One-dimensional Potential Well Research Articles

Towards the Analytical Generalization of the Transcendental Energy Equation, Group Velocity, and Effective Mass in One-Dimensional Periodic Potential Wells with a Computational Application to Common Coupled Potentials

The analytical generalization for N periodic potential wells coupled to a probe rectangular-like potential and a zero potential is extremely important in the study of one-dimensional periodic potentials in solid state physics, e.g., in the calculation of transport, optical, and magnetic properties. These findings raise the possibility of calculating equations for the generalization of N arbitrary potentials related to any potential V(x) using special functions as a solution. In this work, a novel analytical generalization of the transcendental energy equation, group velocity, and effective mass for N-coupled potentials to a probe one-dimensional potential V=V(x) was proposed. Initially, two well-known linear periodic potentials V=V(x) were employed to obtain analytical solutions for rectangular-like and Dirac-delta potentials. Python libraries were used to easily represent the equations for one or two rectangular-like potentials coupled with an arbitrary potential, highlighting the transcendental energy, group velocity, and effective mass. The results showed that the group velocity behavior changed its orientation due to the sign of the potential, whereas the width of the potential V(x) strongly influenced the group velocity behavior. The effective mass was also modified by the potential shapes, and their combinations, both effective mass and group velocity, exhibited similar physical behaviors to those found in ordinary rectangular-like potentials.

Модель поведения квантовой частицы в потенциальной яме с подвижной стенкой

In this paper, we find an analytical expression for the energy spectrum of a particle in a one-dimensional potential well with infinitely high walls. One of the walls with the coordinate x = 0 is motionless, the other wall, which has the coordinate x = a at the initial moment of time, moves at a constant speed. The results obtained can be used to modulate the operation of vacuum nanodevices.

Application of the QDST algorithm for the Schrödinger particle simulation in the infinite potential well

This paper examines whether a quantum computer can efficiently simulate the time evolution of the Schrödinger particle in a one-dimensional infinite potential well. In order to solve the Schrödinger equation in the quantum register, an algorithm based on the Quantum Discrete Sine Transform (QDST) is applied. The paper compares the results obtained in this way with the results given by the previous method (based on the QFT algorithm).

Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential

The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential V=V(x). From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form V=V(x) approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.

Model of cognitive activity of the human brain based on the mathematical apparatus of quantum mechanics

Describing the transmission and processing of information by an individual is a fundamental issue in modern cognitive science. Previously, unique scientific models for information transfer between individuals [1], as well as models for cognitive activity [2], were developed. However, numerous presented models lack scalability and formalization, hindering their ability to provide a comprehensive explanation for information transfer processes and their distortion due to interaction with the external communicative environment. One aspect of human cognitive activity is that individuals think not based on a code, as a computer does, but through the interaction of various mental images. Despite the fact that these images have a specific material foundation in the form of electrical and chemical activity in the human brain, describing them using conventional mathematical models presents several difficulties [3]. Therefore, this paper suggests novel techniques for describing information images/representations’ functioning, which simulate an individual’s cognitive activity. These methods rely on the mathematical models of self-oscillatory and conventional quantum physics (including potential wells and virtual particles conventionally used in the physical domain) for characterizing basic interactions [4]. The authors adopt a phenomenological perspective and do not regard cognitive systems as quantum. The aim of this study ject is to establish a model for cognitive function in the human brain using the mathematical principles of self-oscillating quantum mechanics from the perspective of information imaging and representation. Methods. The theory proposes [5] that information images/representations (IR) share characteristic properties with virtual Feymann particles and other elementary particles. The human mind is presented as a one-dimensional potential well with finite walls of varying sizes and an internal potential barrier that simulates the boundary between consciousness and subconsciousness. The authors have applied parametrization based on this theory. As a result, we propose the foundations of the mathematical apparatus based on classical quantum mechanics, followed by the mathematical apparatus of self-oscillating quantum mechanics. The latter, though little known, may allow to predict certain cognitive functions of the human brain by modifying and applying it to non-quantum settings. An equation is derived for the state function of the information image of an individual engaging in cognitive activity. Primary calculations were conducted on the state functions of information images using a computer model. The movement patterns of the information image within and outside were then deduced.

Relativistic quantum heat engine with the presence of minimal length

The presence of a minimal length has a significant effect on relativistic physical systems. This paper discusses how minimal length affects the efficiency of a relativistic quantum heat engine. The working substance chosen is a Dirac particle trapped in a one-dimensional infinite potential well. In this paper, we calculate the efficiency of a quantum heat engine in three thermodynamic cycles, namely the Carnot, Otto, and Brayton cycles. The engine efficiency is calculated analytically and numerically. In this research, the minimal length is a correction factor for the relativistic energy. The result is that the minimal length could increase or decrease the efficiency of the relativistic quantum heat engine on the small potential width according to the particle mass, the expansion parameter, and the thermodynamic cycle.

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.

Composite basis set of plane wave and Gaussian function or spline function

By combining plane waves with Gaussian or spline functions, a new composite basis set is constructed in this work. As a non local basis vector, the plane wave basis group needs a large number of plane waves to expand all parts of the physical space, including the intermediate regions that are not important to our problems. Our basis set uses the local characteristics of Gaussian function or spline function at the same time, and controls the energy interval by selecting different plane wave vectors, in order to realize the partition solution of Hamiltonian matrix. Orthogonal normalization of composite basis sets is performed by using Gram-Schmidt’s orthogonalization method or Löwdin’s orthogonalization method. Considering the completeness of plane wave vector, a certain value of positive and negative should be selected at the same time. Here, by changing the absolute value of wave vector, we can select the eigenvalue interval to be solved. The plane wave with a specific wave vector value is equivalent to a trial solution in the region with gentle potential energy. The algorithm automatically combines local Gaussian or spline functions to match the difference in wave vector value between the trial solution and the strict solution. By selecting the absolute value of the wave vector in the plane wave function, the calculation of large Hamiltonian matrices turns into the calculation of multiple small matrices, together with reducing the basis numbers in the region where the electron potential changes smoothly, therefore, we can significantly reduce the computational time. As an example, we apply this basis set to a one-dimensional finite depth potential well. It can be found that our method significantly reduce the number of basis vectors used to expand the wave function while maintaining a suitable degree of computational accuracy. We also study the influence of different parameters on calculation accuracy. Finally, the above calculation method can be directly applied to the density functional theory (DFT) calculation of plasmons in silver nanoplates or other metal nanostructures. Given a reasonable tentative initial state, the ground state electron density distribution of the system can be solved by self-consistent solution through using DFT theory, and then the electromagnetic field distribution and optical properties of the system can be solved by using time-dependent density functional theory.

Illustrations of loosely bound and resonant states in atomic nuclei

Using the one-dimensional potential well with realistic parameters for atomic nuclei, we illustrate the movement of the poles of the S-matrix and the transmission coefficient when the well supports an anti-bound state. We calculate the phase shift of the atomic nuclei 5He using the three-dimensional potential well and compare it with the experimental one. The paper gives an introduction to some of the properties found in realistic loosely bound and resonant nuclear systems, using mathematics accessible to undergraduate students.

Excitation and ionization of a particle in a one-dimensional potential well of zero radius by an extremely short light pulse

The Migdal sudden perturbation approximation is used to solve the problem of excitation and ionization particles in a one-dimensional potential of zero radius with an extremely short pulse. There is only one energy level in such a one-dimensional the delta-shaped potential well. It is shown that for pulse durations shorter than the characteristic period of oscillations of the wave function of the particle in the bound state, the population of the level (and the probability of ionization) is determined by the ratio of the electric area of the pulse to the characteristic "scale" of the area inversely proportional to the size of localization of the particle in a bound state. Keywords: unipolar pulses, electrical pulse area, atomic scale of electric area.

Coherent control and creation of population gratings for a pair of attosecond pulses in a resonant medium based on one-dimensional rectangular quantum wells

Attosecond pulses can be used to create and control coherence in resonant media, since their duration is shorter than the population relaxation times T1 and medium polarization T2. Previously, the possibility of creating and ultrafast control of electromagnetically induced gratings (EMIG) of atomic populations in a resonant medium was shown using a sequence of extremely short light pulses, when the pulses coherently interact with the medium and do not simultaneously overlap in the medium. These studies were carried out in various approximations, when a finite number of energy levels of the medium is taken into account, or when the pulse amplitude is small. In this paper, based on a direct numerical solution of the time dependent Schredinger equation without the indicated approximations, we study the possibility of ultrafast coherent control of populations and the creation of an EMIG by a pair of attosecond pulses in a multilevel resonant medium with a low density of particles. The medium is modeled using a one-dimensional rectangular potential well with infinitely high walls. The studies performed show the possibility of ultrafast coherent control of the properties of resonant media based on quantum wells using attosecond pulses. Keywords: electromagnetically induced gratings, coherent interaction, extremely short pulses, unipolar pulses, attosecond pulses, medium coherence.

Когерентное управление и создание решеток населенностей парой аттосекундных импульсов в резонансной среде на основе одномерных прямоугольных квантовых ям

Attosecond pulses can be used to create and control coherence in resonant media, since their duration is shorter than the population relaxation times T_1 and medium polarization T_2. Previously, the possibility of creating and ultrafast control of electromagnetically induced gratings (EMIG) of atomic populations in a resonant medium was shown using a sequence of extremely short light pulses, when the pulses coherently interact with the medium and do not simultaneously overlap in the medium. These studies were carried out in various approximations, when a finite number of energy levels of the medium is taken into account, or when the pulse amplitude is small. In this paper, based on a direct numerical solution of the time dependent Schrödinger equation without the indicated approximations, we study the possibility of ultrafast coherent control of populations and the creation of an EMIG by a pair of attosecond pulses in a multilevel resonant medium with a low density of particles. The medium is modeled using a one-dimensional rectangular potential well with infinitely high walls. The studies performed show the possibility of ultrafast coherent control of the properties of resonant media based on quantum wells using attosecond pulses.

Quantum-Mechanical Substantiation of the Periodic System of Isotopes. Models of Nuclear Orbitals

The topic of this article lies in the field of problems: substantiating the periodic system of isotopes and the principle of multilevel periodicity using quantum mechanical calculations, combining strong and electromagnetic interactions, and searching for the fundamental cause of periodicity in general. This article is a theoretical section and a continuation of the article: "Periodic system of isotopes", in which the system was checked against 10 types of experimental data, the periodic change of properties at the level of nuclei and the vertical symmetry of subgroups of isotopes were found. Periodic system of isotopes was constructed with the help of a special algorithm, the principle of multilevel periodicity of the atom, from the electrons to the nucleus. As a description of the multilevel periodicity, this paper presents a unified system of quantum numbers, which is used to describe both electron and nucleon shells (binomial probabilistic interpretation). With the binomial interpretation the problem of a particle in a one-dimensional potential well has been solved; quantummechanical calculations for the probability functions of the orbitals and periods of both electrons and nucleons have been performed - characteristic equations have been obtained, the projections of electronic orbitals have been reproduced and the binomial interpretation has been shown to correspond to the family of spherical harmonics. For the electron orbitals the calculation and analysis of solutions of the Schrödinger equation for the binomial interpretation of quantum numbers have been performed. The spatial nature of quantum numbers, for this interpretation, in the form of degrees of freedom is shown. Based on the principle of multilevel periodicity, expressions are derived and planar projections of nucleon nucleon orbitals are constructed, and similarity of the forms with electron orbitals is analyzed and revealed. A critical analysis of the modern spherical coordinate system was made, possible errors in the construction of electron orbitals were shown and, taking into account the drawbacks, two alternative spherical coordinate systems were proposed, for which Lame coefficients were calculated and Laplace equations were derived. As a search for the fundamental cause of multilevel periodicity, a spatial model with changing degrees of freedom 0-n is presented, its manifestation in nature (crystal forms) is found; a number of experiments are proposed; the predictions about the applicability of the multilevel periodicity principle in quark theory are made

Quantum cycle in relativistic non-commutative space with generalized uncertainty principle correction

Quantum heat cycles and quantum refrigerators are analyzed using various quantum systems as their working mediums. For example, to evaluate the efficiency and the work done of the Carnot cycle in the quantum regime, one can consider the harmonic oscillator as a working medium. For all these well-defined working substances (which are analyzed in commutative space structure), the efficiency of the engine is not up to the mark of the Carnot efficiency. So, one inevitable question arise, can one observe a catalytic effect on the efficiency of the engines and refrigerators when the space structure is changed? In this paper, two different working substances in non-commutative spacetime with relativistic and generalized uncertainty principle corrections have been considered for the analysis of the efficiency of the heat engine cycles. The efficiency of the quantum heat engine gets a boost for higher values of the non-commutative parameter with a harmonic oscillator as the working substance. In the case of the second working medium (one-dimensional infinite potential well), the efficiency shows a constant result in the non-commutative space structure.

Josephson oscillations in split one-dimensional Bose gases

We consider the non-equilibrium dynamics of a weakly interacting Bose gas tightly confined to a highly elongated double well potential. We use a self-consistent time-dependent Hartree--Fock approximation in combination with a projection of the full three-dimensional theory to several coupled one-dimensional channels. This allows us to model the time-dependent splitting and phase imprinting of a gas initially confined to a single quasi one-dimensional potential well and obtain a microscopic description of the ensuing damped Josephson oscillations.

Algebraic structures and position-dependent mass Schrödinger equation from group entropy theory

Based on the group entropy theory, in this work, we generalize the algebra of real numbers (referred to as G-algebra) along with its associated calculus, thus obtaining the algebraic structures corresponding to the Tsallis and the $$\kappa $$ -statistics. From a G-deformed translation operator, we obtain its associated Schrodinger equation, that corresponds to a particle with an effective position-dependent mass determined by the G-algebra. The q-deformed (standard) Schrodinger equation results in a special case for the Tsallis (Boltzmann–Gibbs) group class. We illustrate the results with the one-dimensional potential well for the $$\kappa $$ and the Tsallis classes and we obtain a family of potentials associated with the group entropy classes by first principles.

Modeling Cognitive Activity of the Human Brain by the Mathematical Apparatus of Quantum Mechanics

This paper discusses the possible approaches to modeling the cognitive activity of the human brain using the mathematical apparatus of quantum mechanics (primarily – potential wells and virtual particles) in terms of the theory of information representations. The article briefly describes the proposed theory of information representations, draws analogies, and identifies common features of information representations of the human mind and Feynman’s virtual particles. The human mind is represented as a one-dimensional potential well with finite walls of different sizes and internal potential barrier simulating the boundary between consciousness and subconsciousness. This creates a foundation for a mathematical apparatus that can make it possible to forecast particular cognitive functions of the human brain. The results of these studies can be used to create predictive models of various cognitive disorders (diseases) and to be used in diagnostics.

The Generalized OTOC from Supersymmetric Quantum Mechanics—Study of Random Fluctuations from Eigenstate Representation of Correlation Functions

The concept of the out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further, we demonstrate an equivalent formalism of computation using a general time-independent Hamiltonian having well-defined eigenstate representation for integrable Supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism, we consider the two well-known models, viz. Harmonic Oscillator and one-dimensional potential well within the framework of Supersymmetry. For the Harmonic Oscillator case, we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both microcanonical and canonical ensembles in quantum mechanics without Supersymmetry. On the other hand, for the One-Dimensional PotentialWell problem, we found significantly different time scales and the other parameter dependence compared to the results obtained from non-Supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model-independent Hamiltonian, along with the previously mentioned well-cited models.

Thermodynamic optimization for a quantum thermoacoustic refrigeration micro-cycle

A model of quantum thermoacoustic refrigeration micro-cycle (QTARMC) is established in which heat leakage is considered. A single particle contained in a one-dimensional harmonic potential well is studied, and the system consists of countless replicas. Each particle is confined in its own potential well, whose occupation probabilities can be expressed by the thermal equilibrium Gibbs distributions. Based on the Schrodinger equation, the expressions of coefficient of performance (COP) and cooling rate for the refrigerator are obtained. Effects of heat leakage on the optimal performance are discussed. The optimal performance region of the refrigeration cycle is obtained by the using of Q objective function. The results obtained can enrich the thermoacoustic theory and expand the application of quantum thermodynamics.

Quantum Lenoir Engine with a Single Particle System in a One Dimensional Infinite Potential Well

Lenoir engine based on the quantum system has been studied theoretically to increase the thermal efficiency of the ideal gas. The quantum system used is a single particle (as a working fluid instead of gas in a piston tube) in a one-dimensional infinite potential well with a wall that is free to move. The analogy of the appropriate variables between classical and quantum systems makes the three processes for the classical Lenoir engine applicable to the quantum system. The thermal efficiency of the quantum Lenoir engine is found to have the same formulation as the classical one. The higher heat capacity ratio in the quantum system increases the thermal efficiency of the quantum Lenoir engine by 56.29% over the classical version at the same compression ratio of 4.41.