Quantum Mechanic z

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Contents

Dedication List of Figures List of Tables Preface

o To the Studento Acknowledgmentso Comments and Feedback

I. Special Relativityo 1. Special Relativity [Draft]

1.1 Overview of Relativity 1.1.1 A note on the history of the theory 1.1.2 The mass-energy relation 1.1.3 The universal speed of light 1.1.4 Disagreements about space and time

1.2 The Lorentz Transformation 1.2.1 The transformation formulae 1.2.2 Proper time and distance 1.2.3 Subluminal and superluminal effects 1.2.4 Four-vectors 1.2.5 Index notation 1.2.6 Group property

1.3 Relativistic Mechanics 1.3.1 Intro to relativistic mechanics 1.3.2 Lagrangian mechanics

II. Basic Quantum Mechanicso 2. Mathematical Prerequisites

2.1 Complex Numbers 2.2 Functions as Vectors 2.3 The Dot, oops, INNER Product 2.4 Operators 2.5 Eigenvalue Problems 2.6 Hermitian Operators 2.7 Additional Points

2.7.1 Dirac notation 2.7.2 Additional independent variables

o 3. Basic Ideas of Quantum Mechanics 3.1 The Revised Picture of Nature 3.2 The Heisenberg Uncertainty Principle
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3.3 The Operators of Quantum Mechanics 3.4 The Orthodox Statistical Interpretation

3.4.1 Only eigenvalues 3.4.2 Statistical selection

3.5 A Particle Confined Inside a Pipe

3.5.1 The physical system 3.5.2 Mathematical notations 3.5.3 The Hamiltonian 3.5.4 The Hamiltonian eigenvalue problem 3.5.5 All solutions of the eigenvalue problem 3.5.6 Discussion of the energy values 3.5.7 Discussion of the eigenfunctions 3.5.8 Three-dimensional solution 3.5.9 Quantum confinement

o 4. Single-Particle Systems 4.1 The Harmonic Oscillator

4.1.1 The Hamiltonian

4.1.2 Solution using separation of variables 4.1.3 Discussion of the eigenvalues 4.1.4 Discussion of the eigenfunctions 4.1.5 Degeneracy 4.1.6 Noneigenstates

4.2 Angular Momentum 4.2.1 Definition of angular momentum 4.2.2 Angular momentum in an arbitrary direction 4.2.3 Square angular momentum 4.2.4 Angular momentum uncertainty

4.3 The Hydrogen Atom 4.3.1 The Hamiltonian 4.3.2 Solution using separation of variables 4.3.3 Discussion of the eigenvalues 4.3.4 Discussion of the eigenfunctions

4.4 Expectation Value and Standard Deviation 4.4.1 Statistics of a die 4.4.2 Statistics of quantum operators 4.4.3 Simplified expressions 4.4.4 Some examples

4.5 The Commutator 4.5.1 Commuting operators 4.5.2 Noncommuting operators and their commutator 4.5.3 The Heisenberg uncertainty relationship 4.5.4 Commutator reference

4.6 The Hydrogen Molecular Ion 4.6.1 The Hamiltonian 4.6.2 Energy when fully dissociated 4.6.3 Energy when closer together
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4.6.4 States that share the electron 4.6.5 Comparative energies of the states 4.6.6 Variational approximation of the ground state 4.6.7 Comparison with the exact ground state

o 5. Multiple-Particle Systems

5.1 Wave Function for Multiple Particles 5.2 The Hydrogen Molecule

5.2.1 The Hamiltonian 5.2.2 Initial approximation to the lowest energy state 5.2.3 The probability density 5.2.4 States that share the electrons 5.2.5 Variational approximation of the ground state 5.2.6 Comparison with the exact ground state

5.3 Two-State Systems 5.4 Spin 5.5 Multiple-Particle Systems Including Spin

5.5.1 Wave function for a single particle with spin

5.5.2 Inner products including spin 5.5.3 Commutators including spin 5.5.4 Wave function for multiple particles with spin 5.5.5 Example: the hydrogen molecule 5.5.6 Triplet and singlet states

5.6 Identical Particles 5.7 Ways to Symmetrize the Wave Function 5.8 Matrix Formulation 5.9 Heavier Atoms

5.9.1 The Hamiltonian eigenvalue problem 5.9.2 Approximate solution using separation of variables 5.9.3 Hydrogen and helium 5.9.4 Lithium to neon 5.9.5 Sodium to argon 5.9.6 Potassium to krypton 5.9.7 Full periodic table

5.10 Pauli Repulsion 5.11 Chemical Bonds

5.11.1 Covalent sigma bonds 5.11.2 Covalent pi bonds 5.11.3 Polar covalent bonds and hydrogen bonds 5.11.4 Promotion and hybridization 5.11.5 Ionic bonds 5.11.6 Limitations of valence bond theory

o 6. Macroscopic Systems 6.1 Intro to Particles in a Box 6.2 The Single-Particle States 6.3 Density of States 6.4 Ground State of a System of Bosons
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6.5 About Temperature 6.6 Bose-Einstein Condensation

6.6.1 Rough explanation of the condensation 6.7 Bose-Einstein Distribution 6.8 Blackbody Radiation

6.9 Ground State of a System of Electrons 6.10 Fermi Energy of the Free-Electron Gas 6.11 Degeneracy Pressure 6.12 Confinement and the DOS 6.13 Fermi-Dirac Distribution 6.14 Maxwell-Boltzmann Distribution 6.15 Thermionic Emission 6.16 Chemical Potential and Diffusion 6.17 Intro to the Periodic Box 6.18 Periodic Single-Particle States 6.19 DOS for a Periodic Box

6.20 Intro to Electrical Conduction 6.21 Intro to Band Structure

6.21.1 Metals and insulators 6.21.2 Typical metals and insulators 6.21.3 Semiconductors 6.21.4 Semimetals 6.21.5 Electronic heat conduction 6.21.6 Ionic conductivity

6.22 Electrons in Crystals 6.22.1 Bloch waves 6.22.2 Example spectra 6.22.3 Effective mass 6.22.4 Crystal momentum 6.22.5 Three-dimensional crystals

6.23 Semiconductors 6.24 The P-N Junction 6.25 The Transistor 6.26 Zener and Avalanche Diodes 6.27 Optical Applications

6.27.1 Atomic spectra 6.27.2 Spectra of solids 6.27.3 Band gap effects 6.27.4 Effects of crystal imperfections 6.27.5 Photoconductivity 6.27.6 Photovoltaic cells 6.27.7 Light-emitting diodes

6.28 Thermoelectric Applications 6.28.1 Peltier effect 6.28.2 Seebeck effect 6.28.3 Thomson effect
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o 7. Time Evolution 7.1 The Schrdinger Equation

7.1.1 The equation 7.1.2 Solution of the equation 7.1.3 Energy conservation

7.1.4 Stationary states 7.1.5 The adiabatic approximation

7.2 Time Variation of Expectation Values 7.2.1 Newtonian motion 7.2.2 Energy-time uncertainty relation

7.3 Conservation Laws and Symmetries 7.4 Conservation Laws in Emission

7.4.1 Conservation of energy 7.4.2 Combining angular momenta and parities 7.4.3 Transition types and their photons 7.4.4 Selection rules

7.5 Symmetric Two-State Systems

7.5.1 A graphical example 7.5.2 Particle exchange and forces 7.5.3 Spontaneous emission

7.6 Asymmetric Two-State Systems 7.6.1 Spontaneous emission revisited

7.7 Absorption and Stimulated Emission 7.7.1 The Hamiltonian 7.7.2 The two-state model

7.8 General Interaction with Radiation 7.9 Position and Linear Momentum

7.9.1 The position eigenfunction 7.9.2 The linear momentum eigenfunction

7.10 Wave Packets 7.10.1 Solution of the Schrdinger equation. 7.10.2 Component wave solutions 7.10.3 Wave packets 7.10.4 Group velocity 7.10.5 Electron motion through crystals

7.11 Almost Classical Motion 7.11.1 Motion through free space 7.11.2 Accelerated motion 7.11.3 Decelerated motion 7.11.4 The harmonic oscillator

7.12 Scattering 7.12.1 Partial reflection 7.12.2 Tunneling

7.13 Reflection and Transmission Coefficientso 8. The Meaning of Quantum Mechanics

8.1 Schrdingers Cat
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8.2 Instantaneous Interactions 8.3 Global Symmetrization 8.4 A story by Wheeler 8.5 Failure of the Schrdinger Equation? 8.6 The Many-Worlds Interpretation

8.7 The Arrow of Time

III. Gateway Topicso 9. Numerical Procedures

9.1 The Variational Method 9.1.1 Basic variational statement 9.1.2 Differential form of the statement 9.1.3 Example application using Lagrangian multipliers

9.2 The Born-Oppenheimer Approximation

9.2.1 The Hamiltonian 9.2.2 The basic Born-Oppenheimer approximation 9.2.3 Going one better

9.3 The Hartree-Fock Approximation 9.3.1 Wave function approximation 9.3.2 The Hamiltonian 9.3.3 The expectation value of energy 9.3.4 The canonical Hartree-Fock equations 9.3.5 Additional points

9.3.5.1 Meaning of the orbital energies 9.3.5.2 Asymptotic behavior

9.3.5.3 Hartree-Fock limit 9.3.5.4 Configuration interaction

o 10. Solids 10.1 Molecular Solids 10.2 Ionic Solids 10.3 Metals

10.3.1 Lithium 10.3.2 One-dimensional crystals 10.3.3 Wave functions of one-dimensional crystals 10.3.4 Analysis of the wave functions 10.3.5 Floquet (Bloch) theory 10.3.6 Fourier analysis 10.3.7 The reciprocal lattice 10.3.8 The energy levels 10.3.9 Merging and splitting bands 10.3.10 Three-dimensional metals

10.4 Covalent Materials 10.5 Free-Electron Gas

10.5.1 Lattice for the free electrons
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10.5.2 Occupied states and Brillouin zones 10.6 Nearly-Free Electrons

10.6.1 Energy changes due to a weak lattice potential 10.6.2 Discussion of the energy changes

10.7 Additional Points

10.7.1 About ferromagnetism 10.7.2 X-ray diffraction

o 11. Basic and Quantum Thermodynamics 11.1 Temperature 11.2 Single-Particle versus System States 11.3 How Many System Eigenfunctions? 11.4 Particle-Energy Distribution Functions 11.5 The Canonical Probability Distribution 11.6 Low Temperature Behavior 11.7 The Basic Thermodynamic Variables 11.8 Intro to the Second Law

11.9 The Reversible Ideal 11.10 Entropy 11.11 The Big Lie of Distinguishable Particles 11.12 The New Variables 11.13 Microscopic Meaning of the Variables 11.14 Application to Particles in a Box

11.14.1 Bose-Einstein condensation 11.14.2 Fermions at low temperatures 11.14.3 A generalized ideal gas law 11.14.4 The ideal gas 11.14.5 Blackbody radiation 11.14.6 The Debye model

11.15 Specific Heatso 12. Angular momentum

12.1 Introduction 12.2 The fundamental commutation relations 12.3 Ladders 12.4 Possible values of angular momentum 12.5 A warning about angular momentum 12.6 Triplet and singlet states 12.7 Clebsch-Gordan coefficients 12.8 Some important results 12.9 Momentum of partially filled shells 12.10 Pauli spin matrices 12.11 General spin matrices 12.12 The Relativistic Dirac Equation

o 13. Electromagnetism 13.1 The Electromagnetic Hamiltonian 13.2 Maxwells Equations 13.3 Example Static Electromagnetic Fields
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13.3.1 Point charge at the origin 13.3.2 Dipoles 13.3.3 Arbitrary charge distributions 13.3.4 Solution of the Poisson equation 13.3.5 Currents

13.3.6 Principle of the electric motor 13.4 Particles in Magnetic Fields 13.5 Stern-Gerlach Apparatus 13.6 Nuclear Magnetic Resonance

13.6.1 Description of the method 13.6.2 The Hamiltonian 13.6.3 The unperturbed system 13.6.4 Effect of the perturbation

o 14. Nuclei [Unfinished Draft] 14.1 Fundamental Concepts 14.2 The Simplest Nuclei

14.2.1 The proton

14.2.2 The neutron 14.2.3 The deuteron 14.2.4 Property summary

14.3 Modeling the Deuteron 14.4 Overview of Nuclei 14.5 Magic numbers 14.6 Radioactivity

14.6.1 Decay rate 14.6.2 Other definitions

14.7 Mass and energy 14.8 Binding energy 14.9 Nucleon separation energies 14.10 Liquid drop model

14.10.1 Nuclear radius 14.10.2 von Weizscker formula 14.10.3 Explanation of the formula 14.10.4 Accuracy of the formula

14.11 Alpha Decay 14.11.1 Decay mechanism 14.11.2 Comparison with data 14.11.3 Forbidden decays 14.11.4 Why alpha decay?

14.12 Shell model 14.12.1 Average potential 14.12.2 Spin-orbit interaction 14.12.3 Example occupation levels 14.12.4 Shell model with pairing 14.12.5 Configuration mixing 14.12.6 Shell model failures
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