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Full Description
Chemistry and physics share a common mathematical foundation. From elementary calculus to vector analysis and group theory, Mathematics for Chemistry and Physics aims to provide a comprehensive reference for students and researchers pursuing these scientific fields. The book is based on the authors many classroom experience.Designed as a reference text, Mathematics for Chemistry and Physics will prove beneficial for students at all university levels in chemistry, physics, applied mathematics, and theoretical biology. Although this book is not computer-based, many references to current applications are included, providing the background to what goes on "behind the screen" in computer experiments.
Contents
Preface1 Variables and Functions1.1 Introduction1.2 Functions1.3 Classification and Properties of Functions1.4 Exponential and Logarithmic Functions1.5 Applications of Exponential and Logarithmic Functions1.6 Complex Numbers1.7 Circular Trigonometric Functions1.8 Hyperbolic FunctionsProblems2 Limits, Derivatives and Series2.1 Definition of a Limit2.2 Continuity2.3 The Derivative2.4 Higher Derivatives2.5 Implicit and Parametric Relations2.6 The Extrema of a Function and Its Critical Points2.7 The Differential2.8 The Mean-Value Theorem and l'Hospital's Rule2.9 Taylor's Series2.10 Binomial Expansion2.11 Tests of Series Convergence2.12 Functions of Several Variables2.13 Exact DifferentialsProblems3 Integration3.1 The Indefinite Integral3.2 Integration Formulas3.3 Methods of Integration3.3.1 Integration by Substitution3.3.2 Integration by Parts3.3.3 Integration of Partial Fractions3.4 Definite Integrals3.4.1 Definition3.4.2 Plane Area3.4.3 Line Integrals3.4.4 Fido and his Master3.4.5 The Gaussian and Its Moments3.5 Integrating Factors3.6 Tables of IntegralsProblems4 Vector Analysis4.1 Introduction4.2 Vector Addition4.3 Scalar Product4.4 Vector Product4.5 Triple Products4.6 Reciprocal Bases4.7 Differentiation of Vectors 4.8 Scalar and Vector Fields4.9 The Gradient4.10 The Divergence4.11 The Curl or Rotation4.12 The Laplacian4.13 Maxwell's Equations4.14 Line Integrals4.15 Curvilinear CoordinatesProblems5 Ordinary Differential Equations5.1 First-Order Differential Equations5.2 Second-Order Differential Equations5.2.1 Series Solution5.2.2 The Classical Harmonic Oscillator5.2.3 The Damped Oscillator5.3 The Differential Operator5.3.1 Harmonic Oscillator5.3.2 Inhomogeneous Equations5.3.3 Forced Vibrations5.4 Applications in Quantum Mechanics5.4.1 The Particle in a Box5.4.2 Symmetric Box5.4.3 Rectangular Barrier: The Tunnel Effect5.4.4 The Harmonic Oscillator in Quantum Mechanics5.5 Special Functions5.5.1 Hermite Polynomials5.5.2 Associated Legendre Polynomials5.5.3 The Associated Laguerre Polynomials5.5.4 The Gamma Function5.5.5 Bessel Functions5.5.6 Mathieu Functions5.5.7 The Hypergeometric FunctionsProblems6 Partial Differential Equations6.1 The Vibrating String6.1.1 The Wave Equation6.1.2 Separation of Variables6.1.3 Boundary Conditions6.1.4 Initial Conditions6.2 The Three-Dimensional Harmonic Oscillator6.2.1 Quantum-Mechanical Applications6.2.2 Degeneracy6.3 The Two-Body Problem6.3.1 Classical Mechanics6.3.2 Quantum Mechanics6.4 Central Forces6.4.1 Spherical Coordinates6.4.2 Spherical Harmonics6.5 The Diatomic Molecule6.5.1 The Rigid Rotator6.5.2 The Vibrating Rotator6.5.3 Centrifugal Forces6.6 The Hydrogen Atom6.6.1 Energy6.6.2 Wavefunctions and The Probability Density6.7 Binary Collisions6.7.1 Conservation of Angular Momentum6.7.2 Conservation of Energy6.7.3 Interaction Potential: LJ (6-12)6.7.4 Angle of Deflection6.7.5 Quantum Mechanical Description: The Phase ShiftProblems7 Operators and Matrices7.1 The Algebra of Operators7.2 Hermitian Operators and Their Eigenvalues7.3 Matrices7.4 The Determinant7.5 Properties of Determinants7.6 Jacobians7.7 Vectors and Matrices7.8 Linear Equations7.9 Partitioning of Matrices7.10 Matrix Formulation of the Eigenvalue Problem7.11 Coupled Oscillators7.12 Geometric Operations7.13 The Matrix Method in Quantum Mechanics7.14 The Harmonic OscillatorProblems8 Group Theory8.1 Definition of a Group8.2 Examples8.3 Permutations8.4 Conjugate Elements and Classes8.5 Molecular Symmetry8.6 The Character8.7 Irreducible Representations8.8 Character Tables8.9 Reduction of a Representation: The "Magic Formula"8.10 The Direct Product Representation8.11 Symmetry-Adapted Functions: Projection Operators8.12 Hybridization of Atomic Orbitals8.13 Crystal SymmetryProblems9 Molecular Mechanics9.1 Kinetic Energy9.2 Molecular Rotation9.2.1 Euler's Angles9.2.2 Classification of Rotators9.2.3 Angular Momenta9.2.4 The Symmetric Top in Quantum Mechanics9.3 Vibrational Energy9.3.1 Kinetic Energy9.3.2 Internal Coordinates: The G Matrix9.3.3 Potential Energy9.3.4 Normal Coordinates9.3.5 Secular Determinant9.3.6 An Example: The Water Molecule9.3.7 Symmetry Coordinates9.3.8 Application to Molecular Vibrations9.3.9 Form of Normal Modes9.4 Nonrigid Molecules9.4.1 Molecular Inversion9.4.2 Internal Rotation9.4.3 Molecular Conformation: The Molecular Mechanics MethodProblems10 Probability and Statistics10.1 Permutations10.2 Combinations10.3 Probability10.4 Stirling's Approximation10.5 Statistical Mechanics10.6 The Lagrange Multipliers10.7 The Partition Function10.8 Molecular Energies10.8.1 Translation10.8.2 Rotation10.8.3 Vibration10.9 Quantum Statistics10.9.1 The Indistinguishability of Identical Particles10.9.2 The Exclusion Principle10.9.3 Fermi-Dirac Statistics10.9.4 Bose-Einstein Statistics10.10 Ortho- and Para-HydrogenProblems11 Integral Transforms11.1 The Fourier Transform11.1.1 Convolution11.1.2 Fourier Transform Pairs11.2 The Laplace Transform11.2.1 Examples of Simple Laplace Transforms11.2.2 The Transform of Derivatives11.2.3 Solution of Differential Equations11.2.4 Laplace Transforms: Convolution and Inversion11.2.5 Green's FunctionsProblems12 Approximation Methods in Quantum Mechanics12.1 The Born-Oppenheimer Approximation12.2 Perturbation Theory: Stationary States12.2.1 Nondegenerate Systems12.2.2 First-Order Approximation12.2.3 Second-Order Approximation12.2.4 The Anharmonic Oscillator12.2.5 Degenerate Systems12.2.6 The Stark Effect of the Hydrogen Atom12.3 Time-Dependent Perturbations12.3.1 The SchrOdinger Equation12.3.2 Interaction of Light and Matter12.3.3 Spectroscopic Selection Rules12.4 The Variation Method12.4.1 The Variation Theorem12.4.2 An Example: The Particle in a Box12.4.3 Linear Variation Functions12.4.4 Linear Combinations of Atomic Orbitals (LCAO)12.4.5 The HUckel ApproximationProblems13 Numerical Analysis13.1 Errors13.1.1 The Gaussian Distribution13.1.2 The Poisson Distribution13.2 The Method of Least Squares13.3 Polynomial Interpolation and Smoothing13.4 The Fourier Transform13.4.1 The Discrete Fourier Transform (DFT)13.4.2 The Fast Fourier Transform (FFT)13.4.3 An Application: Interpolation and Smoothing13.5 Numerical Integration13.5.1 The Trapezoid Rule13.5.2 Simpson's Rule13.5.3 The Method of Romberg13.6 Zeros of Functions13.6.1 Newton's Method13.6.2 The Bisection Method13.6.3 The Roots: An ExampleProblemsAppendicesI The Greek AlphabetII Dimensions and UnitsIII Atomic OrbitalsIV Radial Wavefunctions for Hydrogenlike SpeciesV The Laplacian Operator in Spherical CoordinatesVI The Divergence TheoremVII Determination of the Molecular Symmetry GroupVIII Character Tables for Some of the More Common Point GroupsIX Matrix Elements for the Harmonic OscillatorX Further ReadingApplied MathematicsChemical PhysicsAuthor IndexSubject Index
