Mathematics for Chemistry and Physics

  • ID: 1767896
  • Book
  • 424 Pages
  • Elsevier Science and Technology
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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.

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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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Turrell, George
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