Mathematical Physics with Applications, Problems and Solutions

"This book gives a comprehensive account of mathematical physics, assigning a prominent role to the applications of the relevant mathematics to specific parts of physics ranging over fluid mechanics, electromagnetic theory, special relativity, quantum mechanics, quantum optics, random processes, linear response theory, scattering theory, and so on. Rather than emphasizing formal proofs of propositions, the results are motivated, stated, elaborated and remarked upon. Their inter-relationships and connections are brought out, and several of their applications are discussed in some detail. There are about 370 exercises and problems interspersed in the text itself at appropriate junctures, so as to facilitate the logical flow of the material without interruption. Wherever applicable, solutions to the problems, or answers with hints as to how to arrive at them, along with relevant remarks, are given at the end of each chapter. This book can also be used as a wide-ranging and fairly self-contained text for self-study, and will be well received by students of physics at all levels (senior undergraduate, master's and doctoral), as well as those who are now teachers of the subject"--Page 4 of cover Preface......Page 6 Contents......Page 10 About the Author......Page 26 1.1.1 Features of Interest in a Function......Page 27 1.1.3 A Family of Ovals......Page 28 1.1.4 A Family of Spirals......Page 29 1.2 Maps of the Unit Interval......Page 31 2.1.1 The Basic Gaussian Integral......Page 34 2.1.2 A Couple of Higher Dimensional Examples......Page 35 2.2 Stirling's Formula......Page 37 2.3 The Dirichlet Integral and Its Descendants......Page 38 2.4 Solutions......Page 41 3.1 Functions Represented by Integrals......Page 43 3.1.2 The Error Function......Page 44 3.1.3 Fresnel Integrals......Page 45 3.1.4 The Gamma Function......Page 46 3.1.5 Connection to Gaussian Integrals......Page 47 3.2 Interchange of the Order of Integration......Page 49 3.3 Solutions......Page 52 4.1 The Step Function......Page 53 4.2.1 Defining Relations......Page 55 4.2.2 Sequences of Functions Tending to the δ-Function......Page 57 4.2.4 Fourier Representation of the δ-Function......Page 58 4.2.5 Properties of the δ-Function......Page 59 4.2.6 The Occurrence of the δ-Function in Physical Problems......Page 62 4.3 Solutions......Page 64 5.1.1 What Are Scalars and Vectors?......Page 66 5.1.2 Rotations and the Index Notation......Page 67 5.1.3 Isotropic Tensors......Page 70 5.1.4 Dot and Cross Products in Three Dimensions......Page 73 5.1.5 The Gram Determinant......Page 75 5.1.6 Levi-Civita Symbol in d Dimensions......Page 76 5.2.1 Proper and Improper Rotations......Page 77 5.2.2 Scalars and Pseudoscalars; Polar and Axial Vectors......Page 79 5.2.3 Transformation Properties of Physical Quantities......Page 80 5.3.1 Spherical or Irreducible Tensors......Page 82 5.3.2 Stress, Strain, and Stiffness Tensors......Page 84 5.3.3 Moment of Inertia......Page 87 5.3.4 The Euler Top......Page 89 5.3.5 Multipole Expansion; Quadrupole Moment......Page 90 5.3.6 The Octupole Moment......Page 92 5.4 Solutions......Page 93 6.1.1 Cylindrical and Spherical Polar Coordinates......Page 95 6.1.2 Elliptic and Parabolic Coordinates......Page 98 6.1.3 Polar Coordinates in d Dimensions......Page 99 6.2.1 The Gradient of a Scalar Field......Page 101 6.2.2 The Flux and Divergence of a Vector Field......Page 103 6.2.3 The Circulation and Curl of a Vector Field......Page 105 6.2.4 Some Physical Aspects of the Curl of a Vector Field......Page 108 6.2.5 Any Vector Field is the Sum of a Curl and a Gradient......Page 110 6.2.6 The Laplacian Operator......Page 112 6.2.7 Why Do div, curl, and del-Squared Occur so Frequently?......Page 114 6.3 Solutions......Page 116 7.1.1 Hydrodynamic Variables......Page 119 7.1.2 Equation of Motion......Page 120 7.2.1 Euler's Equation......Page 122 7.2.2 Barotropic Flow......Page 123 7.2.3 Bernoulli's Principle in Steady Flow......Page 124 7.2.4 Irrotational Flow and the Velocity Potential......Page 125 7.3.1 Vortex Lines......Page 126 7.3.2 Equations in Terms of v Alone......Page 128 7.4.1 The Viscous Force in a Fluid......Page 129 7.4.2 The Navier–Stokes Equation......Page 130 7.5 Solutions......Page 132 8.1.1 The Fundamental Theorem of Calculus......Page 134 8.1.2 Stokes' Theorem......Page 135 8.1.3 Green's Theorem......Page 136 8.1.4 A Topological Restriction; ``Exact'' Versus ``Closed''......Page 137 8.1.5 Gauss's Theorem......Page 139 8.1.6 Green's Identities and Reciprocity Relation......Page 140 8.1.7 Comment on the Generalized Stokes' Theorem......Page 141 8.2.1 Mean Value Property......Page 142 8.2.2 Harmonic Functions Have No Absolute Maxima or Minima......Page 144 8.2.3 What Is the Significance of the Laplacian?......Page 145 8.3.1 Critical Points and the Poincaré Index......Page 148 8.3.2 Degenerate Critical Points and Unfolding Singularities......Page 151 8.4 Solutions......Page 154 9.1.1 Maxwell's Field Equations......Page 158 9.1.2 The Scalar and Vector Potentials......Page 160 9.1.3 Gauge Invariance and Choice of Gauge......Page 161 9.1.4 The Coulomb Gauge......Page 163 9.1.5 Electrostatics......Page 164 9.1.6 Magnetostatics......Page 165 9.1.7 The Lorenz Gauge......Page 166 9.2.1 The Principle and the Postulate of Relativity......Page 168 9.2.2 Boost Formulas......Page 169 9.2.3 Collinear Boosts: Velocity Addition Rule......Page 170 9.2.4 Rapidity......Page 172 9.2.5 Lorentz Scalars and Four-Vectors......Page 173 9.2.6 Matrices Representing Lorentz Transformations......Page 175 9.3.2 The Electromagnetic Field Tensor......Page 177 9.3.3 Transformation Properties of E and B......Page 178 9.3.4 Lorentz Invariants of the Electromagnetic Field......Page 180 9.4 Solutions......Page 181 10.1.1 Definition of a Linear Vector Space......Page 184 10.1.2 The Dual of a Linear Space......Page 185 10.1.4 Basis Sets and Dimensionality......Page 186 10.2.1 Gram–Schmidt Orthonormalization......Page 188 10.2.2 Expansion of an Arbitrary Vector......Page 190 10.2.3 Basis Independence of the Inner Product......Page 191 10.3.1 The Cauchy–Schwarz Inequality......Page 192 10.3.3 The Gram Determinant Inequality......Page 194 10.4 Solutions......Page 195 11.1.1 Expansion of a (2times2) Matrix......Page 198 11.1.2 Basic Properties of the Pauli Matrices......Page 199 11.2.1 Occurrence and Definition......Page 201 11.2.2 The Exponential of an Arbitrary (2times2) Matrix......Page 202 11.3.1 Generators of Infinitesimal Rotations and Their Algebra......Page 204 11.3.2 The General Rotation Matrix......Page 207 11.3.3 The Finite Rotation Formula for a Vector......Page 210 11.4.1 The Characteristic Equation......Page 211 11.4.2 Gershgorin's Circle Theorem......Page 212 11.4.3 The Cayley–Hamilton Theorem......Page 214 11.4.4 The Resolvent of a Matrix......Page 215 11.5 A Generalization of the Gaussian Integral......Page 216 11.6 Inner Product in the Linear Space of Matrices......Page 217 11.7 Solutions......Page 218 12.1.1 Representation of Operators......Page 221 12.1.2 Projection Operators......Page 222 12.2.1 Definitions and Eigenvalues......Page 224 12.2.2 The Eigenvalues of a Rotation Matrix in d Dimensions......Page 225 12.2.3 The General Form of a (2times2) Unitary Matrix......Page 226 12.3 Diagonalization of a Matrix and all That......Page 228 12.3.1 Eigenvectors, Nullspace, and Nullity......Page 229 12.3.2 The Rank of a Matrix and the Rank-Nullity Theorem......Page 230 12.3.3 Degenerate Eigenvalues and Defective Matrices......Page 231 12.3.4 When Can a Matrix Be Diagonalized?......Page 232 12.3.5 The Minimal Polynomial of a Matrix......Page 233 12.3.6 Simple Illustrative Examples......Page 234 12.3.7 Jordan Normal Form......Page 237 12.3.9 Circulant Matrices......Page 239 12.3.10 A Simple Illustration: A 3-state Random Walk......Page 241 12.4.1 Mutually Commuting Matrices in Quantum Mechanics......Page 243 12.4.2 The Lie Algebra of (n timesn) Matrices......Page 244 12.5.1 Right and Left Eigenvectors of a Matrix......Page 247 12.5.2 An Illustration......Page 249 12.6 Solutions......Page 252 13.1 The Space ell2 of Square-Summable Sequences......Page 259 13.2.2 Continuous Basis......Page 261 13.2.3 Weight Functions: A Generalization of mathcalL2......Page 263 13.2.4 mathcalL2(-infty,infty) Functions and Fourier Transforms......Page 265 13.2.5 The Wave Function of a Particle......Page 266 13.3.1 Hilbert Space......Page 268 13.3.2 Linear Manifolds and Subspaces......Page 269 13.4 Solutions......Page 270 14.1.1 Domain, Range, and Inverse......Page 272 14.1.2 Linear Operators, Norm, and Bounded Operators......Page 273 14.2 The Adjoint of an Operator......Page 275 14.2.2 Definition of the Adjoint Operator......Page 276 14.2.3 Symmetric, Hermitian, and Self-adjoint Operators......Page 278 14.3.1 The Momentum Operator of a Quantum Particle......Page 279 14.3.2 The Adjoint of the Derivative Operator in mathcalL2(-infty,infty)......Page 280 14.3.3 When Is -i(d/dx) Self-adjoint in mathcalL2[a,b]?......Page 282 14.3.5 Deficiency Indices......Page 283 14.3.6 The Radial Momentum Operator in d 2 Dimensions......Page 285 14.4.1 The Operators xpmip......Page 286 14.4.2 Oscillator Ladder Operators and Coherent States......Page 288 14.4.3 Eigenvalues and Non-normalizable Eigenstates of x and p......Page 291 14.4.4 Matrix Representations for Unbounded Operators......Page 293 14.5 Solutions......Page 294 15.1.1 The Heisenberg Algebra......Page 300 15.1.2 Some Other Basic Operator Algebras......Page 302 15.2 Useful Operator Identities......Page 304 15.2.2 Hadamard's Lemma......Page 305 15.2.4 The Zassenhaus Formula......Page 307 15.2.5 The Baker–Campbell–Hausdorff Formula......Page 308 15.3.1 Angular Momentum Operators......Page 309 15.3.2 Representation of Rotations by SU(2) Matrices......Page 311 15.3.3 Connection Between the Groups SO(3) and SU(2)......Page 313 15.3.4 The Parameter Space of SU(2)......Page 315 15.3.5 The Parameter Space of SO(3)......Page 317 15.3.6 The Parameter Space of SO(2)......Page 320 15.4.1 The Displacement Operator and Coherent States......Page 322 15.4.2 The Squeezing Operator and the Squeezed Vacuum......Page 326 15.4.3 Values of z That Produce Squeezing in x or p......Page 328 15.4.4 The Squeezing Operator and the Group SU(1,1)......Page 330 15.4.5 SU(1,1) Generators in Terms of Pauli Matrices......Page 332 15.5 Solutions......Page 334 16.1.1 Introduction......Page 338 16.1.2 Orthogonality and Completeness......Page 339 16.1.3 Expansion and Inversion Formulas......Page 342 16.1.4 Uniqueness and Explicit Representation......Page 343 16.2.1 Polynomials of the Hypergeometric Type......Page 344 16.2.2 The Hypergeometric Differential Equation......Page 346 16.2.3 Rodrigues Formula and Generating Function......Page 349 16.2.4 Class I.Hermite Polynomials......Page 350 16.2.5 Linear Harmonic Oscillator Eigenfunctions......Page 352 16.2.6 Oscillator Coherent State Wave Functions......Page 353 16.2.7 Class II.Generalized Laguerre Polynomials......Page 354 16.2.8 Class III.Jacobi Polynomials......Page 356 16.3.1 Ultraspherical Harmonics......Page 357 16.3.2 Chebyshev Polynomials of the 1st Kind......Page 358 16.3.3 Chebyshev Polynomials of the Second Kind......Page 360 16.4.1 Basic Properties......Page 361 16.4.2 Pn(x) by Gram–Schmidt Orthonormalization......Page 364 16.4.3 Expansion in Legendre Polynomials......Page 365 16.4.4 Expansion of xn in Legendre Polynomials......Page 366 16.4.5 Legendre Function of the Second Kind......Page 367 16.4.6 Associated Legendre Functions......Page 368 16.4.7 Spherical Harmonics......Page 369 16.4.8 Expansion of the Coulomb Kernel......Page 371 16.5 Solutions......Page 373 17.1.1 Dirichlet Conditions......Page 379 17.1.2 Orthonormal Basis......Page 380 17.1.3 Fourier Series Expansion and Inversion Formula......Page 381 17.1.4 Parseval's Formula for Fourier Series......Page 383 17.1.5 Simplified Formulas When (a,b) = (-π,π)......Page 384 17.2.1 Uniform Convergence of Fourier Series......Page 385 17.2.2 Large-n Behavior of Fourier Coefficients......Page 386 17.2.3 Periodic Array of δ-Functions: The Dirac Comb......Page 387 17.3.1 Some Examples......Page 388 17.3.3 Fourier Series Expansions of cosαx and sinαx......Page 389 17.4 Solutions......Page 391 18.1.1 Fourier Transform and Inverse Fourier Transform......Page 393 18.1.2 Parseval's Formula for Fourier Transforms......Page 394 18.1.4 Examples of Fourier Transforms......Page 395 18.1.5 Relative ``Spreads'' of a Fourier Transform Pair......Page 397 18.1.7 Generalized Parseval Formula......Page 398 18.2.1 Iterates of the Fourier Transform Operator......Page 399 18.2.2 Eigenvalues and Eigenfunctions of mathcalF......Page 400 18.2.3 The Adjoint of an Integral Operator......Page 402 18.3 Generalization to Several Dimensions......Page 403 18.4 The Poisson Summation Formula......Page 404 18.4.1 Derivation of the Formula......Page 405 18.4.2 Some Illustrative Examples......Page 406 18.5 Solutions......Page 409 19.1.1 Mean and Variance......Page 411 19.1.2 Bernoulli Trials and the Binomial Distribution......Page 413 19.1.3 Number Fluctuations in a Classical Ideal Gas......Page 415 19.1.4 The Geometric Distribution......Page 416 19.1.5 Photon Number Distribution in Blackbody Radiation......Page 417 19.2.1 From the Binomial to the Poisson Distribution......Page 420 19.2.2 Photon Number Distribution in Coherent Radiation......Page 421 19.2.3 Photon Number Distribution in the Squeezed Vacuum State......Page 423 19.2.4 The Sum of Poisson-Distributed Random Variables......Page 424 19.2.5 The Difference of Two Poisson-Distributed Random Variables......Page 425 19.3 The Negative Binomial Distribution......Page 428 19.4.1 Random Walk on a Linear Lattice......Page 430 19.5 Solutions......Page 433 20.1.1 Probability Density and Cumulative Distribution......Page 438 20.1.2 The Moment-Generating Function......Page 439 20.1.3 The Cumulant-Generating Function......Page 441 20.1.4 Application to the Discrete Distributions......Page 442 20.1.5 The Characteristic Function......Page 443 20.1.6 The Additivity of Cumulants......Page 444 20.2.1 The Normal Density and Distribution......Page 445 20.2.2 Moments and Cumulants of a Gaussian Distribution......Page 446 20.2.3 Simple Functions of a Gaussian Random Variable......Page 448 20.2.4 Mean Collision Rate in a Dilute Gas......Page 450 20.3.2 The Central Limit Theorem......Page 451 20.3.3 An Explicit Illustration of the Central Limit Theorem......Page 452 20.4.1 From Random Flights to Diffusion......Page 454 20.4.2 The Probability Density for Short Random Flights......Page 459 20.5.1 What Is a Stable Distribution?......Page 463 20.5.2 The Characteristic Function of Stable Distributions......Page 464 20.5.3 Three Important Cases: Gaussian, Cauchy, and Lévy......Page 465 20.5.4 Some Connections Between the Three Cases......Page 467 20.6.1 Divisibility of a Random Variable......Page 468 20.6.2 Infinite Divisibility Does Not Imply Stability......Page 470 20.7 Solutions......Page 471 21.1 Multiple-Time Joint Probabilities......Page 477 21.2.1 The Two-Time Conditional Probability......Page 478 21.2.2 The Master Equation......Page 480 21.2.3 Formal Solution of the Master Equation......Page 482 21.2.4 The Stationary Distribution......Page 484 21.3 The Autocorrelation Function......Page 486 21.4.1 The Stationary Distribution......Page 488 21.4.2 Solution of the Master Equation......Page 489 21.5.1 The Poisson Pulse Process and Radioactive Decay......Page 491 21.5.2 Biased Random Walk on a Linear Lattice......Page 494 21.5.3 Connection with the Skellam Distribution......Page 496 21.6.1 Master Equation for the Conditional density......Page 497 21.6.2 The Fokker–Planck Equation......Page 498 21.6.3 The Autocorrelation Function for a Continuous Process......Page 500 21.7.1 The Ornstein–Uhlenbeck Process......Page 501 21.7.2 The Ornstein–Uhlenbeck Distribution......Page 502 21.7.3 Velocity Distribution in a Classical Ideal Gas......Page 504 21.7.5 Diffusion of a Harmonically Bound Particle......Page 506 21.8 Solutions......Page 508 22.1.1 Complex Numbers......Page 510 22.2.1 Stereographic Projection......Page 512 22.2.2 Maps of Circles on the Riemann Sphere......Page 515 22.2.3 A Metric on the Extended Complex Plane......Page 516 22.3.1 The Cauchy–Riemann Conditions......Page 518 22.3.2 The Real and Imaginary Parts of an Analytic Function......Page 520 22.4 The Derivative of an Analytic Function......Page 521 22.5.1 Radius and Circle of Convergence......Page 523 22.5.2 An Instructive Example......Page 524 22.5.3 Behavior on the Circle of Convergence......Page 527 22.5.4 Lacunary Series......Page 528 22.6.1 Representation of Entire Functions......Page 529 22.6.2 The Order of an Entire Function......Page 530 22.7 Solutions......Page 532 23.1 Cauchy's Integral Theorem......Page 536 23.2.1 Simple Pole; Residue at a Pole......Page 537 23.2.3 Essential Singularity......Page 540 23.2.4 Laurent Series......Page 541 23.2.5 Singularity at Infinity......Page 542 23.2.6 Accumulation Points......Page 543 23.2.7 Meromorphic Functions......Page 544 23.3.1 A Basic Formula......Page 546 23.3.2 Cauchy's Residue Theorem......Page 547 23.3.3 The Dirichlet Integral; Cauchy Principal Value......Page 549 23.3.4 The ``iε-Prescription'' for a Singular Integral......Page 551 23.3.5 Residue at Infinity......Page 553 23.4 Summation of Series Using Contour Integration......Page 555 23.5.1 The Generating Function......Page 558 23.5.2 Hemachandra-Fibonacci Numbers......Page 560 23.5.3 Catalan Numbers......Page 561 23.5.4 Connection with Wigner's Semicircular Distribution......Page 563 23.6 Mittag-Leffler Expansion of Meromorphic Functions......Page 564 23.7 Solutions......Page 567 24.1.1 Linear, Causal, Retarded Response......Page 575 24.1.2 Frequency-Dependent Response......Page 576 24.1.3 Symmetry Properties of the Dynamic Susceptibility......Page 578 24.2.1 Derivation of the Relations......Page 579 24.2.2 Complex Admittance of an LCR Circuit......Page 582 24.2.3 Subtracted Dispersion Relations......Page 584 24.2.4 Hilbert Transform Pairs......Page 585 24.2.5 Discrete and Continuous Relaxation Spectra......Page 587 24.3 Solutions......Page 589 25.1.1 What Is Analytic Continuation?......Page 591 25.1.2 The Permanence of Functional Relations......Page 593 25.2.1 Stripwise Analytic Continuation of Γ(z)......Page 595 25.2.3 Logarithmic Derivative of Γ(z)......Page 598 25.2.4 Infinite Product Representation of Γ(z)......Page 599 25.2.5 Connection with the Riemann Zeta Function......Page 600 25.2.6 The Beta Function......Page 602 25.2.7 Reflection Formula for Γ(z)......Page 604 25.2.8 Legendre's Doubling Formula......Page 605 25.3 Solutions......Page 606 26.1.1 Branch Points and Branch Cuts......Page 608 26.1.2 Types of Branch Points......Page 610 26.1.3 Contour Integrals in the Presence of Branch Points......Page 612 26.2.1 The Gamma Function......Page 615 26.2.2 The Beta Function......Page 617 26.2.3 The Riemann Zeta Function......Page 618 26.2.4 Connection with Bernoulli Numbers......Page 620 26.2.5 The Legendre Functions Pν(z) and Qν(z)......Page 622 26.3.1 End Point and Pinch Singularities......Page 626 26.3.2 Singularities of the Legendre Functions......Page 630 26.4 Solutions......Page 631 27.1 Conformal Mapping......Page 636 27.2.1 Definition......Page 637 27.2.2 Fixed Points......Page 638 27.2.3 The Cross-Ratio and Its Invariance......Page 639 27.3.1 Normal Forms in Different Cases......Page 642 27.3.2 Iterates of a Möbius Transformation......Page 643 27.3.3 Classification of Möbius Transformations......Page 645 27.3.4 The Isometric Circle......Page 647 27.4.1 The Möbius Group......Page 648 27.4.2 The Möbius Group Over the Reals......Page 650 27.4.3 The Invariance Group of the Unit Circle......Page 651 27.4.4 The Group of Cross-Ratios......Page 652 27.5 Solutions......Page 653 28.1.1 Definition of the Laplace Transform......Page 657 28.1.2 Transforms of Some Simple Functions......Page 658 28.1.3 The Convolution Theorem......Page 660 28.1.4 Laplace Transforms of Derivatives......Page 661 28.2.1 The Mellin Formula......Page 662 28.2.2 LCR Circuit Under a Sinusoidal Applied Voltage......Page 663 28.3.1 Differential Equations and Power Series Representations......Page 665 28.3.2 Generating Functions and Integral Representations......Page 667 28.3.3 Spherical Bessel Functions......Page 669 28.3.4 Laplace Transforms of Bessel Functions......Page 670 28.4.1 Random Walk in d Dimensions......Page 672 28.4.2 The First-Passage-Time Distribution......Page 673 28.5 Solutions......Page 677 29.2 Green Functions......Page 679 29.2.1 Green Function for an Ordinary Differential Operator......Page 680 29.2.2 An Illustrative Example......Page 681 29.3.1 Poisson's Equation in Three Dimensions......Page 683 29.3.2 The Solution for G(3)(r, r')......Page 684 29.3.3 Solution of Poisson's Equation......Page 686 29.4.1 Simplification of the Fundamental Green Function......Page 687 29.4.2 Power Counting and a Divergence Problem......Page 689 29.4.3 Dimensional Regularization......Page 691 29.4.4 A Direct Derivation......Page 694 29.5.1 Dimensional Regularization......Page 695 29.5.2 Direct Derivation......Page 696 29.5.3 An Alternative Regularization......Page 697 29.6 Solutions......Page 698 30.1.1 Fick's Laws of Diffusion......Page 700 30.1.2 Further Remarks on Linear Response......Page 701 30.1.3 The Fundamental Solution in d Dimensions......Page 702 30.1.4 Solution for an Arbitrary Initial Distribution......Page 704 30.1.5 Moments of the Distance Travelled in Time t......Page 705 30.2.1 Continuum Limit of a Biased Random Walk......Page 706 30.2.2 Free Diffusion on an Infinite Line......Page 708 30.2.3 Absorbing and Reflecting Boundary Conditions......Page 709 30.2.4 Finite Boundaries: Solution by the Method of Images......Page 710 30.2.5 Finite Boundaries: Solution by Separation of Variables......Page 712 30.2.6 Survival Probability and Escape-Time Distribution......Page 713 30.2.7 Equivalence of the Solutions......Page 715 30.3.1 The Smoluchowski Equation......Page 717 30.3.2 Equilibrium Barometric Distribution......Page 718 30.3.3 The Time-Dependent Solution......Page 719 30.4.1 Connection with the Free-Particle Propagator......Page 721 30.4.2 Spreading of a Quantum Mechanical Wave Packet......Page 722 30.4.3 The Wave Packet in Momentum Space......Page 725 30.5 Solutions......Page 726 31.1.1 Formal Solution as a Fourier Transform......Page 729 31.1.2 Simplification of the Formal Solution......Page 732 31.2.1 The Green Function in (1+1) Dimensions......Page 734 31.2.2 The Green Function in (2+1) Dimensions......Page 735 31.2.3 The Green Function in (3+1) Dimensions......Page 737 31.2.4 Retarded Solution of the Wave Equation......Page 738 31.3 Remarks on Propagation in Dimensions d > 3......Page 739 31.4 Solutions......Page 741 32.1.1 Equation of the First Kind......Page 742 32.1.2 Equation of the Second Kind......Page 743 32.1.3 Degenerate Kernels......Page 745 32.1.4 The Eigenvalues of a Degenerate Kernel......Page 746 32.1.5 Iterative Solution: Neumann Series......Page 747 32.2.1 The Scattering Amplitude......Page 750 32.2.2 Integral Equation for Scattering......Page 752 32.2.3 Green Function for the Helmholtz Operator......Page 753 32.2.4 Formula for the Scattering Amplitude......Page 755 32.2.5 The Born Approximation......Page 757 32.2.6 Yukawa and Coulomb Potentials; Rutherford's Formula......Page 758 32.3.1 The Physical Idea Behind Partial Wave Analysis......Page 760 32.3.2 Expansion of a Plane Wave in Spherical Harmonics......Page 761 32.3.3 Partial Wave Scattering Amplitude and Phase Shift......Page 762 32.3.4 The Optical Theorem......Page 764 32.4.1 The Fredholm Formulas......Page 765 32.4.2 Remark on the Application to the Scattering Problem......Page 766 32.5 Volterra Integral Equations......Page 768 32.6 Solutions......Page 771 BookmarkTitle:......Page 776 Index......Page 778 This textbook is aimed at advanced undergraduate and graduate students interested in learning the fundamental mathematical concepts and tools widely used in different areas of physics. The author draws on a vast teaching experience, and presents a comprehensive and self-contained text which explains how mathematics intertwines with and forms an integral part of physics in numerous instances. Rather than emphasizing rigorous proofs of theorems, specific examples and physical applications (such as fluid dynamics, electromagnetism, quantum mechanics, etc.) are invoked to illustrate and elaborate upon the relevant mathematical techniques. The early chapters of the book introduce different types of functions, vectors and tensors, vector calculus, and matrices. In the subsequent chapters, more advanced topics like linear spaces, operator algebras, special functions, probability distributions, stochastic processes, analytic functions, Fourier series and integrals, Laplace transforms, Green's functions and integral equations are discussed. The book also features about 400 exercises and solved problems interspersed throughout the text at appropriate junctures, to facilitate the logical flow and to test the key concepts. Overall this book will be a valuable resource for a wide spectrum of students and instructors of mathematical physics.