Contents

• Dedication
• List of Figures
• List of Tables
• Preface
  • To the Student
  • Acknowledgments
  • Comments and Feedback
  
  
• 1. Mathematical Prerequisites
  • 1.1 Complex Numbers
  • 1.2 Functions as Vectors
  • 1.3 The Dot, oops, INNER Product
  • 1.4 Operators
  • 1.5 Eigenvalue Problems
  • 1.6 Hermitian Operators
  • 1.7 Additional Points
    • 1.7.1 Dirac notation
    • 1.7.2 Additional independent variables
  
  
• 2. Basic Ideas of Quantum Mechanics
  • 2.1 The Revised Picture of Nature
  • 2.2 The Heisenberg Uncertainty Principle
  • 2.3 The Operators of Quantum Mechanics
  • 2.4 The Orthodox Statistical Interpretation
    • 2.4.1 Only eigenvalues
    • 2.4.2 Statistical selection
  • 2.5 A Particle Confined Inside a Pipe
    • 2.5.1 The physical system
    • 2.5.2 Mathematical notations
    • 2.5.3 The Hamiltonian
    • 2.5.4 The Hamiltonian eigenvalue problem
    • 2.5.5 All solutions of the eigenvalue problem
    • 2.5.6 Discussion of the energy values
    • 2.5.7 Discussion of the eigenfunctions
    • 2.5.8 Three-dimensional solution
    • 2.5.9 Quantum confinement
  • 2.6 The Harmonic Oscillator
    • 2.6.1 The Hamiltonian
    • 2.6.2 Solution using separation of variables
    • 2.6.3 Discussion of the eigenvalues
    • 2.6.4 Discussion of the eigenfunctions
    • 2.6.5 Degeneracy
    • 2.6.6 Non-eigenstates
  
  
• 3. Single-Particle Systems
  • 3.1 Angular Momentum
    • 3.1.1 Definition of angular momentum
    • 3.1.2 Angular momentum in an arbitrary direction
    • 3.1.3 Square angular momentum
    • 3.1.4 Angular momentum uncertainty
  • 3.2 The Hydrogen Atom
    • 3.2.1 The Hamiltonian
    • 3.2.2 Solution using separation of variables
    • 3.2.3 Discussion of the eigenvalues
    • 3.2.4 Discussion of the eigenfunctions
  • 3.3 Expectation Value and Standard Deviation
    • 3.3.1 Statistics of a die
    • 3.3.2 Statistics of quantum operators
    • 3.3.3 Simplified expressions
    • 3.3.4 Some examples
  • 3.4 The Commutator
    • 3.4.1 Commuting operators
    • 3.4.2 Noncommuting operators and their commutator
    • 3.4.3 The Heisenberg uncertainty relationship
    • 3.4.4 Commutator reference [Reference]
  • 3.5 The Hydrogen Molecular Ion
    • 3.5.1 The Hamiltonian
    • 3.5.2 Energy when fully dissociated
    • 3.5.3 Energy when closer together
    • 3.5.4 States that share the electron
    • 3.5.5 Comparative energies of the states
    • 3.5.6 Variational approximation of the ground state
    • 3.5.7 Comparison with the exact ground state
  
  
• 4. Multiple-Particle Systems
  • 4.1 Wave Function for Multiple Particles
  • 4.2 The Hydrogen Molecule
    • 4.2.1 The Hamiltonian
    • 4.2.2 Initial approximation to the lowest energy state
    • 4.2.3 The probability density
    • 4.2.4 States that share the electrons
    • 4.2.5 Variational approximation of the ground state
    • 4.2.6 Comparison with the exact ground state
  • 4.3 Two-State Systems
  • 4.4 Spin
  • 4.5 Multiple-Particle Systems Including Spin
    • 4.5.1 Wave function for a single particle with spin
    • 4.5.2 Inner products including spin
    • 4.5.3 Commutators including spin
    • 4.5.4 Wave function for multiple particles with spin
    • 4.5.5 Example: the hydrogen molecule
    • 4.5.6 Triplet and singlet states
  • 4.6 Identical Particles
  • 4.7 Ways to Symmetrize the Wave Function
  • 4.8 Matrix Formulation
  • 4.9 Heavier Atoms [Descriptive]
    • 4.9.1 The Hamiltonian eigenvalue problem
    • 4.9.2 Approximate solution using separation of variables
    • 4.9.3 Hydrogen and helium
    • 4.9.4 Lithium to neon
    • 4.9.5 Sodium to argon
    • 4.9.6 Potassium to krypton
  • 4.10 Pauli Repulsion [Descriptive]
  • 4.11 Chemical Bonds [Descriptive]
    • 4.11.1 Covalent sigma bonds
    • 4.11.2 Covalent pi bonds
    • 4.11.3 Polar covalent bonds and hydrogen bonds
    • 4.11.4 Promotion and hybridization
    • 4.11.5 Ionic bonds
    • 4.11.6 Limitations of valence bond theory
  
  
• 5. Time Evolution
  • 5.1 The Schrödinger Equation
    • 5.1.1 Introduction to the equation
    • 5.1.2 Some examples
    • 5.1.3 Energy conservation [Descriptive]
    • 5.1.4 Stationary states [Descriptive]
    • 5.1.5 Particle exchange [Descriptive]
    • 5.1.6 Energy-time uncertainty relation [Descriptive]
    • 5.1.7 Time variation of expectation values [Descriptive]
    • 5.1.8 Newtonian motion [Descriptive]
    • 5.1.9 The adiabatic approximation [Descriptive]
    • 5.1.10 Heisenberg picture [Descriptive]
  • 5.2 Conservation Laws and Symmetries
  • 5.3 Unsteady Perturbations of Systems
    • 5.3.1 Schrödinger equation for a two-state system
    • 5.3.2 Spontaneous and stimulated emission
    • 5.3.3 Effect of a single wave
      • 5.3.3.1 The wave
      • 5.3.3.2 The Hamiltonian coefficients
    • 5.3.4 Forbidden transitions
    • 5.3.5 Selection rules
    • 5.3.6 Angular momentum conservation
    • 5.3.7 Parity
    • 5.3.8 Absorption of a single weak wave
    • 5.3.9 Absorption of incoherent radiation
    • 5.3.10 Spontaneous emission of radiation
  • 5.4 Position and Linear Momentum
    • 5.4.1 The position eigenfunction
    • 5.4.2 The linear momentum eigenfunction
  • 5.5 Wave Packets in Free Space
    • 5.5.1 Solution of the Schrödinger equation.
    • 5.5.2 Component wave solutions
    • 5.5.3 Wave packets
    • 5.5.4 Group velocity
  • 5.6 Almost Classical Motion [Descriptive]
    • 5.6.1 Motion through free space
    • 5.6.2 Accelerated motion
    • 5.6.3 Decelerated motion
    • 5.6.4 The harmonic oscillator
  • 5.7 WKB Theory of Nearly Classical Motion
  • 5.8 Scattering
    • 5.8.1 Partial reflection
    • 5.8.2 Tunneling
  • 5.9 Reflection and Transmission Coefficients
  
  
• 6. Numerical Procedures
  • 6.1 The Variational Method
    • 6.1.1 Basic variational statement
    • 6.1.2 Differential form of the statement
    • 6.1.3 Example application using Lagrangian multipliers
  • 6.2 The Born-Oppenheimer Approximation
    • 6.2.1 The Hamiltonian
    • 6.2.2 The basic Born-Oppenheimer approximation
    • 6.2.3 Going one better
  • 6.3 The Hartree-Fock Approximation
    • 6.3.1 Wave function approximation
    • 6.3.2 The Hamiltonian
    • 6.3.3 The expectation value of energy
    • 6.3.4 The canonical Hartree-Fock equations
    • 6.3.5 Additional points
      • 6.3.5.1 Meaning of the orbital energies
      • 6.3.5.2 Asymptotic behavior
      • 6.3.5.3 Hartree-Fock limit
      • 6.3.5.4 Configuration interaction
  
  
• 7. Solids
  • 7.1 Molecular Solids [Descriptive]
  • 7.2 Ionic Solids [Descriptive]
  • 7.3 Introduction to Band Structure [Descriptive]
  • 7.4 Metals [Descriptive]
    • 7.4.1 Lithium
    • 7.4.2 One-dimensional crystals
    • 7.4.3 Wave functions of one-dimensional crystals
    • 7.4.4 Analysis of the wave functions
    • 7.4.5 Floquet (Bloch) theory
    • 7.4.6 Fourier analysis
    • 7.4.7 The reciprocal lattice
    • 7.4.8 The energy levels
    • 7.4.9 Electrical conduction
    • 7.4.10 Merging and splitting bands
    • 7.4.11 Three-dimensional metals
  • 7.5 Covalent Materials [Descriptive]
  • 7.6 Confined Free Electrons
    • 7.6.1 The Hamiltonian eigenvalue problem
    • 7.6.2 Solution by separation of variables
    • 7.6.3 Discussion of the solution
    • 7.6.4 A numerical example
    • 7.6.5 The density of states and confinement
    • 7.6.6 Relation to Bloch functions
  • 7.7 Free Electrons in a Lattice
    • 7.7.1 The lattice structure
    • 7.7.2 Occupied states and Brillouin zones
  • 7.8 Nearly-Free Electrons
    • 7.8.1 Energy changes due to a weak lattice potential
    • 7.8.2 Discussion of the energy changes
  • 7.9 Quantum Statistical Mechanics
  • 7.10 Additional Points [Descriptive]
    • 7.10.1 Thermal properties
    • 7.10.2 Ferromagnetism
    • 7.10.3 X-ray diffraction
  
  
• 8. Basic and Quantum Thermodynamics
  • 8.1 Temperature
  • 8.2 Single-Particle and System Eigenfunctions
  • 8.3 How Many System Eigenfunctions?
  • 8.4 Particle-Energy Distribution Functions
  • 8.5 The Canonical Probability Distribution
  • 8.6 Low Temperature Behavior
  • 8.7 The Basic Thermodynamic Variables
  • 8.8 Introduction to the Second Law
  • 8.9 The Reversible Ideal
  • 8.10 Entropy
  • 8.11 The Big Lie of Distinguishable Particles
  • 8.12 The New Variables
  • 8.13 Microscopic Meaning of the Variables
  • 8.14 Application to Particles in a Box
    • 8.14.1 Bose-Einstein condensation
    • 8.14.2 Fermions at low temperatures
    • 8.14.3 A generalized ideal gas law
    • 8.14.4 The ideal gas
    • 8.14.5 Blackbody radiation
    • 8.14.6 The Debye model
  
  
• 9. Electromagnetism
  • 9.1 All About Angular Momentum
    • 9.1.1 The fundamental commutation relations
    • 9.1.2 Ladders
    • 9.1.3 Possible values of angular momentum
    • 9.1.4 A warning about angular momentum
    • 9.1.5 Triplet and singlet states
    • 9.1.6 Clebsch-Gordan coefficients
    • 9.1.7 Some important results
    • 9.1.8 Momentum of partially filled shells
    • 9.1.9 Pauli spin matrices
    • 9.1.10 General spin matrices
  • 9.2 The Relativistic Dirac Equation
  • 9.3 The Electromagnetic Hamiltonian
  • 9.4 Maxwell’s Equations [Descriptive]
  • 9.5 Example Static Electromagnetic Fields
    • 9.5.1 Point charge at the origin
    • 9.5.2 Dipoles
    • 9.5.3 Arbitrary charge distributions
    • 9.5.4 Solution of the Poisson equation
    • 9.5.5 Currents
    • 9.5.6 Principle of the electric motor
  • 9.6 Particles in Magnetic Fields
  • 9.7 Stern-Gerlach Apparatus [Descriptive]
  • 9.8 Nuclear Magnetic Resonance
    • 9.8.1 Description of the method
    • 9.8.2 The Hamiltonian
    • 9.8.3 The unperturbed system
    • 9.8.4 Effect of the perturbation
  
  
• 10. Some Additional Topics
  • 10.1 Perturbation Theory
    • 10.1.1 Basic perturbation theory
    • 10.1.2 Ionization energy of helium
    • 10.1.3 Degenerate perturbation theory
    • 10.1.4 The Zeeman effect
    • 10.1.5 The Stark effect
    • 10.1.6 The hydrogen atom fine structure
      • 10.1.6.1 Fine structure
      • 10.1.6.2 Weak and intermediate Zeeman effect
      • 10.1.6.3 Lamb shift
      • 10.1.6.4 Hyperfine splitting
  • 10.2 Quantum Field Theory in a Nanoshell
    • 10.2.1 Occupation numbers
    • 10.2.2 Annihilation and creation operators
      • 10.2.2.1 Definition
      • 10.2.2.2 The caHermitians
      • 10.2.2.3 Examples
      • 10.2.2.4 More single particle states
    • 10.2.3 Quantization of radiation
      • 10.2.3.1 Classical energy
      • 10.2.3.2 Quantization
      • 10.2.3.3 Photon spin
      • 10.2.3.4 Traveling waves
    • 10.2.4 Spontaneous emission
    • 10.2.5 Field operators
    • 10.2.6 An example using field operators
  
  
• 11. The Interpretation of Quantum Mechanics
  • 11.1 Schrödinger’s Cat
  • 11.2 Instantaneous Interactions
  • 11.3 Global Symmetrization
  • 11.4 Failure of the Schrödinger Equation?
  • 11.5 The Many-Worlds Interpretation
  • 11.6 The Arrow of Time
  
  
• A. Notes
  • A.1 Why another book on quantum mechanics?
  • A.2 History and wishlist
  • A.3 Lagrangian mechanics
    • A.3.1 Introduction
    • A.3.2 Generalized coordinates
    • A.3.3 Lagrangian equations of motion
      • A.3.3.1 Derivation
    • A.3.4 Hamiltonian dynamics
      • A.3.4.1 Derivation
  • A.4 Special relativity
    • A.4.1 History
    • A.4.2 Overview of relativity
    • A.4.3 Lorentz transformation
      • A.4.3.1 Derivation
    • A.4.4 Proper time and distance
    • A.4.5 Subluminal and superluminal effects
    • A.4.6 Four-vectors
    • A.4.7 Index notation
    • A.4.8 Group property
      • A.4.8.1 Derivation
    • A.4.9 Intro to relativistic mechanics
    • A.4.10 Lagrangian mechanics
      • A.4.10.1 Derivation
  • A.5 Completeness of Fourier modes
  • A.6 Derivation of the Euler formula
  • A.7 Nature and real eigenvalues
  • A.8 Are Hermitian operators really like that?
  • A.9 Are linear momentum operators Hermitian?
  • A.10 Why boundary conditions are tricky
  • A.11 Extension to three-dimensional solutions
  • A.12 Derivation of the harmonic oscillator solution
  • A.13 More on the harmonic oscillator and uncertainty
  • A.14 Derivation of a vector identity
  • A.15 Derivation of the spherical harmonics
  • A.16 The effective mass
  • A.17 The hydrogen radial wave functions
  • A.18 Inner product for the expectation value
  • A.19 Why commuting operators have a common set of eigenvectors
  • A.20 The generalized uncertainty relationship
  • A.21 Derivation of the commutator rules
  • A.22 Is the variational approximation best?
  • A.23 Solution of the hydrogen molecular ion
  • A.24 Accuracy of the variational method
  • A.25 Positive molecular ion wave function
  • A.26 Molecular ion wave function symmetries
  • A.27 Solution of the hydrogen molecule
  • A.28 Hydrogen molecule ground state and spin
  • A.29 Number of boson states
  • A.30 Shielding approximation limitations
  • A.31 Why the s states have the least energy
  • A.32 Why energy eigenstates are stationary
  • A.33 Better description of two-state systems
  • A.34 The evolution of expectation values
  • A.35 The virial theorem
  • A.36 The energy-time uncertainty relationship
  • A.37 The adiabatic theorem
    • A.37.1 Derivation of the theorem
    • A.37.2 Some implications
  • A.38 Symmetry eigenvalue conservation
  • A.39 The two-state approximation of radiation
  • A.40 Selection rules
  • A.41 About spectral broadening
  • A.42 Derivation of the Einstein B coefficients
  • A.43 Parseval and the Fourier inversion theorem
  • A.44 Derivation of group velocity
  • A.45 Details of the animations
  • A.46 Derivation of the WKB approximation
  • A.47 WKB solution near the turning points
  • A.48 Three-dimensional scattering
    • A.48.1 Partial wave analysis
    • A.48.2 The Born approximation
    • A.48.3 The Born series
  • A.49 The evolution of probability
  • A.50 A basic description of Lagrangian multipliers
  • A.51 The generalized variational principle
  • A.52 Spin degeneracy
  • A.53 Derivation of the approximation
  • A.54 Why a single Slater determinant is not exact
  • A.55 Simplification of the Hartree-Fock energy
  • A.56 Integral constraints
  • A.57 Generalized orbitals
  • A.58 Derivation of the Hartree-Fock equations
  • A.59 Why the Fock operator is Hermitian
  • A.60 “Correlation energy”
  • A.61 Explanation of the London forces
  • A.62 Ambiguities in the definition of electron affinity
  • A.63 Why Floquet theory should be called so
  • A.64 Superfluidity versus BEC
  • A.65 Explanation of Hund’s first rule
  • A.66 The mechanism of ferromagnetism
  • A.67 Number of system eigenfunctions
  • A.68 The fundamental assumption of quantum statistics
  • A.69 A problem if the energy is given
  • A.70 Derivation of the particle energy distributions
  • A.71 The canonical probability distribution
  • A.72 Analysis of the ideal gas Carnot cycle
  • A.73 The recipe of life
  • A.74 Checks on the expression for entropy
  • A.75 Chemical potential and distribution functions
  • A.76 The Fermi-Dirac integral for small temperature
  • A.77 Physics of the fundamental commutation relations
  • A.78 Multiple angular momentum components
  • A.79 Components of vectors are less than the total vector
  • A.80 The spherical harmonics with ladder operators
  • A.81 Why angular momenta components can be added
  • A.82 Why the Clebsch-Gordan tables are bidirectional
  • A.83 How to make Clebsch-Gordan tables
  • A.84 Machine language version of the Clebsch-Gordan tables
  • A.85 The triangle inequality
  • A.86 Momentum of shells
  • A.87 Awkward questions about spin
  • A.88 More awkwardness about spin
  • A.89 Emergence of spin from relativity
  • A.90 Electromagnetic evolution of expectation values
  • A.91 Existence of magnetic monopoles
  • A.92 More on Maxwell’s third law
  • A.93 Various electrostatic derivations.
    • A.93.1 Existence of a potential
    • A.93.2 The Laplace equation
    • A.93.3 Egg-shaped dipole field lines
    • A.93.4 Ideal charge dipole delta function
    • A.93.5 Integrals of the current density
    • A.93.6 Lorentz forces on a current distribution
    • A.93.7 Field of a current dipole
    • A.93.8 Biot-Savart law
  • A.94 Energy due to orbital motion in a magnetic field
  • A.95 Energy due to electron spin in a magnetic field
  • A.96 Setting the record straight on alignment
  • A.97 Solving the NMR equations
  • A.98 Derivation of perturbation theory
  • A.99 Stark effect on the hydrogen ground state
  • A.100 Dirac fine structure Hamiltonian
  • A.101 Classical spin-orbit derivation
  • A.102 Expectation powers of r for hydrogen
  • A.103 A tenth of a googol in universes
  
  
• Bibliography
• B. Web Pages
• C. Notations
• Index