Mathematics for Chemistry & Physics book cover

Mathematics for Chemistry & Physics

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.

Audience
Undergraduate and graduate students in chemistry, chemical physics, materials science, and physics. Researchers in chemistry/physics and theoretical biologists.

Hardbound, 424 Pages

Published: December 2001

Imprint: Academic Press

ISBN: 978-0-12-705051-5

Contents


  • Preface

    1 Variables and Functions

    1.1 Introduction

    1.2 Functions

    1.3 Classification and Properties of Functions

    1.4 Exponential and Logarithmic Functions

    1.5 Applications of Exponential and Logarithmic Functions

    1.6 Complex Numbers

    1.7 Circular Trigonometric Functions

    1.8 Hyperbolic Functions

    Problems

    2 Limits, Derivatives and Series

    2.1 Definition of a Limit

    2.2 Continuity

    2.3 The Derivative

    2.4 Higher Derivatives

    2.5 Implicit and Parametric Relations

    2.6 The Extrema of a Function and Its Critical Points

    2.7 The Differential

    2.8 The Mean-Value Theorem and l’Hospital’s Rule

    2.9 Taylor's Series

    2.10 Binomial Expansion

    2.11 Tests of Series Convergence

    2.12 Functions of Several Variables

    2.13 Exact Differentials

    Problems

    3 Integration

    3.1 The Indefinite Integral

    3.2 Integration Formulas

    3.3 Methods of Integration

    3.3.1 Integration by Substitution

    3.3.2 Integration by Parts

    3.3.3 Integration of Partial Fractions

    3.4 Definite Integrals

    3.4.1 Definition

    3.4.2 Plane Area

    3.4.3 Line Integrals

    3.4.4 Fido and his Master

    3.4.5 The Gaussian and Its Moments

    3.5 Integrating Factors

    3.6 Tables of Integrals

    Problems

    4 Vector Analysis

    4.1 Introduction

    4.2 Vector Addition

    4.3 Scalar Product

    4.4 Vector Product

    4.5 Triple Products

    4.6 Reciprocal Bases

    4.7 Differentiation of Vectors

    4.8 Scalar and Vector Fields

    4.9 The Gradient

    4.10 The Divergence

    4.11 The Curl or Rotation

    4.12 The Laplacian

    4.13 Maxwell's Equations

    4.14 Line Integrals

    4.15 Curvilinear Coordinates

    Problems

    5 Ordinary Differential Equations

    5.1 First-Order Differential Equations

    5.2 Second-Order Differential Equations

    5.2.1 Series Solution

    5.2.2 The Classical Harmonic Oscillator

    5.2.3 The Damped Oscillator

    5.3 The Differential Operator

    5.3.1 Harmonic Oscillator

    5.3.2 Inhomogeneous Equations

    5.3.3 Forced Vibrations

    5.4 Applications in Quantum Mechanics

    5.4.1 The Particle in a Box

    5.4.2 Symmetric Box

    5.4.3 Rectangular Barrier: The Tunnel Effect

    5.4.4 The Harmonic Oscillator in Quantum Mechanics

    5.5 Special Functions

    5.5.1 Hermite Polynomials

    5.5.2 Associated Legendre Polynomials

    5.5.3 The Associated Laguerre Polynomials

    5.5.4 The Gamma Function

    5.5.5 Bessel Functions

    5.5.6 Mathieu Functions

    5.5.7 The Hypergeometric Functions

    Problems

    6 Partial Differential Equations

    6.1 The Vibrating String

    6.1.1 The Wave Equation

    6.1.2 Separation of Variables

    6.1.3 Boundary Conditions

    6.1.4 Initial Conditions

    6.2 The Three-Dimensional Harmonic Oscillator

    6.2.1 Quantum-Mechanical Applications

    6.2.2 Degeneracy

    6.3 The Two-Body Problem

    6.3.1 Classical Mechanics

    6.3.2 Quantum Mechanics

    6.4 Central Forces

    6.4.1 Spherical Coordinates

    6.4.2 Spherical Harmonics

    6.5 The Diatomic Molecule

    6.5.1 The Rigid Rotator

    6.5.2 The Vibrating Rotator

    6.5.3 Centrifugal Forces

    6.6 The Hydrogen Atom

    6.6.1 Energy

    6.6.2 Wavefunctions and The Probability Density

    6.7 Binary Collisions

    6.7.1 Conservation of Angular Momentum

    6.7.2 Conservation of Energy

    6.7.3 Interaction Potential: LJ (6-12)

    6.7.4 Angle of Deflection

    6.7.5 Quantum Mechanical Description: The Phase Shift

    Problems

    7 Operators and Matrices

    7.1 The Algebra of Operators

    7.2 Hermitian Operators and Their Eigenvalues

    7.3 Matrices

    7.4 The Determinant

    7.5 Properties of Determinants

    7.6 Jacobians

    7.7 Vectors and Matrices

    7.8 Linear Equations

    7.9 Partitioning of Matrices

    7.10 Matrix Formulation of the Eigenvalue Problem

    7.11 Coupled Oscillators

    7.12 Geometric Operations

    7.13 The Matrix Method in Quantum Mechanics

    7.14 The Harmonic Oscillator

    Problems

    8 Group Theory

    8.1 Definition of a Group

    8.2 Examples

    8.3 Permutations

    8.4 Conjugate Elements and Classes

    8.5 Molecular Symmetry

    8.6 The Character

    8.7 Irreducible Representations

    8.8 Character Tables

    8.9 Reduction of a Representation: The “Magic Formula”

    8.10 The Direct Product Representation

    8.11 Symmetry-Adapted Functions: Projection Operators

    8.12 Hybridization of Atomic Orbitals

    8.13 Crystal Symmetry

    Problems

    9 Molecular Mechanics

    9.1 Kinetic Energy

    9.2 Molecular Rotation

    9.2.1 Euler's Angles

    9.2.2 Classification of Rotators

    9.2.3 Angular Momenta

    9.2.4 The Symmetric Top in Quantum Mechanics

    9.3 Vibrational Energy

    9.3.1 Kinetic Energy

    9.3.2 Internal Coordinates: The G Matrix

    9.3.3 Potential Energy

    9.3.4 Normal Coordinates

    9.3.5 Secular Determinant

    9.3.6 An Example: The Water Molecule

    9.3.7 Symmetry Coordinates

    9.3.8 Application to Molecular Vibrations

    9.3.9 Form of Normal Modes

    9.4 Nonrigid Molecules

    9.4.1 Molecular Inversion

    9.4.2 Internal Rotation

    9.4.3 Molecular Conformation: The Molecular Mechanics Method

    Problems

    10 Probability and Statistics

    10.1 Permutations

    10.2 Combinations

    10.3 Probability

    10.4 Stirling's Approximation

    10.5 Statistical Mechanics

    10.6 The Lagrange Multipliers

    10.7 The Partition Function

    10.8 Molecular Energies

    10.8.1 Translation

    10.8.2 Rotation

    10.8.3 Vibration

    10.9 Quantum Statistics

    10.9.1 The Indistinguishability of Identical Particles

    10.9.2 The Exclusion Principle

    10.9.3 Fermi-Dirac Statistics

    10.9.4 Bose-Einstein Statistics

    10.10 Ortho- and Para-Hydrogen

    Problems

    11 Integral Transforms

    11.1 The Fourier Transform

    11.1.1 Convolution

    11.1.2 Fourier Transform Pairs

    11.2 The Laplace Transform

    11.2.1 Examples of Simple Laplace Transforms

    11.2.2 The Transform of Derivatives

    11.2.3 Solution of Differential Equations

    11.2.4 Laplace Transforms: Convolution and Inversion

    11.2.5 Green's Functions

    Problems

    12 Approximation Methods in Quantum Mechanics

    12.1 The Born-Oppenheimer Approximation

    12.2 Perturbation Theory: Stationary States

    12.2.1 Nondegenerate Systems

    12.2.2 First-Order Approximation

    12.2.3 Second-Order Approximation

    12.2.4 The Anharmonic Oscillator

    12.2.5 Degenerate Systems

    12.2.6 The Stark Effect of the Hydrogen Atom

    12.3 Time-Dependent Perturbations

    12.3.1 The Schr¨Odinger Equation

    12.3.2 Interaction of Light and Matter

    12.3.3 Spectroscopic Selection Rules

    12.4 The Variation Method

    12.4.1 The Variation Theorem

    12.4.2 An Example: The Particle in a Box

    12.4.3 Linear Variation Functions

    12.4.4 Linear Combinations of Atomic Orbitals (LCAO)

    12.4.5 The H¨Uckel Approximation

    Problems

    13 Numerical Analysis

    13.1 Errors

    13.1.1 The Gaussian Distribution

    13.1.2 The Poisson Distribution

    13.2 The Method of Least Squares

    13.3 Polynomial Interpolation and Smoothing

    13.4 The Fourier Transform

    13.4.1 The Discrete Fourier Transform (DFT)

    13.4.2 The Fast Fourier Transform (FFT)

    13.4.3 An Application: Interpolation and Smoothing

    13.5 Numerical Integration

    13.5.1 The Trapezoid Rule

    13.5.2 Simpson's Rule

    13.5.3 The Method of Romberg

    13.6 Zeros of Functions

    13.6.1 Newton's Method

    13.6.2 The Bisection Method

    13.6.3 The Roots: An Example

    Problems

    Appendices

    I The Greek Alphabet

    II Dimensions and Units

    III Atomic Orbitals

    IV Radial Wavefunctions for Hydrogenlike Species

    V The Laplacian Operator in Spherical Coordinates

    VI The Divergence Theorem

    VII Determination of the Molecular Symmetry Group

    VIII Character Tables for Some of the More Common Point Groups

    IX Matrix Elements for the Harmonic Oscillator

    X Further Reading

    Applied Mathematics

    Chemical Physics

    Author Index

    Subject Index

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