Finite Square-well Potential Research Articles

Finite Square Well and Quantum State Discrimination

In this paper, the finite square well and its application are investigated, namely the quantum state discrimination. The finite square well is treated in all standard textbooks on introductory quantum mechanics. It is used as a simple ‘model of departure’ in many areas of physics. In atomic and molecular physics, it may be used as a model of an electron moving in the mean field of a linear molecule It also arises as the partial wave radial equation for a spherically symmetric, finite square-well potential. The Schrodinger equation for finite square well is solved. The domain is divided into three regions by the existing potential V 0, so for convenience, those three regions are named Region I, Region II, and Region III, respectively. Specifically, the potential of Region I and III are V 0, and that of Region II is 0. Also, a constant is needed to make this wave function normalized. Then since it is a wave function, it is necessary to make sure the function is continuous and differentiable everywhere within the domain. Besides, the wave function needs to be either odd or even just like infinite square well. After that, the wave functions for the three intervals can be obtained, and the exact quantum state is distinguished from a group of different quantum states. In the end, two wave functions are obtained; one for even form and one for odd form. Then all the energy level of the corresponding function with different wavelength needs to be found and listed. Local discrimination of orthogonal quantum states has attracted much attention during the last twenty years. The results are applied to the quantum-information task of state discrimination, by using the obtained six states in finite square well. It is assumed that all the quantum states are locally distinguishable, and the six states are distinguished using the hypothesis of quantum measurements.

Implementation of a quantum algorithm to estimate the energy of a particle in a finite square well potential on IBM quantum computer

In this paper, we implement a quantum algorithm -on IBM quantum devices, IBM QASM simulator and PPRC computer cluster -to find the energy values of the ground state and the first excited state of a particle in a finite square-well potential. We use the quantum phase estimation technique and the iterative one to execute the program on PPRC cluster and IBM devices, respectively. Our results obtained from executing the quantum circuits on the IBM classical devices show that our circuits succeed at simulating the system. However, duo to scattered results, we execute only the iterative phase estimation part of the circuit on the 5 qubit quantum devices to reduce the circuit size and obtain low-scattered results.

An infinite square-well potential as a limiting case of a square-well potential in a minimal-length scenario

One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same as one found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement with those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution operator.

Brownian cluster dynamics with short range patchy interactions: its application to polymers and step-growth polymerization.

We present a novel simulation technique derived from Brownian cluster dynamics used so far to study the isotropic colloidal aggregation. It now implements the classical Kern-Frenkel potential to describe patchy interactions between particles. This technique gives access to static properties, dynamics and kinetics of the system, even far from the equilibrium. Particle thermal motions are modeled using billions of independent small random translations and rotations, constrained by the excluded volume and the connectivity. This algorithm, applied to a single polymer chain leads to correct static and dynamic properties, in the framework where hydrodynamic interactions are ignored. By varying patch angles, various local chain flexibilities can be obtained. We have used this new algorithm to model step-growth polymerization under various solvent qualities. The polymerization reaction is modeled by an irreversible aggregation between patches while an isotropic finite square-well potential is superimposed to mimic the solvent quality. In bad solvent conditions, a competition between a phase separation (due to the isotropic interaction) and polymerization (due to patches) occurs. Surprisingly, an arrested network with a very peculiar structure appears. It is made of strands and nodes. Strands gather few stretched chains that dip into entangled globular nodes. These nodes act as reticulation points between the strands. The system is kinetically driven and we observe a trapped arrested structure. That demonstrates one of the strengths of this new simulation technique. It can give valuable insights about mechanisms that could be involved in the formation of stranded gels.

Vorticity of threshold strength in neutron drip-line nuclei

By adopting a finite square-well potential, the analytic expressions of vorticity strengths, which are related to transverse currents, are derived for some selected E1 and E2 transitions. From these analytic expressions it can be seen clearly that, like the threshold strengths, the vorticity strengths are also sharply distributed just above the threshold in nuclei near the drip-line and mainly come from the orbits with small binding energy.

Electron–electron interaction effects in heliumlike atoms confined in finite external square-well potential

A B-spline-based configuration interaction method is used to compute the energy levels of the ground and a few excited states of heliumlike atoms confined in a finite external square-well potential, as a function of the depth of the confining shell potential. The electron probability density and the dependence of the energy levels in the shell potential are used to account for the electron-electron interaction when the atoms are submitted to such an environment.

Finite-well potential in the 3D nonlinear Schrödinger equation: application to Bose-Einstein condensation

Using variational and numerical solutions we show that stationary negative-energy localized (normalizable) bound states can appear in the three-dimensional nonlinear Schr\"odinger equation with a finite square-well potential for a range of nonlinearity parameters. Below a critical attractive nonlinearity, the system becomes unstable and experiences collapse. Above a limiting repulsive nonlinearity, the system becomes highly repulsive and cannot be bound. The system also allows nonnormalizable states of infinite norm at positive energies in the continuum. The normalizable negative-energy bound states could be created in BECs and studied in the laboratory with present knowhow.

Analytical study of resonant transport of Bose-Einstein condensates

We study the stationary nonlinear Schr\"odinger equation, or Gross-Pitaevskii equation, for a one-dimensional finite square-well potential. By neglecting the mean-field interaction outside the potential well it is possible to discuss the transport properties of the system analytically in terms of ingoing and outgoing waves. Resonances and bound states are obtained analytically. The transmitted flux shows a bistable behavior. Novel crossing scenarios of eigenstates similar to beak-to-beak structures are observed for a repulsive mean-field interaction. It is proven that resonances transform to bound states due to an attractive nonlinearity and vice versa for a repulsive nonlinearity, and the critical nonlinearity for the transformation is calculated analytically. The bound-state wave functions of the system satisfy an oscillation theorem as in the case of linear quantum mechanics. Furthermore, the implications of the eigenstates on the dymamics of the system are discussed.

Role of low-lcomponent in deformed wave functions near the continuum threshold

The structure of deformed single-particle wave functions in the vicinity of zero energy limit is studied using a schematic model with a quadrupole deformed finite square-well potential. For this purpose, we expand the single-particle wave functions in multipoles and seek for the bound state and the Gamow resonance solutions. We find that, for the ${K}^{\ensuremath{\pi}}={0}^{+}$ states, where K is the $z$-component of the orbital angular momentum, the probability of each multipole components in the deformed wave function is connected between the negative energy and the positive energy regions asymptotically, although it has a discontinuity around the threshold. This implies that the ${K}^{\ensuremath{\pi}}={0}^{+}$ resonant level exists physically unless the $l=0$ component is inherently large when extrapolated to the well bound region. The dependence of the multipole components on deformation is also discussed.

Analytical solution of the finite quantum square-well problem

The bounded positive- and negative-parity spectrum E(?)n of the symmetric, one-dimensional, finite quantum square-well potential is computed exactly, explicitly and analytically in the form E(?)n = f(?)(n), where f(?) are known functions.

Direct Radiative Capture of Neutron in Drip-Line Nuclei

The analytic expressions of radial matrix elements⟨ℓc = 0|r|ℓh = 1⟩, ⟨ℓc = 1|r|ℓh = 0⟩ and ⟨ℓc = 1|r|ℓh = 2⟩ in a finite square-well potential are derived.Based on these analytic expressions of radial matrix elements, theneutron direct radiative capture (DRC) processes leading to boundp-orbit from incident s-wave, and leading to s- and d-orbits fromincident p-wave are discussed. For the DRC processes leading toloosely bound orbits, the dominant contributions to the radialmatrix elements come from the outer region of nuclear potential radiusR.

Nuclear halo and its scaling laws

We have proposed a procedure to extract the probability for valence particle being out of the binding potential from the measured nuclear asymptotic normalization coefficients. With this procedure, available data regarding the nuclear halo candidates are systematically analyzed and a number of halo nuclei are confirmed. Based on these results we have got a much relaxed condition for nuclear halo occurrence. Furthermore, we have presented the scaling laws for the dimensionless quantity $<r^{2}>/R^{2}$ of nuclear halo in terms of the analytical expressions of the expectation value for the operator $r^{2}$ in a finite square-well potential.

General series solution for finite square-well energy levels for use in wave-packet studies

We develop a series solution for the bound-state energy levels of the quantum-mechanical one-dimensional finite square-well potential. We show that this general solution is useful for local approximations of the energy spectrum (which target a particular energy range of the potential well for high accuracy), for global approximations of the energy spectrum (which provide analytic expressions of reasonable accuracy for the entire range of bound states), and for numerical methods. This solution also provides an analytic description of dynamical phenomena; with it, we compute the time scales of classical motion, revivals, and super-revivals for wave-packet states excited in the well.

Analytical investigation of revival phenomena in the finite square-well potential

We present an analytical investigation of revival phenomena in the finite square-well potential. The classical motion, revival, and super-revival time scales are derived exactly for wave packets excited in the finite well. These time scales exhibit a richer dependence on wave-packet energy and on potential-well depth than has been found in other quantum systems: They explain, for example, the difficulties in exciting wave packets with strong classical features at the bottom of a finite well, or with clearly resolved super-revivals in a shallow well. In the proper regions of validity, the time scales predict the instances of wave-packet reformation extremely accurately. Revivals at the bottom of the well are explored as a ``universal'' limit of the general theory, which offers the clearest connection with the series of fractional and full revivals seen in the dynamics of the infinite square-well potential.

On the energy levels of a finite square-well potential

The theory of Cauchy integrals and the Riemann problem is used to derive an explicit formula for the bound-state energy levels of the finite square-well potential problem and is used to obtain a simple asymptotic expression for them.

Superrevivals in the quantum dynamics of a particle confined in a finite square-well potential

We examine the revival features in wave packet dynamics of a particle confined in a finite square well potential. The possibility of tunneling modifies the revival pattern as compared to an infinite square well potential. We study the dependence of the revival times on the depth of the square well and predict the existence of superrevivals. The nature of these superrevivals is compared with similar features seen in the dynamics of wavepackets in an anharmonic oscillator potential.

Luminescence excitation spectroscopy on Ga0.47In0.53As/ Al0.48In0.52As quantum-well heterostructures.

We have performed low-temperature (5\char21{}30 K) photoluminescence excitation spectroscopy on ${\mathrm{Ga}}_{0.47}$${\mathrm{In}}_{0.53}$As/${\mathrm{Al}}_{0.48}$ ${\mathrm{In}}_{0.52}$As quantum-well heterostructures lattice matched to InP which were grown by molecular-beam epitaxy. Excitation spectroscopy using a tunable KCl:Tl color-center laser allows us to measure the energy shift of the first electron to heavy-hole subband transition in absorption for quantum-well widths down to 10 nm. In samples with well widths of 25 and 35 nm, higher subband transitions are also identified in the excitation spectrum. Comparison of the experimental subband energy shifts with calculated data shows good agreement for well widths down to 10 nm, when a finite square-well potential with conduction- and valence-band discontinuities of 0.5 and 0.2 eV, respectively, is assumed. This band offset is consistent with data determined by the C-V profiling technique.

Integral equations and scattering solutions for a square-well potential

s-wave scattering solutions of the Schrödinger equation with an arbitrary central local potential are considered. Green’s functions and integral equations are derived for scattering solutions subject to a variety of boundary conditions. Exact solutions are obtained for the case of a finite spherical square-well potential, and properties of these solutions are discussed.

Explicit results for the quantum-mechanical energy states basic to a finite square-well potential

The theory of complex variables is used to establish explicit expressions for the discrete energy states relevant to a square-well potential.

Quantum mechanics in a discrete space-time

A substitution rule which governs the replacement of derivatives by differences is derived, which leads to equations which have reasonable eigenvalues and commutation relationships for several simple physical problems. A dynamical equation is suggested and solved for a finite square-well potential.