Differential Equations In Quantum Mechanics Research Articles

Abundant Exact Travelling Wave Solutions for a Fractional Massive Thirring Model Using Extended Jacobi Elliptic Function Method

The fractional massive Thirring model is a coupled system of nonlinear PDEs emerging in the study of the complex ultrashort pulse propagation analysis of nonlinear wave functions. This article considers the NFMT model in terms of a modified Riemann–Liouville fractional derivative. The novel travelling wave solutions of the considered model are investigated by employing an effective analytic approach based on a complex fractional transformation and Jacobi elliptic functions. The extended Jacobi elliptic function method is a systematic tool for restoring many of the well-known results of complex fractional systems by identifying suitable options for arbitrary elliptic functions. To understand the physical characteristics of NFMT, the 3D graphical representations of the obtained propagation wave solutions for some free physical parameters are randomly drawn for a different order of the fractional derivatives. The results indicate that the proposed method is reliable, simple, and powerful enough to handle more complicated nonlinear fractional partial differential equations in quantum mechanics.

Differential equations of quantum mechanics

We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems.

Students’ difficulties with partial differential equations in quantum mechanics

Students in upper division quantum mechanics encounter many difficulties understanding when to and then applying separation of variables technique to solve problems involving partial differential equations.

RETRACTED ARTICLE: Research on statistical algorithm optimization of fractional differential equations of quantum mechanics in ecological compensation

Aiming at the shortcomings of the current fractional calculus Fourier transform applied to image processing, it is necessary to artificially specify the differential order. This paper proposes an adaptive fractional differential, which can be applied to the aerobics view with higher real-time requirements. The adaptive fractional differential derivative Fourier transform selection can represent the fractal dimension of texture detail complexity as a parameter adaptive method to determine the order of the differential, but the commonly used fractional box dimension calculation method has relatively rough results. Its algorithmic shortcomings and an improved algorithm are proposed to make the fractal dimension, obtained by the improved algorithm, more accurate. An image reconstruction application based on the improved adaptive tenor small frame sparse regularization algorithm was developed, which verified that feature matching can be performed with corner points. Experiments show that the improved fractal-based adaptive fractional-order differential Fourier transform constructed in this paper has better experimental results than the integer-order differential one in edge detection and reconstruction of aerobics view images.

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.

The Joy and Pain of Skew Symmetry

In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems.

Soliton Solutions and Numerical Treatment of the Nonlinear Schrodinger’s Equation Using Modified Adomian Decomposition Method

In this paper, the improved Adomian decomposition method (ADM) is applied to the nonlinear Schr&oumldinger’s equation (NLSE), one of the most important partial differential equations in quantum mechanics that governs the propagation of solitons through optical fibers. The performance and the accuracy of our improved method are supported by investigating several numerical examples that include initial conditions. The obtained results are compared with the exact solutions. It is shown that the method does not need linearization, weak or perturbation theory to obtain the solutions.

Solutions of Fractional Partial Differential Equations of Quantum Mechanics

AbstractThe aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox’sH-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

On the solution of the asymptotic differential equations in quantum mechanics

The solution of the asymptotic quantum mechanical differential equations in the neighbourhood of singular points is suggested to overcome some complexities occurring with the technique of the indicial equations. After separation of the variables, the differential equations are solved in their asymptotic regions, the general regular solutions in the whole interval being then found by replacing integration constants by functions of the variables and finding the differential equations determining these functions. The resulting differential equations are now free from singularities and can be solved by the usual series expansion, giving simple polynomial solutions for the harmonic oscillator and the hydrogen atom and more complicated solutions for H 2 + .

The separation of a coupled system of differential equations in quantum mechanics

The authors discuss various aspects of the methodology of the separation operation of a coupled system of differential equations in quantum mechanics, namely the Darboux' transformation, the T and N matrices approach for the case of symmetric and non-symmetric coupling of the equations. The analysis is also extended to systems with more than two equations with, in each case, specific examples illustrating their implementation in practical situations.

Integration of operator differential equations.

In a previous paper we introduced a method for obtaining exact solutions to the operator differential equations of quantum mechanics. In that paper we showed how to solve some simple quantum-mechanical models and we suggested that the method could be used to obtain exact solutions to the operator differential equations of more complicated models, such as the anharmonic oscillator whose Hamiltonian is $H=\frac{1}{2}{p}^{2}+\frac{1}{4}{q}^{4}$. In this paper we further sharpen the formalism and introduce the concept of a minimal solution. We then obtain the exact minimal solution to the operator differential equations arising from two different anharmonic-oscillator models whose Hamiltonians are $H=\frac{1}{2}{p}^{2}+\frac{1}{4}{q}^{4}$ and $H=\frac{1}{4}{p}^{4}+\frac{1}{4}{q}^{4}$.

Measurement theory and stochastic differential equations in quantum mechanics.

Continuous (in time) measurements can be introduced in quantum mechanics by using operation-valued measures and quantum stochastic calculus. In this paper quantum stochastic calculus is used for showing the connections between measurement theory and open-system theory. In particular, it is shown how continuous measurements are strictly related to the concept of output channels, introduced in the framework of quantum stochastic differential equations by Gardiner and Collet.

The multigrid method for accelerated solution of the discretized Schrödinger equation

The use of the new multigrid method for the numerical solution of the discretized Schrödinger equation is discussed. A simple approach is chosen to circumvent the difficulties associated with the fact that the Schrödinger differential operator is not positive definite. The scheme is applied to a two-dimensional model problem which includes the essential features present in actual scattering problems. The results show that significant savings can be achieved both in computational work and core memory requirements. The multigrid method should greatly increase the utility of meshing procedures for solving the partial differential equations of quantum mechanics.