Mathematical methods for physicists

"The book covers different aspects of mathematical methods for Physics. It is designed for graduate courses but a part of it can also be used by undergraduate students. The leitmotiv of the book is the search for a common mathematical framework for a wide class of apparently disparate physical phenomena. An important role, within this respect, is provided by a nonconventional formulation of special functions and polynomials. The proposed methods simplify the understanding of the relevant technicalities and yield a unifying view to their applications in Physics as well as other branches of science. The chapters are not organized through the mathematical study of specific problems in Physics, rather they are suggested by the formalism itself. For example, it is shown how the matrix formalism is useful to treat ray Optics, atomic systems evolution, QED, QCD and Feynman diagrams. The methods presented here are simple but rigorous. They allow a fairly substantive tool of analysis for a variety of topics and are useful for beginners as well as the more experienced researchers."-- Read more... Contents 8 Abstract 14 Chapter 1 Matrices, Exponential Operators and Physical Applications 16 1.1 Introduction 16 1.2 Pauli Matrices 21 1.3 Applications of 2 27 1.4 Cabibbo Angle and See-Saw Mechanism 44 1.5 Gell-Mann and Pauli Matrices 48 1.6 Concluding Remarks 58 Chapter 2 Ordinary and Partial Di erential Equations, Evolution Operator Method and Applications 78 2.1 Ordinary Di erential Equations, Matrices and Exponential Operators 78 2.2 Partial Di erential Equations and Exponential Operators, I 81 2.3 Partial Di erential Equations and Exponential Operators, II 87 2.4 Operator Ordering 88 2.5 Schr odinger Equation and Paraxial Wave Equation of Classical Optics 94 2.6 Examples of Fokker-Planck, Schr odinger and Liouville Equations 99 2.7 Concluding Remarks 103 Chapter 3 Hermite Polynomials and Applications 116 3.1 Introduction 116 3.2 Hermite Polynomials Generating Function 119 3.3 Hermite Polynomials as an Orthogonal Basis 124 3.4 Hermite Polynomials in Quantum Mechanics: 128 3.5 Quantum Mechanics Applications 132 3.6 Coherent or Quasi-Classical States of Harmonic Oscillators 136 3.7 Jaynes-Cummings Model 142 3.8 Classical Optics and Hermite Polynomials 144 Chapter 4 Laguerre Polynomials, Integral Operators and Applications 150 4.1 Introduction 150 4.2 Laguerre Polynomials Generating Function 155 4.3 Orthogonality Properties of Laguerre Polynomials 156 4.4 Bessel Functions 159 4.5 Associated Laguerre Polynomials 163 4.6 Legendre Polynomials 166 4.7 Miscellaneous Applications and Comments 168 4.8 App el Polynomials and Final Comments 173 Chapter 5 Exercises and Complements I 182 5.1 Pauli and Jones Matrices and Mueller Calculus 182 5.2 Magnetic Lenses and Matrix Description 189 5.3 Miscellanea on the MatrixFormalism and Solution of Evolution Problems 197 5.4 Lorentz Transformation 201 5.5 Hyperbolic Trigonometry and Special Relativity 204 5.6 A Touch on Elliptic Functions 209 5.7 Concluding Comments 222 Chapter 6 Exercises and Complements II 232 6.1 Ordinary Di erential Equations and Matrices 232 6.2 Crofton-Glaisher Identities and Heat Type Equations 245 6.3 Gamma Function and De nite Integrals 250 6.4 Complex Variable Method and Evaluation of Integrals 259 6.5 Fourier Transform 265 6.6 Fourier Transform and the Solution of Di erential Equations 274 6.7 Fourier-Type Transforms 277 Chapter 7 Exercises and Complements III 288 7.1 Second Solution of Hermite Equation 288 7.2 Higher Orders Hermite Polynomials 290 7.3 Multi-Index Hermite Polynomials 295 7.4 Creation-Annihilation Operators Algebra and Physical Applications 299 7.5 Eisenstein Integers 305 7.6 Harmonic Oscillator Hamiltonian Formal Aspects and Further Miscellaneous Considerations 310 7.7 Time-Dependent Hamiltonians 313 7.8 Dyson Series 314 7.9 Special Polynomials and Perturbation Theory 322 Chapter 8 Exercises and Complements IV 334 8.1 Sturm-Liouville Problem 334 8.2 Green's Functions 339 8.3 Laguerre Polynomials, Associated Operators and PDE 341 8.4 App el Polynomials, Associated Operators and Partial Di erential Equations 346 8.5 Riemann Function 352 8.6 Bessel Special Functions 357 Chapter 9 Special Functions, Umbral Methods and Applications 370 9.1 Introduction to Umbral Methods and Relevant Applications 370 9.2 Further Comments on Umbral Methods, In nite Integrals and Borel Transform 377 9.3 Borel Transform and Applications 383 9.4 Umbral Formalism and Laguerre Polynomials 393 9.5 Umbral Formalism and Hermite Polynomials 395 9.6 Umbral Formalism and Operator Ordering 397 9.7 Mittag-Le er Function and Fractional Calculus Application 401 9.8 Formalism of Negative Derivative and De nite Integrals 407 9.9 Umbral Formalism, Dual Numbers and Super-Gaussian Beam Transport 409 Chapter 10 A Glimpse into the Math of the Feynman Diagrams 426 10.1 Introduction 426 10.2 Fermi Golden Rule 432 10.3 Feynman Diagrams: Introductory Rules 438 10.4 Virtual Particles and Propagators 442 10.5 Space and Time like Feynman Diagrams 445 10.6 Dirac Gamma Matrices 446 10.7 Mathematics of the Dirac Equation 454 10.8 A Touch on Quantum Electrodynamics 460 10.9 Formal Point of View to the Dimensions and Units in Physics 465 Index 478 MATHEMATICAL,METHODS,FOR,PHYSICISTS "The book covers different aspects of mathematical methods for Physics. It is designed for graduate courses but a part of it can also be used by undergraduate students. The leitmotiv of the book is the search for a common mathematical framework for a wide class of apparently disparate physical phenomena. An important role, within this respect, is provided by a nonconventional formulation of special functions and polynomials. The proposed methods simplify the understanding of the relevant technicalities and yield a unifying view to their applications in Physics as well as other branches of science."--Back cover