A textbook of quantum mechanics

Cover
Half Title
About the Authors
Title Page
Copyright
Contents
Preface to the Second Edition
Preface to the First Edition
Chapter 1: Towards Quantum Mechanics
	A. CONCEPTS OF CLASSICAL MECHANICS
		1.1 Mechanics of Material Systems
		1.2 Electromagnetic Fields and Light
	B. INADEQUACY OF CLASSICAL CONCEPTS
		(i) Macroscopic Statistical Phenomena
			1.3 Black Body Radiation; Planck’s Quantum Hypothesis
			1.4 Specific Heats of Solids
		(ii) Electromagnetic Radiation—Wave-Particle Duality
			1.5 The Photoelectric Effect
			1.6 The Compton Effect
		(iii) Atomic Structure and Atomic Spectra
			1.7 The Rutherford Atom Model
			1.8 Bohr’s Postulates
			1.9 Bohr’s Theory of the Hydrogen Spectrum
			1.10 Bohr-Sommerfeld Quantum Rules; Degeneracy
			1.11 Space Quantization
			1.12 Limitations of the Old Quantum Theory
		(iv) Matter Waves
			1.13 De Broglie’s Hypothesis
			1.14 The Motion of a Free Wave Packet; Classical Approximation and the Uncertainty Principle
			1.15 Uncertainties Introduced in the Process of Measurement
			1.16 Approximate Classical Motion in Slowly Varying Fields
			1.17 Diffraction Phenomena: Interpretation of the Wave-Particle Dualism
			1.18 Complementarity
			1.19 The Formulation of Quantum Mechanics
			1.20 Photons: The Quantization of Fields
Chapter 2: The Schrödinger Equation and Stationary States
	A. THE SCHRÖDINGER EQUATION
		2.1 A Free Particle in One Dimension
		2.2 Generalization to Three Dimensions
		2.3 The Operator Correspondence and the Schrödinger Equation for a Particle Subject to Forces
	B. PHYSICAL INTERPRETATION AND CONDITIONS ON ψ
		2.4 Normalization and Probability Interpretation
		2.5 Non-normalizable Wave Functions and Box Normalization
		2.6 Conservation of Probability
		2.7 Expectation Values; Ehrenfest’s Theorem
		2.8 Admissibility Conditions on the Wave Function
	C. STATIONARY STATES AND ENERGY SPECTRA
		2.9 Stationary States; The Time-Independent Schödinger Equation
		2.10 A particle in a Square Well Potential
		2.11 Bound States in a Square Well: (E < 0)
		2.12 The Square Well: Non-localized States (E > 0)
		2.13 Square Potential Barrier
		2.14 One-dimensional delta-function well
		2.15 Multiple Potential Wells: Splitting of Energy Levels; Energy Bands
		Problems
Chapter 3: General Formalism of Wave Mechanics
	3.1 The Schrödinger Equation and the Probability Interpretation for an N-Particle System
	3.2 The Fundamental Postulates of Wave Mechanics
	3.3 The Adjoint of an Operator and Self-Adjointness
	3.4 The Eigenvalue Problem; Degeneracy
	3.5 Eigenvalues and Eigenfunctions of Self-Adjoint Operators
	3.6 The Dirac Delta Function
	3.7 Observables: Completeness and Normalization of Eigenfunctions
	3.8 Closure
	3.9 Physical Interpretation of Eigenvalues, Eigenfunctions and Expansion Coefficients
	3.10 Momentum Eigenfunctions; Wave Functions in Momentum Space
	3.11 The Uncertainty Principle
	3.12 States with Minimum Value for Uncertainty Product
	3.13 Commuting Observables; Removal of Degeneracy
	3.14 Evolution of System with Time; Constants of the Motion
	3.15 Non-Interacting and Interacting Systems
	3.16 Systems of Identical Particles
	Problems
Chapter 4: Exactly Soluble Eigenvalue Problems
	A. THE SIMPLE HARMONIC OSCILLATOR
		4.1 The Schrödinger Equation and Energy Eigenvalues
		4.2 The Energy Eigenfunctions
		4.3 Properties of Stationary States
		4.4 The Abstract Operator Method
		4.5 Coherent States
	B. ANGULAR MOMENTUM AND PARITY
		4.6 The Angular Momentum Operators
		4.7 The Eigenvalue Equation for L2; Separation of Variables
		4.8 Admissibility Conditions on Solutions; Eigenvalues
		4.9 The Eigenfunctions: Spherical Harmonics
		4.10 Physical Interpretation
		4.11 Parity
		4.12 Angular Momentum in Stationary States of Systems with Spherical Symmetry
	C. THREE-DIMENSIONAL SQUARE WELL POTENTIAL
		4.13 Solutions in the Interior Region
		4.14 Solution in the Exterior Region and Matching
	D. THE HYDROGEN ATOM
		4.15 Solution of the Radial Equation; Energy Levels
		4.16 Stationary State Wave Functions
		4.17 Discussion of Bound States
		4.18 Solution in Terms of Confluent Hypergeometric Functions; Nonlocalized States
		4.19 Solution in Parabolic Coordinates
	E. OTHER PROBLEMS IN THREE DIMENSIONS
		4.20 The Anisotropic Oscillator
		4.21 The Isotropic Oscillator
		4.22 Normal Modes of Coupled Systems of Particles
		4.23 A Charged Particle in a Uniform Mag­netic Field
		4.24 Integer Quantum Hall Effect
		Problems
Chapter 5: Approximation Methods for Stationary States
	A. PERTURBATION THEORY FOR DISCRETE LEVELS
		5.1 Equations in Various Orders of Perturbation Theory
		5.2 The Non-Degenerate Case
		5.3 The Degenerate Case — Removal of Degeneracy
		5.4 The Effect of an Electric Field on the Energy Levels of an Atom (Stark Effect)
		5.5 Two-Electron Atoms
	B. THE VARIATION METHOD
		5.6 Upper Bound on Ground State Energy
		5.7 Application to Excited States
		5.8 Trial Function Linear in Variational Parameters
		5.9 The Hydrogen Molecule
		5.10 Exchange Interaction
	C. THE WKB APPROXIMATION
		5.11 The One-Dimensional Schrödinger Equation
		5.12 The Bohr-Sommerfeld Quantum Condition
		5.13 WKB Solution of the Radial Wave Equation
		Problems
Chapter 6: Scattering Theory
	A. THE SCATTERING CROSS-SECTION: GENERAL CONSIDERATIONS
		6.1 Kinematics of the Scattering Process: Differential and Total Cross-Sections
		6.2 Wave Mechanical Picture of Scattering: The Scattering Amplitude
		6.3 Green’s Functions; Formal Expression for Scattering Amplitude
	B. THE BORN AND EIKONAL APPROXIMATIONS
		6.4 The Born Approximation
		6.5 Validity of the Born Approximation
		6.6 The Born Series
		6.7 The Eikonal Approximation
	C. PARTIAL WAVE ANALYSIS
		6.8 Asymptotic Behaviour of Partial Waves: Phase Shifts
		6.9 The scattering Amplitude in Terms of Phase Shifts
		6.10 The Differential and Total Cross-Sections; Optical Theorem
		6.11 Phase Shifts: Relation to the Potential
		6.12 Potentials of Finite Range
		6.13 Low Energy Scattering
	D. EXACTLY SOLUBLE PROBLEMS
		6.14 Scattering by a Square Well Potential
		6.15 Scattering by a Hard Sphere
		6.16 Scattering by a Coulomb Potential
	E. MUTUAL SCATTERING OF TWO PARTICLES
		6.17 Reduction of the Two-Body Problem: The Centre of Mass Frame
		6.18 Transformation from Centre of Mass to Laboratory Frame of Reference
		6.19 Collisions between Identical Particles
		Problems
Chapter 7: Representations, Transformations and Symmetries
	7.1 Quantum States; State Vectors and Wave Functions
	7.2 The Hilbert Space of State Vectors; Dirac Notation
	7.3 Dynamical Variables and Linear Operators
	7.4 Representations
	7.5 Continuous Basis—The Schrödinger Representation
	7.6 Degeneracy; Labelling by Commuting Observables
	7.7 Change of Basis; Unitary Transformations
	7.8 Unitary Transformations Induced by Change of Coordinate System: Translations
	7.9 Unitary Transformation Induced by Rotation of Coordinate System
	7.10 The Algebra of Rotation Generators
	7.11 Transformation of Dynamical Variables
	7.12 Symmetries and Conservation Laws
	7.13 Space Inversion
	7.14 Time Reversal
	Problems
Chapter 8: Angular momentum
	8.1 The Eigenvalue Spectrum
	8.2 Matrix Representaion of K in the jm Basis
	8.3 Spin Angular Momentum
	8.4 Non-relativistic Hamiltonian with spin; Diamagnetism
	8.5 Addition of Angular Momenta
	8.6 Clebsch-Gordan Coefficients
	8.7 Spin Wave Functions for a System of Two Spin – 1/2 Practicles
	8.8 Identical Particles with Spin
	8.9 Addition of Spin and Orbital Angular Momenta
	8.10 Spherical Tensors; Tensor Operators
	8.11 The Wigner-Eckart Theorem
	8.12 Projection Theorem for a First Rank Tensor
	Problems
Chapter 9: Evolution with Time
	A. EXACT FORMAL SOLUTIONS: PROPAGATORS
		9.1 The Schrödinger Equation; General Solution
		9.2 Propagators
		9.3 Relation of Retarded Propagator to the Green’s Function of the Time-Independent Schrödinger Equation
		9.4 Alteration of Hamiltonian: Transitions; Sudden Approximation
		9.5 Path Integrals in Quantum Mechanics
		9.6 Aharonov–Bohm effect
	B. Perturbation Theory For Time Evolution Problems
	B. PERTURBATION THEORY FOR TIME EVOLUTION PROBLEMS
		9.7 Perturbative Solution for Transition Amplitude
		9.8 Selection Rules
		9.9 First Order Transitions: Constant Perturbation
		9.10 Transitions in the Second Order: Constant Perturbation
		9.11 Scattering of a Particle by a Potential
		9.12 Inelastic Scattering: Exchange Effects
		9.13 Double Scattering by Two Non-Overlapping Scatterers
		9.14 Harmonic Perturbations
		9.15 Interaction of an Atom with Electromagnetic Radiation
		9.16 The Dipole Approximation: Selection Rules
		9.17 The Einstein Coefficients: Spontaneous Emission
	C. ALTERNATIVE PICTURES OF TIME EVOLUTION
		9.18 The Schrödinger Picture: Transformation to other Pictures
		9.19 The Heisenberg Picture
		9.20 Matrix Mechanics—The Simple Harmonic Oscillator
		9.21 Electromagnetic Wave as Harmonic Oscillator; Quantization: Photons
		9.22 Atom Interacting with Quantized Radiation: Spontaneous Emission
		9.23 The Interaction Picture
		9.24 The Scattering Operator
	D. TIME EVOLUTION OF ENSEMBLES
		9.25 The Density Matrix
		9.26 Spin Density Matrix
		9.27 The Quantum Liouville Equation
		9.28 Magnetic Resonance
		Problems
Chapter 10: Relativistic Wave Equations
	10.1 Generalization of the Schrödinger Equation
	A. THE KLEIN-GORDON EQUATION
		10.2 Plane Wave Solutions; Charge and Current Densities
		10.3 Interaction with Electromagnetic Fields; Hydrogen-like Atom
		10.4 Nonrelativistic Limit
	B. The DIRAC EQUATION
		10.5 Dirac’s Relativistic Hamiltonian
		10.6 Position Probability Density; Expectation Values
		10.7 Dirac Matrices
		10.8 Plane Wave Solutions of the Dirac Equation; Energy Spectrum
		10.9 The Spin of the Dirac Particle
		10.10 Significance of Negative Energy States; Dirac Particle in Electromagnetic Fields
		10.11 Relativistic Electron in a Central Potential: Total Angular Momentum
		10.12 Radial Wave Equations in Coulomb Potential
		10.13 Series Solutions of the Radial Equations: Asymptotic Behaviour
		10.14 Determination of the Energy Levels
		10.15 Exact Radial Wave Functions; Comparison to Non-Relativistic Case
		10.16 Electron in a Magnetic Field—Spin Magnetic Moment
		10.17 The Spin Orbit Energy
		Problems
Appendices
	A. Classical Mechanics
	B. Relativistic Mechanics
	C. The dirac delta Function
	D. Mathematical Appendix
	E. Many-Electron Atoms
	F. Internal Symmetry
	G. Conversion between Gaussian and Rationalized MKSA Systems
Table of Physical Constants
Index