Infinite Well Research Articles

Evanescent Electron Wave-Spin.

This study demonstrates the existence of an evanescent electron wave outside both finite and infinite quantum wells by solving the Dirac equation and ensuring the continuity of the spinor wavefunction at the boundaries. We show that this evanescent wave shares the spin characteristics of the wave confined within the well, as indicated by analytical expressions for the current density across all regions. Our findings suggest that the electron cannot be confined to a mathematical singularity and that quantum information, or quantum entropy, can leak through any quantum confinement. These results emphasize that the electron wave, fully characterized by Lorentz-invariant charge and current densities, should be considered the true and sole entity of the electron.

Non-Hermitian momentum operator for the particle in a box

We construct a discrete non-Hermitian momentum operator, which implements faithfully the non-self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first nontrivial eigenstate. We show how to construct the corresponding Hermitian Hamiltonian for the infinite well as a concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered. Published by the American Physical Society 2024

Square-well Model for Superconducting Pair Potential

We study Andreev reflection in a one-dimensional square-well pair potential. We discuss the history of the model. The current-phase relation is presented as a sum over Matsubara frequencies. How the current arises from bound and continuum levels is found by analytic continuation. We discuss two limiting cases of the square-well potential, the zero-length well and the infinite well. The model is quantitatively valid in some cases but forms the basis for understanding a wide range of problems in inhomogeneous superconductivity and superfluidity.

Warning! The negative divergence of the stress-tensor does not always yield the Ehrenfest force.

It has been assumed that the negative divergence of all stress tensors in common use yields the same force. This work finds that this is untrue, and, in fact, can vary wildly. We demonstrate this for the hydrogen atom, the one-particle isotropic harmonic oscillator, and a particle in an infinite spherical well where the exact density, pair-density, and the first order reduced density matrix are known for ground and excited states without any approximation. The Ehrenfest stress-tensor is introduced as any stress-tensor whose negative divergence will yield the corresponding Ehrenfest force for the same system when the exact wave-function is utilized. Stress-tensors within the literature are examined to show those that are Ehrenfest stress-tensors. Those that differ are demonstrated by how they differ within an exact formulation. The proof that the negative divergence of an Ehrenfest stress-tensor yields the Ehrenfest force is summarized.

Mesin Carnot Quantum dengan Dua Partikel Tidak Berinteraksi dan Tidak Berdekatan

In this article, we study the quantum Carnot engine using a one-dimensional infinite well potential model in which two particles are non-interacting, indistinguishable and have the same mass. We find that the efficiency of this model is the same as that of the one-dimensional infinite potential well model in which there is one particle. Moreover, efficiency can be measured only when L_C/L_A≥2, where L_A and L_C are the potential well widths during isothermal and adiabatic expansion, respectively.

Wigner function dynamics with boundaries expressed as convolution

In the present paper a method of finding the dynamics of the Wigner function of a particle in an infinite quantum well is developed. Starting with the problem of a reflection from an impenetrable wall, the obtained solution is then generalized to the case of a particle confined in an infinite well in arbitrary dimensions. It is known, that boundary value problems in the phase space formulation of the quantum mechanics are surprisingly tricky. The complications arise from nonlocality of the expression involved in calculation of the Wigner function. Several ways of treating such problems were proposed. They are rather complicated and even exotic, involving, for example, corrections to the kinetic energy proportional to the derivatives of the Dirac delta–function. The presented in the manuscript approach is simpler both from analytical point of view and regarding numerical calculation. The solution is brought to a form of convolution of the free particle solution with some function, defined by the shape of the well. This procedure requires calculation of an integral, which can be done by developed analytical and numerical methods.

Fractality of Certain Quantum States

We prove the theorem announced in Phys. Rev. Lett. {\bf 85}:5022, 2001 concerning the existence and properties of fractal states for the Schr\"odinger equation in the infinite one-dimensional well.

Information-theoretic spreading measures of a particle confined in a 3D infinite spherical well

Information-theoretic spreading measures of a particle confined in a 3D infinite spherical well

Infinite spherical well as model of quantum carnot engine

The potential well is a simple example that generally used to present an understanding of quantum mechanics. In this article, we used infinite spherical well model to evaluate the thermodynamic processes in a quantum Carnot engine. The energy of the particles depended on the value of n and l lead to complex calculations. For simplicity we used the φ100 and φ200 quantum states to determine work and efficiency of a quantum Carnot machine. The results obtained show that efficiency depends on the value of which is the ratio of RC and RB.

Numerical Calculations of Energies for an Infinite Potential Well with Sinusoidal Bottom

Abstract: We present an investigation for a particle confined in an infinite well with sinusoidal bottom, using the perturbation theory and numerical solution for the Schrödinger equation to obtain the eigen energies and wavefunctions. Potential strength and potential oscillation dependence of the state are examined and analyzed. It is shown that the particle in a box with sinusoidal bottom does not show up the Klauder phenomenon when the perturbations are gradually reduced to zero. The research results show that the potential oscillation significantly affects certain quantum states and, therefore, the ability to manipulate the energy difference between the states. In addition, our results for the present system converge to their corresponding values for the unperturbed one in the high-potential oscillation limit. Keywords: Infinite well, Perturbation theory, Sinusoidal potential, Numerical calculations, Klauder phenomenon.

Evaluation of Effective Mass in InGaAsN/GaAs Quantum Wells Using Transient Spectroscopy.

Transient spectroscopies are sensitive to charge carriers released from trapping centres in semiconducting devices. Even though these spectroscopies are mostly applied to reveal defects causing states that are localised in the energy gap, these methods also sense-charge from quantum wells in heterostructures. However, proper evaluation of material response to external stimuli requires knowledge of material properties such as electron effective mass in complex structures. Here we propose a method for precise evaluation of effective mass in quantum well heterostructures. The infinite well model is successfully applied to the InGaAsN/GaAs quantum well structure and used to evaluate electron effective mass in the conduction and valence bands. The effective mass m/m0 of charges from the conduction band was 0.093 ± 0.006, while the charges from the valence band exhibited an effective mass of 0.122 ± 0.018.

A simple approach for extending up to the ultra-relativistic limit the theory of a non-relativistic Fermi gas

Background: The theory of a Fermi gas constitutes a useful physical model with multiple applications from solid state to stellar physics. The theory of a non-relativistic Fermi gas is relatively simple when compared with the one for a relativistic Fermi gas. Therefore, it would be useful if a rigorous theory of a relativistic Fermi gas could be constructed like the theory of a non-relativistic Fermi gas. Methods: Such a theory, for T > 0 K, is presented here. It is based on using a recently reported formula for the energies of a relativistic spin-0 particle in a three-dimensional infinite well. The Pauli exclusion principle is used for taking care of the fermion nature of the particles forming a Fermi gas. Results: A unified description of a Fermi gas at low temperatures is presented. This description is valid from the non-relativistic to the ultra-relativistic limits. Moreover, it is simple. The theory follows the relatively simple approach often used for constructing the theory of a non-relativistic Fermi gas. Conclusions: The ultra-relativistic results reported here are often obtained using a different approach than the one used in this work. These results are not new, but they corroborate the correctness of the presented approach. In addition, the relative simplicity of the approach presented in this work allows for teaching the theory of a relativistic Fermi gas in introductory quantum mechanics courses.

Generation of microwave harmonics by non-ohmic surface layers in quantum wells of compound semiconductors at low lattice temperatures

It is well known that the free electron ensemble in the material of a semiconductor structure, at times, may exhibit electrical nonlinearity, when an effectively large electric field is applied. Under the condition, when a microwave field is also superimposed at the input, the efficiency of generation of second harmonic has been estimated here, using the electrical nonlinearity of the surface layers in infinite triangular quantum wells of compound semiconductors at low lattice temperatures. The nonlinear electrical characteristics are obtainable in the widely used framework of the field-dependent effective electron temperature approximation. Some of the explicit low-temperature features, like the degeneracy of the ensemble, and their interaction with the piezoelectric phonons, are taken into account. The efficiency characteristics which are thus obtained for some practically used modulation doped heterostructures of compounds, like InSb, GaAs, and GaN, seem to be interesting. Moreover, the specific low-temperature features are found to effect significant changes in the efficiency characteristics of the layers of such compounds. How the present analysis leaves much scope for further refinement is also discussed.

Exact quantum-mechanical solution for the one-dimensional harmonic oscillator model asymmetrically confined into the infinite well

We constructed a new model of the non-relativistic quantum harmonic oscillator within the canonical approach. It is asymmetrically confined into the infinitely high potential well. Asymmetrical confinement effect is achieved thanks to the introduction of the effective mass that changes with position. It is shown that the model has exact solutions. Analytical expression of its wavefunctions of the stationary states is expressed through the Jacobi polynomials , whereas, energy spectrum is non-equidistant. At the limit, when confinement parameters a and b , defining the position of two walls of the potential well go to infinity, then obtained expressions of the wavefunctions also recover the wavefunctions expressed through the Hermite polynomials. • An exactly solvable new model of the non-relativistic quantum harmonic oscillator asymmetrically confined into the infinitely high potential well. • Asymmetrical confinement effect is achieved thanks to the introduction of the effective mass that changes with position. • Energy spectrum is non-equidistant. • The model has correct limit to the so-called Hermite oscillator when position of both walls of the potential well go to infinity.

Relativistic spin-0 particle in a box: Bound states, wave packets, and the disappearance of the Klein paradox

The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.

Topological effects of a disclination on quantum revivals

In this paper, we discuss the effects of the topology of a disclination on quantum revivals when a quantum particle is subject to an infinite cylindrical well and a logarithmic potential. By using the WKB (Wentzel, Kramers and Brillouin) approximation, we obtain the eigenvalues of energy and show that the influence of the topology of the disclination on them gives rise to an Aharonov–Bohm-type effect. Further, we discuss how the topology of a disclination influences the classical periods and the revival times.

A pedagogical extension of the one-dimensional Schrödinger’s equation to symmetric proximity effect system film sandwiches

This study sought to use Schrödigner’s equation to model superconducting proximity effect systems of symmetric forms. As Werthamer noted [Phys. Rev. 132(6), 2440–2445 (1963)], one to one analogies between the standard superconducting proximity effect equation and the one-dimensional, time-independent Schrödinger’s equation can be made, thus allowing one to model the behavior of proximity effect systems of metallic film sandwiches by solving Schrödinger’s equation. In this project, film systems were modeled by infinite square wells with simple potentials. Schrödinger’s equation was solved for sandwiches of the form S(NS)M and N(SN)M, where S and N represent superconducting and nonsuperconducting metal films, respectively, and M is the number of repeated bilayers, or the period. A comparison of Neumann and Dirichlet boundary conditions was carried out in order to explore their effects. The Dirichlet type produced eigenvalues for S(NS)M and N(SN)M sandwiches that converged for increasing M, but the Neumann type produced eigenvalues for the same structures that approached two different limits as M increased. This last behavior is unexpected as it implies a dependence on the type of the film end layer.

Influence of the adiabatic process on a quantum system demonstrated using the energy conservation principle

The adiabatic process is an old topic, which does not excite energy transitions and entropy changes in quantum systems. For undergraduate students, these two properties are easily accepted intuitively, but the general proof is complicated. Unlike ordinary textbooks, we use the principle of energy conservation in a one-dimensional infinite well to prove these properties. This paper will enable students to deepen their views on quantum mechanics and statistical physics.

Optimum Strategy for Perforating Distribution of Homogeneous Gas-Liquid Two-Phase Flow in Vertical Gas Wells.

Aiming to optimize the total production of homogeneous gas–liquid two-phase flow in perforated vertical gas wells, a new coupled reservoir–wellbore mathematical model is proposed which consists of the crushed zone damage skin factor, pressure gradient, flow rate, and liquid holdup. An optimum strategy consisting of two optimization problems is developed to study the effect of perforation distribution on the production of vertical wells under a steady-state inflow. Then, these optimization models are applied to infinite and finite conductivity gas wells and the optimal distribution of perforation could be determined. The results show that inflow rate profiles could be improved by optimizing the perforation location for vertical gas wells.

Open-circuit voltage photodetector architecture for infrared imagers

We theoretically and experimentally investigate the application of an open-circuit voltage photodetector (VocP) architecture for mid-wave infrared (MWIR, 3–5 μm) detection and imaging. In contrast to conventional reverse-bias (RB) operation of the diode, which generates a photocurrent that is proportional to the photon irradiance, we evaluate the potential of using unbiased diodes that generate an open-circuit voltage, VOC, under illumination. The predicted Noise Equivalent Differential Temperature (NEDT) of a VocP is inferior to conventional RB when we assume an infinite well capacity and fixed integration time, but the prediction reverses when the actual well capacity of a readout integrated circuit (ROIC) is taken into account. Therefore, for a focal plane array (FPA) with a ROIC, we predict superior NEDT for the VocP. To demonstrate this concept, we fabricated and tested a basic VocP unit-cell architecture by connecting the VOC anode of a MWIR photodiode to the gate of an n-type metal-oxide semiconductor transistor that is operated in sub-threshold. Very good agreement is obtained between the analytical model and the observed drain current of the transistor over three orders of photon irradiance (1015–1018 photons/sec-cm2). The decoupling of the diode photocurrent from the integration capacitor in the circuit leads to a lower dark current that allows for longer integration times and improved sensitivity. This potentially can have a great impact on the performance and functionality of FPAs, leading to FPAs with better NEDT at a higher operating temperature, wider dynamic range, and smaller pixel size leading to larger array formats.