Solution Of Schrodinger Equation Research Articles

Numerical solution of Schrodinger equation using compact finite differences method and the cubic spline functions

In this paper, a high-order method for solving the Schrodinger equation is introduced. We apply a compact finite difference approximation for discretizing spatial derivatives and we use the C1-cubic spline collocation method for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. We can obtain both pointwise approximations at the all mesh points and, a cubic spline solution in each space step by the method. Numerical results show that the method is an efficient technique for solving the one-dimensional Schrodinger equation.Â

Approximate Solution of Schrodinger Equation in D-Dimensions for Scarf Hyperbolic plus Non-Central Poschl- Teller Potential Using Nikiforov-Uvarov Method

The approximate analytical solution of Schrodinger equation in D-Dimensions for Scarf hyperbolic plus non-central Pocshl-Teller potential were investigated using Nikiforov- Uvarov method. The approximate bound state energy are given in the close form and the corresponding approximate wave function for arbitary l-state in D-dimensions are formulated in the form of generalized Jacobi Polynomials. Special case is given for the ground state in 3 dimensions. The existence of arbitrary dimensions increase bound state energy system. In the other hand, the existence of arbitrary dimensions decreases the amplitude of wave function. The effect of Scarf Hyperbolic potential increases the bound state energy of system. The effect of non central Poschl-Teller potential decreases the bound state energy of system.

Numerical study of carbon nanotube field effect transistors in presence of carbon–carbon third nearest neighbor interactions

In this paper, for the first time, we have used a more precise Hamiltonian matrix based on first nearest neighbor (1NN) and third nearest neighbor (3NN) carbon–carbon interactions to simulate carbon nanotube field effect transistors (CNTFETs). By taking the interactions with more distant neighbors into account, an improvement in tight-binding picture is gained. A self-consistent solution of Schrodinger equation based on nonequilibrium Green's function (NEGF) formalism coupled to a two-dimensional Poisson's equation for treating the electrostatics of the device has been employed to simulate CNTFETs. A tight-binding Hamiltonian with an atomistic (pz orbitals) mode space basis in the ballistic limits has been used to describe the carbon nanotube (CNT) region. Simulations show that in the presence of 3NN, the energy bandgap of the CNT decreases and consequently the simulated device has lower threshold voltage compared to a simulated device with just 1NN. Short channel effects study demonstrates that neglecting 3NN underestimates the subthreshold swing and overestimates ON/OFF current ratio. All these investigations show that for simulating a CNTFET more precisely, the 3NN interactions can be taken into account in addition to the 1NN.

Electrically-activated source extension graphene nanoribbon field effect transistor: Novel attributes and design considerations for suppressing short channel effects

In this paper a double gate graphene nanoribbon field effect transistor with electrically-activated source extension is proposed. Source region of the proposed structure includes two sections, an electrically-activated extension and a doped section. The electrically extension, which is located between doped source section and gate region, is biased independent of the gate to form a virtual extension for source. The electrically-activated extension creates a step in potential profile which increases the horizontal distance between conduction and valance bands at channel to source junction. This step reduces the probability of band to band tunneling, lowers the leakage current and improves drain induced barrier lowering. The devices have been simulated based on self consistent solution of Poisson and Schrodinger equations within non-equilibrium Green’s function formalism. In addition, the effects of the edge and third nearest neighbor are included for more accurate outcomes. Simulations show that the proposed structure is a more reliable device because of its higher ON/Off current ratio, shorter delay time, and smaller power delay product beside lower subthreshold swing than conventional graphene nanoribbon field effect transistor.

Semiclassical and quantum analysis of a free-particle Hermite wave function

In this paper we discuss a solution of the free particle Schrodinger equation in which the time and space dependence are not separable. The wavefunction is written as a product of exponential terms, Hermite polynomials and a phase. The peaks in the wavefunction decelerate and then accelerate around t = 0. We analyse this behaviour within both a quantum and a semi-classical regime. We show that the acceleration does not represent true acceleration of the particle but can be related to the envelope function of the allowed classical paths. Comparison with other "accelerating" wavefunctions is also made. The analysis provides considerable insight into the meaning of the quantum wavefunction.

Algorithm for computing a wave packet evolution of the time-dependent Schrödinger equation

Abstract. The algorithm implemented as FORTRAN 77 program TIME6T which calculates, with controlled accuracy, the wave-packet evolution of the onedimensional time-dependent Schrodinger equation on a finite time interval is presented. Symmetric implicit operator-difference multi-layer schemes based on decomposition of the evolution operator up to the sixth-order of accuracy with respect to the time step are utilized. This decomposition is obtained via the explicit truncated Magnus expansion and Pade approximations. The additional gauge transformations which provide the symmetry properties needed for discretization, within the framework of the high-order finite-element method, of the evolutionary boundary problem on a finite spatial interval with the first and/or second type boundary conditions are applied. Solution of time-dependent Schrodinger equation for the Poschl-Teller two-center problem is used to illustrate an efficiency of the proposed schemes by comparing the computational error and execution time with those obtained by conventional symmetric splitting exponential operator techniques using the Pade approximations and fast Fourier transform method. The program is applied to the benchmark calculations of the exactly solvable model of a one-dimensional time-dependent oscillator.

Energy of Nonlinear Optical Solitons for Optical Sensors

Abstract The proposal of this work is analysis the energy density of both solutions of nonlinear Schrodinger equation (NLS), spatial and temporal solutions and show that in some cases the correspondent energy for that solutions is proportional to the square of the solutions. This analysis can be strongly applied to the optical fiber sensors. We also present a method to obtain the solution of the generalized NLS based on the real part of the Lagrangian of the system. The method for calculating the power of optical solitons can be employed in the development of optical sensors and in the development of optical fiber sensors sensitive to nonlinear optical effects, and monitoring of non-linear optical scattering along the optical fiber.

The impact of high-k gate dielectric and FIBL on performance of nano DG-MOSFETs with underlapped source/drain regions

The impact of high-k gate dielectrics and fringing induced barrier lowering (FIBL) effects on a nano double gate MOSFET is studied over a wide range of dielectric permittivity using ballistic quantum simulation. The simulations are based on self-consistent solution of 2D Poisson equation and Schrodinger equation with open boundary conditions, within the Non-equilibrium Green's Function formalism. The numerical results show that the use of high-k gate at fixed equivalent oxide thickness (EOT), deteriorates the short channel effects due to FIBL effect. We show that the FIBL can be effectively suppressed by using underlapped source/drain region.

Quantum self-consistent calculation of the differential capacitance of a semiconductor film

A fast scheme for calculating the surface differential capacitance of a semiconductor film with an ohmic contact on the rear side is proposed within the self-consistent solution of the Schrodinger and Poisson equations. The method is considered using germanium as an example. Comparison with the results of the phenomenological calculation revealed the specifics of size-quantization effects on the capacitance-voltage characteristic of the film.

Error estimates for the Coupled Cluster method

The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrodinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the full-CI-limit) in (Schneider, 2009). Recently, we globalized the CC formulation to the full continuous space, giving a root equation for an infinite dimensional, nonlinear Coupled Cluster operator that is equivalent the full electronic Schrodinger equation (Rohwedder, 2011). In this paper, we combine both approaches to prove existence and uniqueness results, quasi-optimality estimates and energy estimates for the CC method with respect to the solution of the full, original Schrodinger equation. The main property used is a local strong monotonicity result for the Coupled Cluster function, and we give two characterizations for situations in which this property holds.

Transmission Canals for the Photon-Assisted Transport of Electron through the Double-Barrier Resonant Tunneling Structure

The resonance and non-resonance transmission canals of double-barrier resonant tunneling structure are established for the electron photon system using the exact solution of one-dimensional non-stationary Schrodinger equation expanded into the Fourier range. It is shown that besides the main and satellite, the mixed quasi-stationary states which cause the appearance of speci c transmission canals with the properties strongly dependent on the intensity and frequency of electromagnetic eld, exist in the nanostructure.

A Simple Scattering Cross-Section Calculation Method

We have calculated the general elastic and inelastic scattering differential and total cross-sections using a simple general solution of radial Schrodinger equation for a particle in a central field, generally. We have obtained the general formulae of the scattering amplitudes and elastic, inelastic (no-elastic) and total scattering cross-sections. We have made some applications. In these applications, the potentials were assumed to have Saxon-Woods shape plus spin-orbit interaction, centrifugal and Coulomb potentials. With these potentials, before for the thermal neutrons; the elastic, inelastic (neutron radiative capture) and total scattering cross-sections of different targets were calculated. Later, total scattering cross-sections for 32He particles of three different energies on 13 targets, from 94Be to 20882Pb were calculated. The calculated results were compared with experimental results. The results calculated have given satisfactory agreement with the available experimental results.

Class of invariants for a time dependent linear potential

General solution of the one-dimensional Schrodinger equation in presence of a time-dependent linear potential is reconsidered in the context of Lewis–Riesenfeld and unitary transformation approaches. Three invariant operators are constructed as limiting cases of a general Hermitian quadratic invariant and their instantaneous eigenfunctions are obtained. Then the corresponding solutions of Schrodinger equation for each invariant operator are derived. These solutions include all known solutions of the system. Furthermore, it is shown how different solutions can be related to each other.

3D numerical modeling of Quantum Dot using homotopy analysis

The 3D numerical modeling of nanoscale InGaAs quantum dot is developed and the characteristics of the device are presented. The exact potential and energy profile of the Quantum Dot are computed by obtaining the solution of 3D Poisson and Schrodinger equations using homotopy analysis. The dark current is estimated by considering the Quantum Dot density, applied voltage, length of quantum dot array, number of quantum dot array and temperature. The results obtained show that the dark current is strongly influenced by Quantum Dot density and applied voltage. The developed model is physics based one and overcomes the limitations of the existing analytical models. The model is validated by comparing the results obtained with the existing models.

English

In this paper, we obtain solution of Schrodinger equation in general Besov spaces. Precise results on and general Besov estimates of the maximal function of the solutions to the Schrodinger equation are given. The obtained results improve some recent results. Further, we shall consider estimates of general -norm and the general Besov type norm of integrals of this kind by means of the general Besov norm of the function f, and give -estimates of their maximal functions. Key words: Maximal function, general Besov norm, Schrodinger equation.

A meshfree method for the solution of two-dimensional cubic nonlinear Schrödinger equation

Abstract In this paper, an efficient numerical technique is developed to approximate the solution of two-dimensional cubic nonlinear Schrodinger equations. The method is based on the nonsymmetric radial basis function collocation method (Kansa's method), within an operator Newton algorithm. In the proposed process, three-dimensional radial basis functions (especially, three-dimensional Multiquadrics (MQ) and Inverse multiquadrics (IMQ) functions) are used as the basis functions. For solving the resulting nonlinear system, an algorithm based on the Newton approach is constructed and applied. In the multilevel Newton algorithm, to overcome the instability of the standard methods for solving the resulting ill-conditioned system an interesting and efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-conditioned system. Finally, the presented method is used for solving some examples of the governing problem. The comparison between the obtained numerical solutions and the exact solutions demonstrates the reliability, accuracy and efficiency of this method.

Solution of Schrodinger equation for Three Dimensional Harmonics Oscillator plus Rosen-Morse Non-central potential using NU Method and Romanovski Polynomials

The energy eigenvalues and eigenfunctions of Schrodinger equation for three dimensional harmonic oscillator potential plus Rosen-Morse non-central potential are investigated using NU method and Romanovski polynomial. The bound state energy eigenvalues are given in a closed form and corresponding radial wave functions are expressed in associated Laguerre polynomials while angular eigen functions are given in terms of Romanovski polynomials. The Rosen-Morse potential is considered to be a perturbation factor to the three dimensional harmonic oscillator potential that causes the increase of radial wave function amplitude and decrease of angular momentum length. Keywords: Schrodinger Equation, Three dimensional Harmonic Oscillator potential, Rosen-morse non-central potential, NU method, Romanovski Polynomials

Approximate Solution Of Schrodinger Equation For Eckart Potential Combined With Trigonormetric Poschl-Teller Non-Central Potential Using Romanovski Polynomials

The approximate analytical solution of Schrodinger equation for Eckart potential combined with trigonometric Poschl-Teller noncentral potential is investigated using Romanovski polynomial. The approximate bound state energy eigenvalues are given in the close form, the corresponding approximate radial eigenfunctions is formulated in term of Romanovski polynomials, and the angular wave function is also expressed in term Romanovski polynomial. The effect of the presence of trigonometric Poschl-Teller potential changes the state of angular wave function level.

Energy Spectra and Wave Function Analysis of q-Deformed Modified Poschl-Teller and Hyperbolic Scarf II Potentials Using NU Method and a Mapping Method

The solution of Schrodinger equations for q-deformed modified Poschl-Teller and hyperbolic Scarf II potentials are investigated using Nikiforov-Uvarov method and a mapping method. By applying a translation of spatial variable non-deformed potentials are mapped on to deformed potential or vice versa and as a product the potential’s parameters scaling obtained. The bound state energy spectra obtained using NU method are given in the close form and the corresponding wave functions are formulated in terms of the Jacobi Polynomials. The energy spectra and the radial wave functions of the system are also produced by a mapping using potential’s parameters scaling from the non-deformed to the deformed potential. Keywords :Energy spectra, wave-function, q-deformed hyperbolic Scarf II, q-deformed modified Poschl-Teller, Nikiforov-Uvarov method, a mapping method DOI: 10.7176/APTA-16-8

Optical properties of nano-multi-layered quantum dot: oscillator strength, absorption coefficient and refractive index

In this paper, the exact solution of Schrodinger equation for multi-layered quantum dot (MLQD) within the effective mass approximation and dielectric continuum model is obtained with finite and infinite confining potential (CP). The MLQD is a nano-structured semiconductor system that consists of a spherical core (GaAs) and a coated spherical shell (Ga $$_{1-x}$$ Al $$_{x}$$ As) as the whole dot is embedded inside a bulk material (Ga $$_{1-y}$$ Al $$_{y}$$ As). Using the obtained energies, wave functions and taking advantage of numeric calculations, the oscillator strength, refractive index and absorbtion coefficient change associated with intersubband electronic transition from the ground state to the first allowed excited state are investigated for different CPs (both finite and infinite) and shell thicknesses. The results show that all values of ground state energy for large core dot radius approach the same value (the energy of bulk material) independent of CPs and shell thicknesses. Also it is shown that the optical properties are strongly affected by the changes in CPs and shell thicknesses.