Mathematical Physics _ Applications and Problems

Preface
Contents
About the Author
1 Warming Up: Functions of a Real Variable
1.1 Sketching Functions
1.1.1 Features of Interest in a Function
1.1.2 Powers of x
1.1.3 A Family of Ovals
1.1.4 A Family of Spirals
1.2 Maps of the Unit Interval
2 Gaussian Integrals, Stirling's Formula, and Some Integrals
2.1 Gaussian Integrals
2.1.1 The Basic Gaussian Integral
2.1.2 A Couple of Higher Dimensional Examples
2.2 Stirling's Formula
2.3 The Dirichlet Integral and Its Descendants
2.4 Solutions
3 Some More Functions
3.1 Functions Represented by Integrals
3.1.1 Differentiation Under the Integral Sign
3.1.2 The Error Function
3.1.3 Fresnel Integrals
3.1.4 The Gamma Function
3.1.5 Connection to Gaussian Integrals
3.2 Interchange of the Order of Integration
3.3 Solutions
4 Generalized Functions
4.1 The Step Function
4.2 The Dirac Delta Function
4.2.1 Defining Relations
4.2.2 Sequences of Functions Tending to the δ-Function
4.2.3 Relation Between δ(x) and θ(x)
4.2.4 Fourier Representation of the δ-Function
4.2.5 Properties of the δ-Function
4.2.6 The Occurrence of the δ-Function in Physical Problems
4.2.7 The δ-Function in Polar Coordinates
4.3 Solutions
5 Vectors and Tensors
5.1 Cartesian Tensors
5.1.1 What Are Scalars and Vectors?
5.1.2 Rotations and the Index Notation
5.1.3 Isotropic Tensors
5.1.4 Dot and Cross Products in Three Dimensions
5.1.5 The Gram Determinant
5.1.6 Levi-Civita Symbol in d Dimensions
5.2 Rotations in Three Dimensions
5.2.1 Proper and Improper Rotations
5.2.2 Scalars and Pseudoscalars; Polar and Axial Vectors
5.2.3 Transformation Properties of Physical Quantities
5.3 Invariant Decomposition of a 2nd Rank Tensor
5.3.1 Spherical or Irreducible Tensors
5.3.2 Stress, Strain, and Stiffness Tensors
5.3.3 Moment of Inertia
5.3.4 The Euler Top
5.3.5 Multipole Expansion; Quadrupole Moment
5.3.6 The Octupole Moment
5.4 Solutions
6 Vector Calculus
6.1 Orthogonal Curvilinear Coordinates
6.1.1 Cylindrical and Spherical Polar Coordinates
6.1.2 Elliptic and Parabolic Coordinates
6.1.3 Polar Coordinates in d Dimensions
6.2 Scalar and Vector Fields and Their Derivatives
6.2.1 The Gradient of a Scalar Field
6.2.2 The Flux and Divergence of a Vector Field
6.2.3 The Circulation and Curl of a Vector Field
6.2.4 Some Physical Aspects of the Curl of a Vector Field
6.2.5 Any Vector Field is the Sum of a Curl and a Gradient
6.2.6 The Laplacian Operator
6.2.7 Why Do div, curl, and del-Squared Occur so Frequently?
6.2.8 The Standard Identities of Vector Calculus
6.3 Solutions
7 A Bit of Fluid Dynamics
7.1 Equation of Motion of a Fluid Element
7.1.1 Hydrodynamic Variables
7.1.2 Equation of Motion
7.2 Flow When Viscosity Is Neglected
7.2.1 Euler's Equation
7.2.2 Barotropic Flow
7.2.3 Bernoulli's Principle in Steady Flow
7.2.4 Irrotational Flow and the Velocity Potential
7.3 Vorticity
7.3.1 Vortex Lines
7.3.2 Equations in Terms of v Alone
7.4 Flow of a Viscous Fluid
7.4.1 The Viscous Force in a Fluid
7.4.2 The Navier–Stokes Equation
7.5 Solutions
8 Some More Vector Calculus
8.1 Integral Theorems of Vector Calculus
8.1.1 The Fundamental Theorem of Calculus
8.1.2 Stokes' Theorem
8.1.3 Green's Theorem
8.1.4 A Topological Restriction; ``Exact'' Versus ``Closed''
8.1.5 Gauss's Theorem
8.1.6 Green's Identities and Reciprocity Relation
8.1.7 Comment on the Generalized Stokes' Theorem
8.2 Harmonic Functions
8.2.1 Mean Value Property
8.2.2 Harmonic Functions Have No Absolute Maxima or Minima
8.2.3 What Is the Significance of the Laplacian?
8.3 Singularities of Planar Vector Fields
8.3.1 Critical Points and the Poincaré Index
8.3.2 Degenerate Critical Points and Unfolding Singularities
8.3.3 Singularities of Three-Vector Fields
8.4 Solutions
9 A Bit of Electromagnetism and Special Relativity
9.1 Classical Electromagnetism
9.1.1 Maxwell's Field Equations
9.1.2 The Scalar and Vector Potentials
9.1.3 Gauge Invariance and Choice of Gauge
9.1.4 The Coulomb Gauge
9.1.5 Electrostatics
9.1.6 Magnetostatics
9.1.7 The Lorenz Gauge
9.2 Special Relativity
9.2.1 The Principle and the Postulate of Relativity
9.2.2 Boost Formulas
9.2.3 Collinear Boosts: Velocity Addition Rule
9.2.4 Rapidity
9.2.5 Lorentz Scalars and Four-Vectors
9.2.6 Matrices Representing Lorentz Transformations
9.3 Relativistic Invariance of Electromagnetism
9.3.1 Covariant Form of the Field Equations
9.3.2 The Electromagnetic Field Tensor
9.3.3 Transformation Properties of E and B
9.3.4 Lorentz Invariants of the Electromagnetic Field
9.3.5 Energy Density and the Poynting Vector
9.4 Solutions
10 Linear Vector Spaces
10.1 Definitions and Basic Properties
10.1.1 Definition of a Linear Vector Space
10.1.2 The Dual of a Linear Space
10.1.3 The Inner Product of Two Vectors
10.1.4 Basis Sets and Dimensionality
10.2 Orthonormal Basis Sets
10.2.1 Gram–Schmidt Orthonormalization
10.2.2 Expansion of an Arbitrary Vector
10.2.3 Basis Independence of the Inner Product
10.3 Some Important Inequalities
10.3.1 The Cauchy–Schwarz Inequality
10.3.2 The Triangle Inequality
10.3.3 The Gram Determinant Inequality
10.4 Solutions
11 A Look at Matrices
11.1 Pauli Matrices
11.1.1 Expansion of a (2times2) Matrix
11.1.2 Basic Properties of the Pauli Matrices
11.2 The Exponential of a Matrix
11.2.1 Occurrence and Definition
11.2.2 The Exponential of an Arbitrary (2times2) Matrix
11.3 Rotation Matrices in Three Dimensions
11.3.1 Generators of Infinitesimal Rotations and Their Algebra
11.3.2 The General Rotation Matrix
11.3.3 The Finite Rotation Formula for a Vector
11.4 The Eigenvalue Spectrum of a Matrix
11.4.1 The Characteristic Equation
11.4.2 Gershgorin's Circle Theorem
11.4.3 The Cayley–Hamilton Theorem
11.4.4 The Resolvent of a Matrix
11.5 A Generalization of the Gaussian Integral
11.6 Inner Product in the Linear Space of Matrices
11.7 Solutions
12 More About Matrices
12.1 Matrices as Operators in a Linear Space
12.1.1 Representation of Operators
12.1.2 Projection Operators
12.2 Hermitian, Unitary, and Positive Definite Matrices
12.2.1 Definitions and Eigenvalues
12.2.2 The Eigenvalues of a Rotation Matrix in d Dimensions
12.2.3 The General Form of a (2times2) Unitary Matrix
12.3 Diagonalization of a Matrix and all That
12.3.1 Eigenvectors, Nullspace, and Nullity
12.3.2 The Rank of a Matrix and the Rank-Nullity Theorem
12.3.3 Degenerate Eigenvalues and Defective Matrices
12.3.4 When Can a Matrix Be Diagonalized?
12.3.5 The Minimal Polynomial of a Matrix
12.3.6 Simple Illustrative Examples
12.3.7 Jordan Normal Form
12.3.8 Other Matrix Decompositions
12.3.9 Circulant Matrices
12.3.10 A Simple Illustration: A 3-state Random Walk
12.4 Commutators of Matrices
12.4.1 Mutually Commuting Matrices in Quantum Mechanics
12.4.2 The Lie Algebra of (n timesn) Matrices
12.5 Spectral Representation of a Matrix
12.5.1 Right and Left Eigenvectors of a Matrix
12.5.2 An Illustration
12.6 Solutions
13 Infinite-Dimensional Vector Spaces
13.1 The Space ell2 of Square-Summable Sequences
13.2 The Space mathcalL2 of Square-Integrable Functions
13.2.1 Definition of mathcalL2
13.2.2 Continuous Basis
13.2.3 Weight Functions: A Generalization of mathcalL2
13.2.4 mathcalL2(-infty,infty) Functions and Fourier Transforms
13.2.5 The Wave Function of a Particle
13.3 Hilbert Space and Subspaces
13.3.1 Hilbert Space
13.3.2 Linear Manifolds and Subspaces
13.4 Solutions
14 Linear Operators on a Vector Space
14.1 Some Basic Notions
14.1.1 Domain, Range, and Inverse
14.1.2 Linear Operators, Norm, and Bounded Operators
14.2 The Adjoint of an Operator
14.2.1 Densely Defined Operators
14.2.2 Definition of the Adjoint Operator
14.2.3 Symmetric, Hermitian, and Self-adjoint Operators
14.3 The Derivative Operator in mathcalL2
14.3.1 The Momentum Operator of a Quantum Particle
14.3.2 The Adjoint of the Derivative Operator in mathcalL2(-infty,infty)
14.3.3 When Is -i(d/dx) Self-adjoint in mathcalL2[a,b]?
14.3.4 Self-adjoint Extensions of Operators
14.3.5 Deficiency Indices
14.3.6 The Radial Momentum Operator in d 2 Dimensions
14.4 Nonsymmetric Operators
14.4.1 The Operators xpmip
14.4.2 Oscillator Ladder Operators and Coherent States
14.4.3 Eigenvalues and Non-normalizable Eigenstates of x and p
14.4.4 Matrix Representations for Unbounded Operators
14.5 Solutions
15 Operator Algebras and Identities
15.1 Operator Algebras
15.1.1 The Heisenberg Algebra
15.1.2 Some Other Basic Operator Algebras
15.2 Useful Operator Identities
15.2.1 Perturbation Series for an Inverse Operator
15.2.2 Hadamard's Lemma
15.2.3 Weyl Form of the Canonical Commutation Relation
15.2.4 The Zassenhaus Formula
15.2.5 The Baker–Campbell–Hausdorff Formula
15.3 Some Physical Applications
15.3.1 Angular Momentum Operators
15.3.2 Representation of Rotations by SU(2) Matrices
15.3.3 Connection Between the Groups SO(3) and SU(2)
15.3.4 The Parameter Space of SU(2)
15.3.5 The Parameter Space of SO(3)
15.3.6 The Parameter Space of SO(2)
15.4 Some More Physical Applications
15.4.1 The Displacement Operator and Coherent States
15.4.2 The Squeezing Operator and the Squeezed Vacuum
15.4.3 Values of z That Produce Squeezing in x or p
15.4.4 The Squeezing Operator and the Group SU(1,1)
15.4.5 SU(1,1) Generators in Terms of Pauli Matrices
15.5 Solutions
16 Orthogonal Polynomials
16.1 General Formalism
16.1.1 Introduction
16.1.2 Orthogonality and Completeness
16.1.3 Expansion and Inversion Formulas
16.1.4 Uniqueness and Explicit Representation
16.1.5 Recursion Relation
16.2 The Classical Orthogonal Polynomials
16.2.1 Polynomials of the Hypergeometric Type
16.2.2 The Hypergeometric Differential Equation
16.2.3 Rodrigues Formula and Generating Function
16.2.4 Class I.Hermite Polynomials
16.2.5 Linear Harmonic Oscillator Eigenfunctions
16.2.6 Oscillator Coherent State Wave Functions
16.2.7 Class II.Generalized Laguerre Polynomials
16.2.8 Class III.Jacobi Polynomials
16.3 Gegenbauer Polynomials
16.3.1 Ultraspherical Harmonics
16.3.2 Chebyshev Polynomials of the 1st Kind
16.3.3 Chebyshev Polynomials of the Second Kind
16.4 Legendre Polynomials
16.4.1 Basic Properties
16.4.2 Pn(x) by Gram–Schmidt Orthonormalization
16.4.3 Expansion in Legendre Polynomials
16.4.4 Expansion of xn in Legendre Polynomials
16.4.5 Legendre Function of the Second Kind
16.4.6 Associated Legendre Functions
16.4.7 Spherical Harmonics
16.4.8 Expansion of the Coulomb Kernel
16.5 Solutions
17 Fourier Series
17.1 Series Expansion of Periodic Functions
17.1.1 Dirichlet Conditions
17.1.2 Orthonormal Basis
17.1.3 Fourier Series Expansion and Inversion Formula
17.1.4 Parseval's Formula for Fourier Series
17.1.5 Simplified Formulas When (a,b) = (-π,π)
17.2 Asymptotic Behavior and Convergence
17.2.1 Uniform Convergence of Fourier Series
17.2.2 Large-n Behavior of Fourier Coefficients
17.2.3 Periodic Array of δ-Functions: The Dirac Comb
17.3 Summation of Series
17.3.1 Some Examples
17.3.2 The Riemann Zeta Function ζ(2k)
17.3.3 Fourier Series Expansions of cosαx and sinαx
17.4 Solutions
18 Fourier Integrals
18.1 Expansion of Nonperiodic Functions
18.1.1 Fourier Transform and Inverse Fourier Transform
18.1.2 Parseval's Formula for Fourier Transforms
18.1.3 Fourier Transform of the δ-Function
18.1.4 Examples of Fourier Transforms
18.1.5 Relative ``Spreads'' of a Fourier Transform Pair
18.1.6 The Convolution Theorem
18.1.7 Generalized Parseval Formula
18.2 The Fourier Transform Operator in mathcalL2
18.2.1 Iterates of the Fourier Transform Operator
18.2.2 Eigenvalues and Eigenfunctions of mathcalF
18.2.3 The Adjoint of an Integral Operator
18.2.4 Unitarity of the Fourier Transformation
18.3 Generalization to Several Dimensions
18.4 The Poisson Summation Formula
18.4.1 Derivation of the Formula
18.4.2 Some Illustrative Examples
18.4.3 Generalization to Higher Dimensions
18.5 Solutions
19 Discrete Probability Distributions
19.1 Some Elementary Distributions
19.1.1 Mean and Variance
19.1.2 Bernoulli Trials and the Binomial Distribution
19.1.3 Number Fluctuations in a Classical Ideal Gas
19.1.4 The Geometric Distribution
19.1.5 Photon Number Distribution in Blackbody Radiation
19.2 The Poisson Distribution
19.2.1 From the Binomial to the Poisson Distribution
19.2.2 Photon Number Distribution in Coherent Radiation
19.2.3 Photon Number Distribution in the Squeezed Vacuum State
19.2.4 The Sum of Poisson-Distributed Random Variables
19.2.5 The Difference of Two Poisson-Distributed Random Variables
19.3 The Negative Binomial Distribution
19.4 The Simple Random Walk
19.4.1 Random Walk on a Linear Lattice
19.4.2 Some Generalizations of the Simple Random Walk
19.5 Solutions
20 Continuous Probability Distributions
20.1 Continuous Random Variables
20.1.1 Probability Density and Cumulative Distribution
20.1.2 The Moment-Generating Function
20.1.3 The Cumulant-Generating Function
20.1.4 Application to the Discrete Distributions
20.1.5 The Characteristic Function
20.1.6 The Additivity of Cumulants
20.2 The Gaussian Distribution
20.2.1 The Normal Density and Distribution
20.2.2 Moments and Cumulants of a Gaussian Distribution
20.2.3 Simple Functions of a Gaussian Random Variable
20.2.4 Mean Collision Rate in a Dilute Gas
20.3 The Gaussian as a Limit Law
20.3.1 Linear Combinations of Gaussian Random Variables
20.3.2 The Central Limit Theorem
20.3.3 An Explicit Illustration of the Central Limit Theorem
20.4 Random Flights
20.4.1 From Random Flights to Diffusion
20.4.2 The Probability Density for Short Random Flights
20.5 The Family of Stable Distributions
20.5.1 What Is a Stable Distribution?
20.5.2 The Characteristic Function of Stable Distributions
20.5.3 Three Important Cases: Gaussian, Cauchy, and Lévy
20.5.4 Some Connections Between the Three Cases
20.6 Infinitely Divisible Distributions
20.6.1 Divisibility of a Random Variable
20.6.2 Infinite Divisibility Does Not Imply Stability
20.7 Solutions
21 Stochastic Processes
21.1 Multiple-Time Joint Probabilities
21.2 Discrete Markov Processes
21.2.1 The Two-Time Conditional Probability
21.2.2 The Master Equation
21.2.3 Formal Solution of the Master Equation
21.2.4 The Stationary Distribution
21.2.5 Detailed Balance
21.3 The Autocorrelation Function
21.4 The Dichotomous Markov Process
21.4.1 The Stationary Distribution
21.4.2 Solution of the Master Equation
21.5 Birth-and-Death Processes
21.5.1 The Poisson Pulse Process and Radioactive Decay
21.5.2 Biased Random Walk on a Linear Lattice
21.5.3 Connection with the Skellam Distribution
21.5.4 Asymptotic Behavior of the Probability
21.6 Continuous Markov Processes
21.6.1 Master Equation for the Conditional density
21.6.2 The Fokker–Planck Equation
21.6.3 The Autocorrelation Function for a Continuous Process
21.7 The Stationary Gaussian Markov Process
21.7.1 The Ornstein–Uhlenbeck Process
21.7.2 The Ornstein–Uhlenbeck Distribution
21.7.3 Velocity Distribution in a Classical Ideal Gas
21.7.4 Solution for an Arbitrary Initial Velocity Distribution
21.7.5 Diffusion of a Harmonically Bound Particle
21.8 Solutions
22 Analytic Functions of a Complex Variable
22.1 Some Preliminaries
22.1.1 Complex Numbers
22.1.2 Equations to Curves in the Plane in Terms of z
22.2 The Riemann Sphere
22.2.1 Stereographic Projection
22.2.2 Maps of Circles on the Riemann Sphere
22.2.3 A Metric on the Extended Complex Plane
22.3 Analytic Functions of z
22.3.1 The Cauchy–Riemann Conditions
22.3.2 The Real and Imaginary Parts of an Analytic Function
22.4 The Derivative of an Analytic Function
22.5 Power Series as Analytic Functions
22.5.1 Radius and Circle of Convergence
22.5.2 An Instructive Example
22.5.3 Behavior on the Circle of Convergence
22.5.4 Lacunary Series
22.6 Entire Functions
22.6.1 Representation of Entire Functions
22.6.2 The Order of an Entire Function
22.7 Solutions
23 More on Analytic Functions
23.1 Cauchy's Integral Theorem
23.2 Singularities
23.2.1 Simple Pole; Residue at a Pole
23.2.2 Multiple pole
23.2.3 Essential Singularity
23.2.4 Laurent Series
23.2.5 Singularity at Infinity
23.2.6 Accumulation Points
23.2.7 Meromorphic Functions
23.3 Contour Integration
23.3.1 A Basic Formula
23.3.2 Cauchy's Residue Theorem
23.3.3 The Dirichlet Integral; Cauchy Principal Value
23.3.4 The ``iε-Prescription'' for a Singular Integral
23.3.5 Residue at Infinity
23.4 Summation of Series Using Contour Integration
23.5 Linear Recursion Relations with Constant Coefficients
23.5.1 The Generating Function
23.5.2 Hemachandra-Fibonacci Numbers
23.5.3 Catalan Numbers
23.5.4 Connection with Wigner's Semicircular Distribution
23.6 Mittag-Leffler Expansion of Meromorphic Functions
23.7 Solutions
24 Linear Response and Analyticity
24.1 The Dynamic Susceptibility
24.1.1 Linear, Causal, Retarded Response
24.1.2 Frequency-Dependent Response
24.1.3 Symmetry Properties of the Dynamic Susceptibility
24.2 Dispersion Relations
24.2.1 Derivation of the Relations
24.2.2 Complex Admittance of an LCR Circuit
24.2.3 Subtracted Dispersion Relations
24.2.4 Hilbert Transform Pairs
24.2.5 Discrete and Continuous Relaxation Spectra
24.3 Solutions
25 Analytic Continuation and the Gamma Function
25.1 Analytic Continuation
25.1.1 What Is Analytic Continuation?
25.1.2 The Permanence of Functional Relations
25.2 The Gamma Function for Complex Argument
25.2.1 Stripwise Analytic Continuation of Γ(z)
25.2.2 Mittag-Leffler Expansion of Γ(z)
25.2.3 Logarithmic Derivative of Γ(z)
25.2.4 Infinite Product Representation of Γ(z)
25.2.5 Connection with the Riemann Zeta Function
25.2.6 The Beta Function
25.2.7 Reflection Formula for Γ(z)
25.2.8 Legendre's Doubling Formula
25.3 Solutions
26 Multivalued Functions and Integral Representations
26.1 Multivalued Functions
26.1.1 Branch Points and Branch Cuts
26.1.2 Types of Branch Points
26.1.3 Contour Integrals in the Presence of Branch Points
26.2 Contour Integral Representations
26.2.1 The Gamma Function
26.2.2 The Beta Function
26.2.3 The Riemann Zeta Function
26.2.4 Connection with Bernoulli Numbers
26.2.5 The Legendre Functions Pν(z) and Qν(z)
26.3 Singularities of Functions Defined by Integrals
26.3.1 End Point and Pinch Singularities
26.3.2 Singularities of the Legendre Functions
26.4 Solutions
27 Möbius Transformations
27.1 Conformal Mapping
27.2 Möbius (or Fractional Linear) Transformations
27.2.1 Definition
27.2.2 Fixed Points
27.2.3 The Cross-Ratio and Its Invariance
27.3 Normal Form of a Möbius Transformation
27.3.1 Normal Forms in Different Cases
27.3.2 Iterates of a Möbius Transformation
27.3.3 Classification of Möbius Transformations
27.3.4 The Isometric Circle
27.4 Group Properties
27.4.1 The Möbius Group
27.4.2 The Möbius Group Over the Reals
27.4.3 The Invariance Group of the Unit Circle
27.4.4 The Group of Cross-Ratios
27.5 Solutions
28 Laplace Transforms
28.1 Definition and Properties
28.1.1 Definition of the Laplace Transform
28.1.2 Transforms of Some Simple Functions
28.1.3 The Convolution Theorem
28.1.4 Laplace Transforms of Derivatives
28.2 The Inverse Laplace Transform
28.2.1 The Mellin Formula
28.2.2 LCR Circuit Under a Sinusoidal Applied Voltage
28.3 Bessel Functions and Laplace Transforms
28.3.1 Differential Equations and Power Series Representations
28.3.2 Generating Functions and Integral Representations
28.3.3 Spherical Bessel Functions
28.3.4 Laplace Transforms of Bessel Functions
28.4 Laplace Transforms and Random Walks
28.4.1 Random Walk in d Dimensions
28.4.2 The First-Passage-Time Distribution
28.5 Solutions
29 Green Function for the Laplacian Operator
29.1 The Partial Differential Equations of Physics
29.2 Green Functions
29.2.1 Green Function for an Ordinary Differential Operator
29.2.2 An Illustrative Example
29.3 The Fundamental Green Function for 2
29.3.1 Poisson's Equation in Three Dimensions
29.3.2 The Solution for G(3)(r, r')
29.3.3 Solution of Poisson's Equation
29.3.4 Connection with the Coulomb Potential
29.4 The Coulomb Potential in d > 3 Dimensions
29.4.1 Simplification of the Fundamental Green Function
29.4.2 Power Counting and a Divergence Problem
29.4.3 Dimensional Regularization
29.4.4 A Direct Derivation
29.5 The Coulomb Potential in d=2 Dimensions
29.5.1 Dimensional Regularization
29.5.2 Direct Derivation
29.5.3 An Alternative Regularization
29.6 Solutions
30 The Diffusion Equation
30.1 The Fundamental Gaussian Solution
30.1.1 Fick's Laws of Diffusion
30.1.2 Further Remarks on Linear Response
30.1.3 The Fundamental Solution in d Dimensions
30.1.4 Solution for an Arbitrary Initial Distribution
30.1.5 Moments of the Distance Travelled in Time t
30.2 Diffusion in One Dimension
30.2.1 Continuum Limit of a Biased Random Walk
30.2.2 Free Diffusion on an Infinite Line
30.2.3 Absorbing and Reflecting Boundary Conditions
30.2.4 Finite Boundaries: Solution by the Method of Images
30.2.5 Finite Boundaries: Solution by Separation of Variables
30.2.6 Survival Probability and Escape-Time Distribution
30.2.7 Equivalence of the Solutions
30.3 Diffusion with Drift: Sedimentation
30.3.1 The Smoluchowski Equation
30.3.2 Equilibrium Barometric Distribution
30.3.3 The Time-Dependent Solution
30.4 The Schrödinger Equation for a Free Particle
30.4.1 Connection with the Free-Particle Propagator
30.4.2 Spreading of a Quantum Mechanical Wave Packet
30.4.3 The Wave Packet in Momentum Space
30.5 Solutions
31 The Wave Equation
31.1 Causal Green Function of the Wave Operator
31.1.1 Formal Solution as a Fourier Transform
31.1.2 Simplification of the Formal Solution
31.2 Explicit Solutions for d =1, 2 and 3
31.2.1 The Green Function in (1+1) Dimensions
31.2.2 The Green Function in (2+1) Dimensions
31.2.3 The Green Function in (3+1) Dimensions
31.2.4 Retarded Solution of the Wave Equation
31.3 Remarks on Propagation in Dimensions d > 3
31.4 Solutions
32 Integral Equations
32.1 Fredholm Integral Equations
32.1.1 Equation of the First Kind
32.1.2 Equation of the Second Kind
32.1.3 Degenerate Kernels
32.1.4 The Eigenvalues of a Degenerate Kernel
32.1.5 Iterative Solution: Neumann Series
32.2 Nonrelativistic Potential Scattering
32.2.1 The Scattering Amplitude
32.2.2 Integral Equation for Scattering
32.2.3 Green Function for the Helmholtz Operator
32.2.4 Formula for the Scattering Amplitude
32.2.5 The Born Approximation
32.2.6 Yukawa and Coulomb Potentials; Rutherford's Formula
32.3 Partial Wave Analysis
32.3.1 The Physical Idea Behind Partial Wave Analysis
32.3.2 Expansion of a Plane Wave in Spherical Harmonics
32.3.3 Partial Wave Scattering Amplitude and Phase Shift
32.3.4 The Optical Theorem
32.4 The Fredholm Solution
32.4.1 The Fredholm Formulas
32.4.2 Remark on the Application to the Scattering Problem
32.5 Volterra Integral Equations
32.6 Solutions
Appendix Bibliography and Further Reading

Index