Book Chapter
An Introduction to Special Functions with
Some Applications to Quantum Mechanics
by
Arizona State University
Tempe, AZ
email: sergei@asu.edu
Lamar University
Beaumont, TX
email: jvegaguzman@lamar.edu
Kamal Barley
Stony Brook University
Stony Brook, NY
email: kamal.barley@stonybrook.edu
https://www.springer.com/gp/book/9783030367435
There is no doubt that next generation of the textbooks with be on-line only.
This is why we are working on a project which brings together traditional texts,
say in pdf, computer graphics, animations and on-line research evaluations,
video and reference sources, all under one web-oriented platform for the benefits
of our students and next generation of researchers. We will concentrate on several
topics in theoretical and mathematical physics, classical and quantum optics and
nonlinear science, based on our original results. Our group includes professors and
ASU graduates.
Table of Contents
1. An Introduction to Special Functions
1.1. Classical Hypergeometric Functions
1.1.1. Method of undetermined coefficients
1.1.2. Some solutions of hypergeometric equations
1.2. Integral Representations
1.2.1. Transformation to the simplest form
1.2.2. Main theorem
1.2.3. Integrals for hypergeometric and Bessel functions
1.3. Classical Orthogonal Polynomials
1.3.1. Main property
1.3.2. Rodrigues formula
1.3.3. Orthogonality
1.3.4. Classification
1.3.5. Functions of the second kind
1.3.6. Complex orthogonality
Exercises
2. Some Problems of Nonrelativistic and Relativistic Quantum Mechanics
2.1. Generalized Equation of Hypergeometric Type
2.2. Classical Orthogonal Polynomials and Eigenvalue Problems
2.2.1. Example: linear harmonic oscillator
2.3. Method of Separation of Variables and Its Extension
2.3.1. Method of Separation of Variables
2.3.2. Dirac-type systems
2.4. Nonrelativistic Coulomb Problem
2.4.1. Radial equation
2.4.2. Quantization
2.4.3. Summary: wave functions and energy levels
2.5. Matrix Elements
2.5.1. General Results
2.5.2. Special cases
3. Relativistic Coulomb Problem
3.1. Dirac Equation
3.2. Relativistic Coulomb Wave Functions and Discrete Energy Levels
3.3. Solution of Dirac Wave Equation for Coulomb Potential
3.3.1. The spinor spherical harmonics
3.3.2. Separation of variables in spherical coordinates
3.3.3. Solution of radial equations
3.3.4. Nonrelativistic limit of the wave functions
4. Symmetry of Quantum Harmonic Oscillators
4.1. Symmetry and "Hidden" Solutions
4.2. Computer Animations
4.3. The Momentum Representation
4.4. The Schrodinger Group for Simple Harmonic Oscillators
4.5. A Complex Parametrization of the Schrodinger Group
4.6. Discussion
5. Expectation Values in Relativistic Coulomb Problems
5.1. Evaluation of the Matrix Elements
5.2. Inversion Formulas
5.3. Recurrence Relations
5.4. Special Expectation Values and Their Applications
5.5. Three-Term Recurrence Relations and Computer Algebra Methods
Appendix A. Evaluation of an Integral
Appendix B. Hypergeometric Series, Discrete Orthogonal Polynomials, and Useful Relations
Appendix C. Dirac Matrices and Inner Product
References
AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials ans Applications
Douala, Cameron
October 5-12, 2018
http://www.aims-volkswagen-workshops.org/
More information coming soon!