Showing papers on "Finite potential well published in 2014"

Solution of the quantum finite square well problem using the Lambert W function

Abstract: We present a solution of the quantum mechanics problem of the allow- able energy levels of a bound particle in a one-dimensional nite square well. The method is a geometric-analytic technique utilizing the confor- mal mapping w ! z = we w between two complex domains. The solution of the nite square well problem can be seen to be described by the im- ages of simple geometric shapes, lines and circles, under this map and its inverse image. The technique can also be described using the Lambert W function. One can work in either of the complex domains, thereby obtain- ing additional insight into the nite square well problem and its bound energy states. There are many opportunities to follow up, and we present the method in a pedagogical manner to stimulate further research in this and related avenues.

A unidirectionally invisible $\mathcal{P}\mathcal{T}$-symmetric complex crystal with arbitrary thickness

Abstract: We introduce a new class of parity-time ()-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e., unidirectionally invisible, regardless of the thickness L of the crystal. The -symmetric complex crystal is synthesized by a supersymmetric (SUSY) transformation of a Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As L is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal -symmetric potential at the symmetry breaking point, where transparency breaks down as .

The Kronig-Penney model extended to arbitrary potentials via numerical matrix mechanics

Abstract: The Kronig-Penney model describes what happens to electron states when a confining potential is repeated indefinitely. This model uses a square well potential; the energies and eigenstates can be obtained analytically for a the single well, and then Bloch's Theorem allows one to extend these solutions to the periodically repeating square well potential. In this work we describe how to obtain simple numerical solutions for the eigenvalues and eigenstates for any confining potential within a unit cell, and then extend this procedure, with virtually no extra effort, to the case of periodically repeating potentials. In this way one can study the band structure effects which arise from differently-shaped potentials. One of these effects is the electron-hole mass asymmetry. More realistic unit cell potentials generally give rise to higher electron-hole mass asymmetries.

Spectrum of the three dimensional fuzzy well

Abstract: We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrodinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high energy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultra-violet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.

Non-equilibrium Liouville and Wigner equations: moment methods and long-time approximations

Abstract: We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external “heat bath” (hb) with negligible dissipation. For the classical equilibrium Boltzmann distribution, Wc,eq, a non-equilibrium three-term hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate long-time thermalization. That gives partial dynamical support to Boltzmann’s Wc,eq, out of the set of classical stationary distributions, Wc;st, also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without hb (by using Wc,eq), the long-time approximate thermalization for three-term hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum one-particle system through the non-equilibrium Wigner function, W. Weq for a repulsive finite square well is reported. W’s (< 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (q). That yields orthogonal polynomials, HQ,n(q), for Weq (and for stationary Wst), non-equilibrium moments, Wn, of W and hierarchies. For the first excited state of the harmonic oscillator, its stationary Wst is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with Weq) for the Wn’s are not three-term ones. As an illustration, we outline a non-equilibrium four-term hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate long-time approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant Weq and a non-equilibrium equation for W are reported: the non-equilibrium hierarchy could plausibly be a three-term one and possibly not far from Gaussian, and thermalization could possibly be justified.

High-fidelity pseudopotentials for the contact interaction

Abstract: The contact interaction is often used in modeling ultracold atomic gases, although it leads to pathological behavior arising from the divergence of the many-body wave function when two particles coalesce. This makes it difficult to use this model interaction in quantum Monte Carlo and other popular numerical methods. Researchers therefore model the contact interaction with pseudopotentials, such as the square well potential, whose scattering properties deviate markedly from those of the contact potential. In this article, we propose a family of pseudopotentials that reproduce the scattering phase shifts of the contact interaction up to a hundred times more accurately than the square well potential. Moreover, the pseudopotentials are smooth, resulting in significant improvements in efficiency when used in numerical calculations.

States of a quantum particle exhibiting complex quasi-energy in a dynamic trap

Abstract: States of a single quantum particle in a one-dimensional dynamic trap, which correspond to complex values of quasi-energy, are found The trap is described by a square potential well with periodic smallamplitude oscillations of barrier position The specific feature of the problem is the fact that the dynamic trap affords simultaneous localization and excitation of the confined particle The complex nature of the quasienergy is caused by finite probability of particle escape from the trap due to finite depth of the potential well Within the framework of the second-order perturbation theory with respect to modulation depth, the dependence of the decay rate of the states on modulation frequency is determined, which proves to be nonmonotonic

The modulation of soliton dynamics in Bose–Einstein condensate by a double square well potential

Abstract: By using multiple-scale method, we analytically study the soliton dynamics of Bose–Einstein condensate (BEC) trapped in a double square well potential. It is shown that dark soliton can be stable, which is generated in BEC trapped in a double square well. The amplitude and the width of dark soliton are related to the initial position of the soliton. Within the double square well, because of the confinement of the potential, the amplitude of the dark soliton is larger, while the width of the dark soliton is smaller. Furthermore, we also find that the amplitude and the width of the soliton are closely related to the depth of the double square well. With the increase of the depth, the amplitude of the dark soliton increases remarkably, while the width of the dark soliton decreases slowly.

Studying and removing effects of fixed topology in a quantum mechanical model

Abstract: At small lattice spacing, or when using e.g. overlap fermions, lattice QCD simulations tend to become stuck in a single topological sector. Physical observables then differ from their full QCD counterparts by 1/V corrections, where V is the spacetime volume. Brower et al. and Aoki et al. have derived equations by means of a saddle point approximation, to determine and to remove these corrections. We extend these equations and apply them to a simple toy model, a quantum mechanical particle on a circle in a square well potential at fixed topology. This model can be solved numerically up to arbitrary precision and allows to explore effects arising due to fixed topology. We investigate the range of validity and accuracy of the above mentioned equations, to remove such fixed topology effects.

Recent progresses on the pseudospin symmetry in single particle resonant states

Abstract: The pseudospin symmetry (PSS) is a relativistic dynamical symmetry directly connected with the small component of the nucleon Dirac wave function Much effort has been made to study this symmetry in bound states Recently, a rigorous justification of the PSS in single particle resonant states was achieved by examining the asymptotic behaviors of the radial Dirac wave functions: The PSS in single particle resonant states in nuclei is conserved exactly when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign Several issues related to the exact conservation and breaking mechanism of the PSS in single particle resonances were investigated by employing spherical square well potentials in which the PSS breaking part can be well isolated in the Jost function A threshold effect in the energy splitting and an anomaly in the width splitting of pseudospin partners were found when the depth of the square well potential varies from zero to a finite value

Wigner Distribution Functions as a Tool for Studying Gas Phase Alkali Metal Plus Noble Gas Collisions

Abstract: : Wave packet propagation methods are used to compute scattering Wigner Distribution Functions (WDF) for the square well potential, square barrier potential, and the H+H2 and OH+CO, and several M+Ng collisions. The scattering WDF are used to interpret how probabilities flow among various potential energy surfaces as a function of time during a collision. Positive values of the scattering WDF correspond to the addition of probability to scatter into the state that corresponds to the asymptotic limit of the potential energy surface of interest. Negative values correspond to the loss of probability to scatter into the state that corresponds to the asymptotic limit of the surface of interest, and zero values correspond to probability associated with the wave packet that is still in the interaction region. The loss of probability on one surface corresponds to the addition of probability on another surface at different times. Bands of oscillating peaks and valleys that form in the structure of the scattering WDF correspond to presence of secondary transmission or reflection with significant probability. The square well frequencies at which probability arrive in a scattering channel corresponds to the depth and width of the well. Scattering WDF were computed for the following combinations: K+He, K+Ne, K+Ar, Rb+He, Rb+Ne, Rb+Ar, Cs+Ne, and Cs+Ar. The scattering WDF revealed that as the mass of the noble gas increased, a significant proportion of probability was transferred from the 2P3/2 pump state to the 2P1/2 lasing state for a larger number of total angular momentum values. Similarily, as the mass of the alkali metal decreased, there was a reduced transfer of probability to make a transition from the 2P3/2 state to the 2P1/2 state. The reduced probability to make a transition from the 2P3/2 to the 2P1/2 manifolds for the M+He collisions is compensated by the large average velocity of He.

A Theoretical Evaluation of Scattering Length for Two Coupled Square Well Potential as a Function of ΔμB

Abstract: An evaluation of the scattering length and effective range of an attractive square well potential for ultracold atomic gases were performed. Our theoretical results showed that for any potential with large positive scattering length has a bound state just below the continuum threshold of energy. The theoretically evaluated results are in good agreement with other theoretical workers.

Barrier penetration effects on thermopower in semiconductor quantum wells

Abstract: Finite confinement effects, due to the penetration of the electron wavefunction into the barriers of a square well potential, on the low–temperature acoustic-phonon-limited thermopower (TP) of 2DEG are investigated. The 2DEG is considered to be scattered by acoustic phonons via screened deformation potential and piezoelectric couplings. Incorporating the barrier penetration effects, the dependences of diffusion TP and phonon drag TP on barrier height are studied. An expression for phonon drag TP is obtained. Numerical calculations of temperature dependences of mobility and TP for a 10 nm InN/In xGa1−xN quantum well for different values of x show that the magnitude and behavior of TP are altered. A decrease in the barrier height from 500 meV by a factor of 5, enhances the mobility by 34% and reduces the TP by 58% at 20 K. Results are compared with those of infinite barrier approximation.

Polarization effect in scattering of Dirac particles by a central potential in the zero-energy limit

Abstract: The scattering of Dirac particles by a central potential is revisited. An analytic analysis is carried out for the behavior of the phase shifts and the cross sections for unpolarized and polarized particles in the zero-energy limit. Special attention is paid to the polarization effect in the zero-energy limit. For incident particles with longitudinal polarization, it turns out that their polarization is in general kept unchanged after scattering, consistent with naive physical judgement. However, when the central potential takes some special form such that it supports a critical-energy solution in some specific angular momentum channel, the situation is dramatically changed, and a considerable part of the scattered particles have their original polarization reversed. For a spherical square well potential with appropriate depth, up to 53.8% of the scattered particles may have their polarization reversed.