Time-independent Schrodinger Equation Research Articles

Simulation Analysis of Factors Affecting the Quantum Tunneling Effect in 1D Double Potential Barriers Structures

The working principle of a large number of modern scientific equipment has been applied to the quantum tunneling effect. Although the explanations for the relevant principles of this effect are clarified, the calculation process is very complex and difficult to understand. Previous studies combine specific applications without theoretical model references. This study adopts the method of using the time-independent Schrodinger equation to solve the wave function, which is more convenient to obtain the expression of transmission probability. Then, one discusses the relationship between transmission probability and potential barrier width, potential barrier spacing, and incident particle energy. The experimental results show that the transmission probability decreases exponentially from 1 to 0 as the potential barrier width gradually increases. As the potential barrier width increases, the transmission probability exhibits a clear periodic oscillation relationship, the oscillation period remains unchanged, and this paper derives the relationship between the oscillation period and the potential barrier width. As the incident energy of particles gradually increases, the transmission probability exhibits periodic changes, and the minimum value of transmission probability increases with the periodic changes. Based on the analysis, the smaller the potential barrier width, the greater the incident energy, and the more likely quantum tunneling occurs.

Numerical solution of the Schrödinger equation using Neural Networks in Python

The motion of quantum mechanical systems in physical sciences is described by partial differential equations, usually of second order with respect to spatial coordinates. The required solutions of the time-dependent type of equations are, in general, functions of the temporal variable t and the state vectors determining the positions of the system’s particles at the time t. Only a small number, however of those differential equations can however be solved analytically, while the majority of them must be solved numerically by applying specific very advanced integration techniques. Among these equations, the fundamental Schrödinger equation offers great insight towards numerical solving. In this work we present an effective method that solves numerically the time-independent Schrodinger equation on the basis of neural networks techniques. Analytical and numerical results for the radial part of this equation with the corresponding energies are then compared so as to estimate the performance of our method.

Algebraic Solution of the Quantum Harmonic Oscillator

The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.

Simulation of Time Independent Schrödinger Equation for Finite Potential Well Using the Graphical Solution Method

This paper investigates applying the approximate method; Graphical Solution Method (GSM), to theoretically solve the Time Independent Schrödinger Equation (TISE) in one dimension for a finite square well using MATLAB. With just few lines of MATLAB coding, calculating and plotting accurate eigenvalues (energy), eigenvectors (wave functions) and the bound eigenstates are possible for the finite square well of a negative potential (depth of the well) of -400 eV and a well width of 0.1 nm for an electron confined to this quantum well. These eigenvalues, eigenvectors and eigenstates are obtained and discussed. The found energy eigenvalues and states are discrete and yield physical acceptable solutions. The even and odd solutions of the TISE are also considered. The graphical solutions for the finite potential well are shown. The locations of discrete eigenvalues for even and odd solutions are also presented. These eigenvalues are tested confirming the correct eigenfunctions. The precision of these solutions depend on well width L and on the interval dx used to integrate the equation. Exact analytical solutions for this case are obtained and compared with results from the GSM. The accuracy and the convergence of the numerical results are easily checked. The results showed that the GSM can be considered as a suitable mean for determining the one dimensional solutions for the finite square well.

Using hermite neural networks to solve the time-independent schrodinger equation

Using hermite neural networks to solve the time-independent schrodinger equation

Fine structure calculations, polarizability and oscillator strengths for C VI ion embedded in Debye plasma applying accurate Numerov method

The dipole polarizabilities, dipole and quadruple oscillator strengths and relativistic energy corrections are calculated for plasma embedded C VI ion which is of relevance in astrophysics and plasma physics. Numerical simulation of Numerov method is used to find the energy eigenvalues and eigenfunctions by solving the time-independent Schrodinger equation. The dependence of dipole and quadruple polarizabilities, oscillator strengths and relativistic energy corrections on Debye length is shown.

Thermal properties of coulomb-type potentials in a magnetic field using the Phase – Integral model

The work examined some Coulomb-type potentials under the influence of magnetic field potential and some select thermodynamic functions. The combination of the chosen Coulomb-type potentials and the magnetic field formed the generalized potentials which were put into the time independent radial Schrodinger equation and solution for the energy eigenvalues obtained using the Phase – Integral model. Four thermodynamic functions were considered and the results showed that an increase in the reduced parameters and magnetic field brings about a depreciation of the generalized potentials and the thermodynamic functions as well as the removal of degeneracy and integer decrease in the energy eigenvalues. The study also showed the change of the property of entropy in the classical domain as it enters the quantum mechanical system.

An alternative method to obtain analytical solutions of the time-independent Schrödinger equation

ABSTRACT An alternative method for obtaining analytical solutions of the time-independent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The differential equation resulting from operating the TISE on the product wave function is compared with a differential equation with a known solution. This comparison produces a system of differential equations for the integrating factor. This system of equations is easily solved for model quantum systems such as the harmonic oscillator, the hydrogen atom and the Morse oscillator. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the TISE, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the TISE of physical models as the harmonic oscillator, the Hydrogen atom, and the Morse oscillator.

Triton Scattering Phase-Shifts for S-wave using Morse Potential

In this paper, the phase-shifts for neutron-dueteron (n-d) scattering have been determined using the molecular Morse potential as theoretical model of interaction. The Triton (n-d) 2S1/2 ground state initially has been chosen as -7.61 MeV to determine the model parameters using variational Monte-Carlo technique in combination with matrix methods numerical approach to solving the time independent Schrodinger equation (TISE). The obtained potential is incorporated into the phase function equation, which is solved using Runge-Kutta (RK) 4,5 order technique, to calculate the phaseshifts at various lab energies below 15 MeV, for which experimental data is available. The results have been compared with those obtained using another molecular potential named Manning-Rosen (MR) and have been observed to fare better. Finally, the Triton ground state has been chosen as its binding energy (BE), given by -8.481795 MeV, as determined from experimental atomic mass evaluation data and the calculations are repeated. It has been found that these phase-shifts from BE data are slightly better matched with experimental ones as compared to those obtained using -7.61 MeV ground state for Triton (n-d two-body system) modeled using Morse potential.

Numerical solution of Schrodinger equation for rotating Morse potential using matrix methods with Fourier sine basis and optimization using variational Monte‐Carlo approach

AbstractIn this paper, time independent Schrodinger equation (TISE) for rotating Morse potential is solved numerically using matrix mechanics approach with Fourier sine basis, to obtain the ro‐vibrational spectra for diatomic molecules, here for HCl. Alongside, an optimization procedure for extraction of best model parameters based on variational Monte‐Carlo technique (VMC) is implemented. The first three vibrational frequencies from simulation are compared with experimental data by calculating mean‐square error, called χ2‐value. Minimizing χ2 by VMC technique is akin to extracting inverted potential based on Morse function that best fits experimental data. Finally, these optimized parameters for Morse potential, along with centrifugal term, are utilized in TISE to obtain ro‐vibrational frequencies for HCl. It has been found that mean % error obtained for ro‐vibrational lines from VMC are two orders of magnitude smaller than those obtained from best parameter fits using multiple regression analysis. The programs have been implemented in Scilab, and matrix methods results have been validated with those obtained using analytical solutions from Nikiforov–Uvarov method and asymptotic iteration method.

Quantum Teaching-Learning-Based Optimization algorithm for sizing optimization of skeletal structures with discrete variables

In terms of optimization algorithm improvement, researchers utilize some strategies to establish the right balance between local and global search phases to escape from a local minimum in some NP-hard problems. The approach of quantum-behaved particles, as a novel strategy, uses uncertainty law and a distinct formulation obtained from solving the time-independent Schrodinger differential equation in the delta-potential-well function to update the solution candidates' positions. New formulation defines the local attractors as probable solutions between the best solution and the others to explore all solution space domain. Also, mentioned formulation employs the difference between the average solution and other ones as a new step size. Both local attractors and new step sizes guarantee diversification besides the inherent intensification. In the present paper, the teacher phase of the Teaching-Learning-Based Optimization algorithm (TLBO), which has been introduced recently, is improved using the formulation obtained from solving the time-independent Schrodinger equation to predict the probable positions of optimal solutions. The results show that QTLBO, an acronym for the Quantum Teaching- Learning- Based Optimization, improves the stability and robustness of the TLBO by defining the quantum teacher phase. The two planner rigid frames and two trusses with strength and serviceability constraints are chosen to verify the quality and performance of QTLBO. Comparing the results obtained from the proposed algorithm with those of the standard version of the TLBO algorithm and other literature methods shows that QTLBO increases the chance of finding a better solution besides improving the statistical criteria compared to the current TLBO.

Charge currents and induced magnetic fields in a bounded two-dimensional hydrogen atom

The present work involves solving the time-independent Schrodinger equation for a two-dimensional (2D) hydrogen atom under the combined effect of static electric field, magnetic field and circular confinement. A study of the influence of external fields as well as spatial confinement on charge currents and induced magnetic fields has been carried out. It has been found that applied magnetic field reduces Stark shifts. The presence of tight spatial confinement leads to increase in the magnitude of currents as well as induced magnetic fields. In fact, induced magnetic field of the order of 400 T has been observed in one of the cases. The results of the work may prove helpful in manipulating these currents and fields.

Static multipole polarizabilites of H-atom in modified ring-shaped potentials

The time-independent Schrodinger equation (TISE) is solved to evaluate energy eigenvalues and eigenfunctions of H-atom confined by one of the radial potentials, which is modified by a ring-shaped potential and further encaged in a spherical boundary. We have considered the following four radial potentials i.e., Debye, Exponential Cosine Screened Coulomb (ECSC), Hulthen( $$\alpha $$ ), and Hulthen(2 $$\alpha $$ ). Static $$2^{l}$$ -pole polarizability of H-atom in different modified ring confinement potential is evaluated for a range of screening parameter ( $$\alpha $$ ). Repulsive ring-shaped potential affects multipole polarizabilities and reduces the critical screening parameter in different modified ring confinement potentials. The confinement parameters $$\alpha $$ and $$\beta $$ considerably affect the amplitude of multipole polarizabilities. Size of the spherical boundary ( $$r_{0}$$ ) crucially affects multipole polarizabilities and overweighs the effect of $$\alpha $$ and $$\beta $$ . The results prove that as the screening parameter increases the polarizabilities corresponding to various radial potentials become almost equal, i.e., the response due to the different radial potentials becomes indistinguishable after a particular value of the screening parameter.

Quantum Trajectory Description of the Time-Independent (Inverse) Fermi Accelerator

We re-examine the (inverse) Fermi accelerator problem by resorting to a quantum trajectory description of the dynamics. Quantum trajectories are generated from the time-independent Schrodinger equation solutions, using a unipolar treatment for the (light) confined particle and a bipolar treatment for the (heavy) movable wall. Analytic results are presented for the exact coupled two-dimensional problem, as well as for the adiabatic and mixed quantum-classical approximations.

Bound States of Spherically Symmetric Potentials

We solve the time-independent Schrodinger equation for spherically symmetric potentials. First, we consider simple cases of a particle on a ring and a particle on a sphere to illustrate the degeneracy arising due to symmetry. We then consider three different spherically symmetric potentials: (i) spherical well potential, (ii) isotropic three-dimensional harmonic oscillator, and (iii) spherically confined isotropic three-dimensional harmonic oscillator. Our discussion mainly focuses on the energy levels of the bound states and the associated degeneracies. Finally, we calculate the heat capacity of endohedral fullerenes using two simple models—particle in a spherical box and confined harmonic oscillator.

A new implicit high-order six-step singularly P-stable method for the numerical solution of Schrödinger equation

In this paper, we present a new implicit six-step singularly P-stable method with vanished phase-lag and its derivatives up to fifth order for the numerical integration of the one-dimensional radial time independent Schrodinger equation. The periodicity region of the method is plotted and the numerical stability and phase properties of the new methods are analyzed. The advantage of the new method in comparison with similar methods—in terms of efficiency, accuracy and stability—have been shown by implementing them in the radial time-independent Schrodinger equation during the resonance problems with the use of the Woods–Saxon potential.

Numerical Solutions of Quantum Mechanical Eigenvalue Problems

A large class of problems in quantum physics involve solution of the time independent Schrodinger equation in one or more space dimensions. These are boundary value problems, which in many cases only have solutions for specific (quantized) values of the total energy. In this article we describe a Python package that ``automagically'' transforms an analytically formulated Quantum Mechanical eigenvalue problem to a numerical form which can be handled by existing (or novel) numerical solvers. We illustrate some uses of this package. The problem is specified in terms of a small set of parameters and selectors (all provided with default values) that are easy to modify, and should be straightforward to interpret. From this the numerical details required by the solver is generated by the package, and the selected numerical solver is executed. In all cases the spatial continuum is replaced by a finite rectangular lattice. We compare common stensil discretizations of the Laplace operator with formulations involving Fast Fourier (and related trigonometric) Transforms. The numerical solutions are based on the NumPy and SciPy packages for Python 3, in particular routines from the \btt{scipy.linalg}, \btt{scipy.sparse.linalg}, and \btt{scipy.fftpack} libraries. These, like most Python resources, are freely available for Linux, MacOS, and MSWindows. We demonstrate that some interesting problems, like the lowest eigenvalues of anharmonic oscillators, can be solved quite accurately in up to 3 space dimensions on a modern laptop --- with some patience in the 3-dimensional case. We demonstrate that a reduction in the lattice distance, for a fixed the spatial volume, does not necessarily lead to more accurate results: A smaller lattice length increases the spectral width of the lattice Laplace operator, which in turn leads to an enhanced amplification of the numerical noise generated by round-off errors.

Two-particles bounded states as a mechanism to weaken the Anderson localization in systems with structural disorder

Abstract We study numerically the effect of the on-site Hubbard interaction U on the Anderson localization of two-interacting particles in a one-dimensional lattice with structural disorder. By solving the time-dependent Schrodinger equation to follow the time evolution of an initially localized two-electron state, we show the inter-particle interaction employing a non monotonic behavior on the Anderson localization. This non monotonic behavior is consistent with the results depicted by several many-body models in random potentials, which suggests an universal phenomenology for disordered systems. By studying the numerical solution of the time-independent Schrodinger equation, we reveal the origin of this non monotonic behavior: competitive aspects of the bounded and unbounded states of two-particle eigenstates. Ultimately, we show the disorder destroying the coherent Bloch oscillations induced by an electric field, even while particles have the proper interaction strength to correlate their motions.

Thermal behaviour of hindered rotor in static electric and laser field.

We solve exactly the time independent Schrödinger equation (TISE) for hindered rotor confined in well shaped potential. Afterwards, the confined system is subjected to interact with laser and static electric field. TISE for the dressed confined rotor is solved using standard numerical method to get energy spectrum and eigenfunctions. Dependence of orientation and alignment parameters on static electric field and confining potential is studied. We also compute thermal properties like entropy and heat capacity of the system under consideration using canonical partition within statistical thermodynamic formalism. We have also studied the effect of external field parameters, confinement strength and temperature on these thermal properties.

Computation of intersubband transition energy and absorption spectra of binary capped GaAs/Al0.42Ga0.58As core-shell quantum dot

Intersubband transition energy is computed for both core-shell quantum dot (CSQD) and binary capped core-shell quantum dot (CCSQD) of cubic geometry by solving the time-independent Schrodinger equation using the finite difference method. The discretization of the structures in all three spatial directions generates sparse Hamiltonian matrices, which are diagonalized to obtain energy eigenstates for the conduction band. The transition energy for the lowest three energy eigenstates is compared for different structural parameters considering GaAs / Al0.42Ga0.58As CSQDs. CSQD capped with AlAs (CCSQD) viz GaAs / Al0.42Ga0.58As / AlAs shows higher eigenstates and transition energy, which decreases with the increase in core thickness. Furthermore, the optical properties of these structures have been investigated which are in concurrence with the obtained eigen energy. The broader tuning range and blueshifted higher absorption coefficient of CCSQD support significant application in quantum dot detectors and lasers.