Homotopy Analysis Method in Nonlinear Differential Equations (Springer & Higher Education Press, Heidelberg, 2012). Beyond Perturbation: Introduction to the Homotopy Analysis Method (Chapman & Hall/CRC Press, Boca Raton, 2003). Dissertation for Doctoral Degree (Shanghai Jiao Tong University, Shanghai, 1992). The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Quantum Mechanics-A Conceptual Approach (John Wiley & Sons, Inc. Quantum Mechanics-An Introduction (4th ed) (Springer-Verlag, Berlin, 2000). The Principles of Quantum Mechanics (4th ed) (Oxford University Press, Oxford, 1958). Perturbation Methods (John Wiley & Sons, New York, 2000). Obviously, this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.ĭ. This HAM-based approach could provide us rigorous theoretical results in quantum mechanics, which can be directly compared with experimental data. A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be one thousand times larger than the possible superior limit of the perturbative approach.
#Time independent schrödinger equation series
Besides, convergent series solution can be obtained even if the disturbance is far from the known status. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters.
It is based on the homotopy analysis method (HAM) that was developed by the author in 1992 for highly nonlinear equations and has been widely applied in many fields. Four examples are presented: an “eigensimple” equation, Bessel's equation of order ? greater than or equal to 0, Hermite's equation, and Legendre's equation.A new non-perturbative approach is proposed to solve time-independent Schrödinger equations in quantum mechanics. It also describes Weyl's theorem, given the Sturm-Liouville equation, and looks at two cases: the limit point and limit circle. The chapter first considers the one-dimensional Schrödinger equation in the standard dimensionless form (with independent variable x) and various relevant theorems, along with the proofs, before discussing bound states, taking into account bound-state theorems and complex eigenvalues. Unlike the classical case, the spectrum may contain both a countable set of eigenvalues and a continuous part. The regular Sturm-Liouville theory was generalized in 1908 by the German mathematician Hermann Weyl on a finite closed interval to second-order differential operators with singularities at the endpoints of the interval. This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. Princeton Series in Applied Mathematics.Appendix F Semiclassical Scattering: A Précis (and a Few More Details).Appendix D Electromagnetic Scattering from a Radially Inhomogeneous Sphere.Appendix C Radially Inhomogeneous Spherically Symmetric Scattering: The Governing Equations.Appendix A Order Notation: The “Big O,” “Little o,” and “∼” Symbols.Chapter Twenty-Eight Back Where We Started.Chapter Twenty-Seven Morphology-Dependent Resonances: The Effective Potential.Chapter Twenty-Six One-Dimensional Jost Solutions: The S-Matrix Revisited.Chapter Twenty-Five The Jost Solutions: Technical Details.Chapter Twenty-Four The S-Matrix and Its Analysis.Part V Special Topics in Scattering Theory.Chapter Twenty-Three A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation.Chapter Twenty-Two The WKB(J) Approximation Revisited.Chapter Twenty-One The Classical-to-Semiclassical Connection.Chapter Twenty Diffraction of Plane Electromagnetic Waves by a Cylinder.Chapter Nineteen Electromagnetic Scattering: The Mie Solution.Chapter Seventeen Scattering of Surface Gravity Waves by Islands, Reefs, and Barriers.Chapter Sixteen Gravitational Scattering.Chapter Fifteen The Classical Connection.Chapter Seven Introduction to the WKB(J) Approximation: All Things Airy.Chapter Five An Improvement over Ray Optics: Airy’s Rainbow.Chapter Four Ray Optics: The Classical Rainbow.Chapter Three Introduction to the Mathematics of Rays.
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