Analytic presentation of a solution of the Schrödinger equation (original) (raw)
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Highly Accurate Analytic Presentation of Solution of the Schr\"{o}dinger Equation
High-precision approximate analytic expressions for energies and wave functions are found for arbitrary physical potentials. The Schrödinger equation is cast into nonlinear Riccati equation, which is solved analytically in first iteration of the quasi-linearization method (QLM). The zeroth iteration is based on general features of the exact solution near the boundaries. The approach is illustrated on the Yukawa potential. The results enable accurate analytical estimates of effects of parameter variations on physical systems.
Annals of Physics, 2007
High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schrö dinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10 À4 and 10 À8 for the energies and, correspondingly, 10 À2 and 10 À4 for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.
Quasilinearization approach to quantum mechanics
Computer Physics Communications, 2003
The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precision of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.
AN ANALYTICAL METHOD FOR SOLVING LINEAR AND NONLINEAR SCHRÖDINGER EQUATIONS
The main aim of this article is to introduce an analytical method called the Natural Homotopy Perturbation Method (NHPM) for solving linear and nonlinear Schrödinger equations. The proposed analytical method is a combination of the Natural transform method (NTM) and homotopy perturbation method (HPM). The analytical method is applied directly without using any linearization, transformation, discretization or taking some restrictive assumptions, and it reduces the computational size and avoids round-off errors.
Quasilinearization approach to computations with singular potentials
Computer Physics Communications, 2008
We pioneered the application of the quasilinearization method (QLM) to the numerical solution of the Schrödinger equation with singular potentials. The spiked harmonic oscillator r 2 + λr −α is chosen as the simplest example of such potential. The QLM has been suggested recently for solving the Schrödinger equation after conversion into the nonlinear Riccati form. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of solutions near the boundaries. We show that the energies of bound state levels in the spiked harmonic oscillator potential which are notoriously difficult to compute for small couplings λ, are easily calculated with the help of QLM for any λ and α with accuracy of twenty significant figures.
Symmetry, 2016
In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark-and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams.
Numerical investigation of quasilinearization method in quantum mechanics
Computer Physics Communications, 2001
The general properties of the quasilinearization method (QLM), particularly its fast convergence, monotonicity and numerical stability are analyzed and verified on the example of scattering length calculations in the variable phase approach to quantum mechanics. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and is not based, unlike perturbation theories, on the existence of some kind of a small parameter. Each approximation of the method sums many orders of the perturbation theory. It is shown that already the first few iterations provide very accurate and numerically stable answers for small and intermediate values of the coupling constant. The number of iterations necessary to reach a given precision only moderately increases for its larger values. The method provides accurate and stable answers for any coupling strengths, including for super singular potentials for which each term of the perturbation theory diverges and the perturbation expansion does not exist even for a very small coupling.
Quasilinearization method and WKB
Computer Physics Communications, 2006
Solutions obtained by the quasilinearization method (QLM) are compared with the WKB solutions. While the WKB method generates an expansion in powers ofh, the quasilinearization method (QLM) approaches the solution of the nonlinear equation obtained by casting the Schrödinger equation into the Riccati form by approximating nonlinear terms by a sequence of linear ones. It does not rely on the existence of any kind of smallness parameter. It also, unlike the WKB, displays no unphysical turning point singularities. It is shown that both energies and wave functions obtained in the first QLM iteration are accurate to a few parts of the percent. Since the first QLM iterate is represented by the closed expression it allows to estimate analytically and precisely the role of different parameters, and influence of their variation on the properties of the quantum systems. The next iterates display very fast quadratic convergence so that accuracy of energies and wave functions obtained after a few iterations is extremely high, reaching 20 significant figures for the energy of the sixth iterate. It is therefore demonstrated that the QLM method could be preferable over the usual WKB method.
A new approach to linear and nonlinear Schrodingerequations
Advances in Differential Equations, 2011
In this paper, a new powerful and efficient technique Reconstruction of Variational Iteration Method (RVIM) is applied to find the exact solutions to the linear and nonlinear Schrödinger equations. The Reconstruction of Variational Iteration Method (RVIM) is independent of small parameters as well as algorithm overcomes the difficulty arising in calculating nonlinear intricately terms. The method is capable of reducing the size of calculation and easily overcomes the difficulty of the perturbation technique or Adomian polynomials. Besides, it provides us with a simple way to ensure the convergence of series solution, so that we can always get enough accuracy in approximations. The present results are in good agreement with the existing literature.
New analytical Solution of Schrödinger Equation
EPL 89, 50004 (2010).
We obtain an analytic solution beyond adiabatic approximation by transferring the 1D Schrödinger equation into the Ricatti equation. Then we show that our solution is more accurate than JWKB approximation. The generalizations of the approach to 3D are suggested, and possible applications of obtained solutions are discussed.