in each domain of local parity symmetry of the potential, there exists an invariant quantity in the form of a non-local current, in addition to the globally invariant probability current. For symmetrically incoming states, both invariant currents vanish if weak commutation of the total local parity operator with the Hamiltonian is established, leading to local parity eigenstates. For asymmetrically incoming states which resonate within locally symmetric potential units, the complete local parity symmetry of the probability density is shown to be necessary and sufficient for the occurrence of perfect transmission. We connect the presence of local parity symmetries on different spatial scales to the occurrence of multiple perfectly transmitting resonances and propose a construction scheme for the design of resonant transparent aperiodic potentials. Our findings are illustrated through application to the analytically tractable case of piecewise constant potentials.

The inverse scattering problem at fixed energy based on the Marchenko equation for an auxiliary Sturm–Liouville operator

A new approach is proposed to the solution of the quantum mechanical inverse scattering problem at a fixed energy. The method relates the fixed energy phase shifts to those arising in an auxiliary Sturm–Liouville problem via the interpolation theory of the Weyl–Titchmarsh m-function. Then a Marchenko equation is solved to obtain the potential. As an illustration test results are shown including the reconstructions of the piecewise constant and Coulomb-type potentials. A byproduct of the new framework is a demonstration of the connection found between the Gelfand–Levitan and Marchenko formalisms.

Transfer Matrix Approach to One-Dimensional Electron Transport in Graphene Sheets with Piecewise Constant Potentials

We studied one-dimensional electron transport in a system composed of two monolayer graphene sheets with an optional arrangement of di erent constant rectangular electrostatic potential barriers between them. We derived a generalized transfer matrix for the electron which passes through this system. Finally, we examined our model by applying it on a well known rectangular shape constant potential barrier and we obtained the same result from our method similar to the others.

-symmetric lattices with spatially extended gain/loss are generically unstable

We illustrate, through a series of prototypical examples, that linear parity-time () symmetric lattices with spatially extended gain/loss are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and piecewise constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact -symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical computations.

Calculating the Energy Spectrum of Complex Low-Dimensional Heterostructures in the Electric Field

An algorithm for solving the steady-state Schrödinger equation for a complex piecewise-constant potential in the presence of the E-field is developed and implemented. The algorithm is based on the consecutive matching of solutions given by the Airy functions at the band boundaries with the matrix rank increasing by no more than two orders, which enables the characteristic solution to be obtained in the convenient form for search of the roots. The algorithm developed allows valid solutions to be obtained for the electric field magnitudes larger than the ground-state energy level, that is, when the perturbation method is not suitable.