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WORLD SCIENTIFIC eBooks

Applied Quantum Mechanics

Walter A. Harrison

Published Jul 1, 2000• 61 citations
Publisher

Abstract

The basic approach: foundations simple cases Hamiltonian mechanics. Electronic structure: atoms molecules crystals. Time dependence: transitions tunnelling transition rates. Statistical physics: statistical mechanics transport theory noise. Electrons and phonons: energy bands electron dynamics lattice vibrations. Quantum optics: operators phonons photons coherent states. Many-body effects: coulomb effects co-operative phenomena magnetism shake-off excitations.

DOI
10.1142/4485

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Cambridge University Press eBooks•Nov 17, 2011•0 citations
The basic approach: foundations simple cases Hamiltonian mechanics. Electronic structure: atoms molecules crystals. Time dependence: transitions tunnelling transition rates. Statistical physics: statistical mechanics transport theory noise. Electrons and phonons: energy bands electron dynamics lattice vibrations. Quantum optics: operators phonons photons coherent states. Many-body effects: coulomb effects co-operative phenomena magnetism shake-off excitations.
BRIEF REVIEW OF CLASSICAL THEORIES Harmonic Oscillator Boltzmann Transport Equation Maxwell's Equations MILESTONE DISCOVERIES AND OLD QUANTUM THEORY Blackbody Radiation and Quantum of Energy Specific Heat of Solids Photoelectric Effects Compton Scattering Duality of Matter Bohr?s H-Atom Theory SCHRODINGER EQUATION AND OPERATOR ALGEBRA Schrodinger Equation Momentum Eigenfunction and Fourier Series Hermitian Operator and Bra Ket Notations Orthgonality and Completeness of Eigenfunctions Basic Postulates of Quantum Mechanics Commutation Relations Conjugate Variables and Uncertainty Relation Operator Equation of Motion and Ehrenfest Theorem PARTICLE IN POTENTIAL WELL Infinite Square Well Potential Particle in 3-D Box Density of States: 1-D, 2-D and 3-D Particle in Quantum Well Particle in Delta Function Potential Well Quantum Well and Wire SCATTERING OF A PARTICLE AT 1-D POTENTIALS Scattering at Step Potential Scattering at Quantum Well TUNNELING AND ITS APPLICATIONS Tunneling across Square Potential Barrier Fowler-Nordheim and Direct Tunneling Resonant Tunneling The Applications of Tunneling PERIODIC POTENTIALS AND ENERGY BANDS 1-D Crystal and Kronig-Penny Model E-k Dispersion and Energy Bands Energy Bands and Resonant Tunneling THE HARMONIC OSCILLATOR Energy Eigenequation The Properties of Energy Eigenfunction The Operator Treatment THE ANGULAR MOMENTUM Angualr Momentum Operators Eigenfunctions and Spherical Harmonics THE HYDROGEN ATOM: THE SCHRODINGER TREATMENT Two-Body Central Force Problem The Hydrogen Atom The Atomic Orbital Virial Theorm and Doppler Shift SYSTEM OF IDENTICAL PARTICLES AND MANY-ELECTRON ATOMS Two Electron System Two Spin 1/2 System The Helium Atom The Periodic Table and the Structure of Atoms MOLECULES AND CHEMICAL BONDS The Ionized Hydrogen Molecule The Hydrogen Molecule Ionic Bond and Van der Waals Attraction THE PERTURBATION THEORY Time Independent Perturbation in Non-Degenerate Systems Time Dependent Perturbation Theory ATOM - FIELD INTERACTION Field Quantization Atom - Field Interaction Driven, Damped Two Level Atom INTERACTION BETWEEN EM WAVES AND OPTICAL MEDIA Attenuation and Dispersion of Waves Density Matrix and Ensemble Averaging Laser Devices QUANTUM STATISTICS General Background and Three Kinds of Particles Statistics for Distinguishable particles Statistics for Fermions and Fermi Distribution Function Statistics for Boson and Bose Einstein Distribution SEMICONDUCTOR STATISTICS Carrier Densities in Intrinsic Semiconductors Carrier Densities in Extrinsic Semiconductors Fermi Level in Extrinsic Semiconductors CHARGE TRANSPORT IN SEMICONDUCTORS Drift and Diffusion Currents Transport Coefficients Equilibrium and Non-Equilibrium Recombination and Generation Currents P-N JUNCTION DIODE The Junction Interface in Equilibrium The Junction Interface Under Bias The Diode I-V Applications of P-N Junction Diodes THE BIPOLAR JUNCTION TRANSISTOR: DEVICE PHYSICS AND TECHNOLOGY Bipolar Junction Transistor: Overview The Physics of Transistor Action Ebers Moll Equations Base Transit Time and Charge Control Model METAL OXIDE SILICON FIELD EFFECT TRANSISTORS (MOSFET) I: OVERVIEW OF DEVICE BEHAVIOR AND APPLICATIONS MOSFET: Overview Charge Control and Metal-Oxide-Silicon System NMOS I-V Applications of MOSFET METAL OXIDE SILICON FIELD EFFECT TRANSISTORS (MOSFET) II: DEVICE SCALING AND SCHOTTKY CONTACT Device Scaling: Physical issues and Limitations Metal - Semiconductor Contacts The I-V Behavior in Metal - Semiconductor Contacts
PREFACE PART I: THE STRUCTURE OF MATTER 1. THE MICROSCOPIC WORLD: ATOMS AND MOLECULES 1.1 Development of the Atomic Theory: Relative Atomic Weights 1.2 Atomic Magnitudes 1.3 The Charge-to-Mass Ratio of the Electron: Thomson's Method 1.4 The Charge of the Electron: Millikan's Method 1.5 Mass Spectrometry 1.6 The Atomic Mass Scale and the Mole 1.7 The Periodic Table 2. ORIGINS OF THE QUANTUM THEORY OF MATTER 2.1 The Franck-Hertz Experiment 2.2 The Photoelectric Effect 2.3 X-rays and Matter 2.4 The Emission Spectra of Atoms 2.5 The Nuclear Atom 2.6 The Problem of Black-Body Radiation 2.7 The Concept of Action 2.8 The Harmonic Oscillator 2.9 Action Quantized: The Heat Capacity of Solids 2.10 Some Orders of Magnitude 2.11 Bohr's Model of the Atom Appendix 2A: Rutherford Scattering 3. MATTER WAVES IN SIMPLE SYSTEMS 3.1 The de Broglie Hypothesis 3.2 The Nature of Waves 3.3 Dispersion Relations and Wave Equations: The Free Particle 3.4 Operators 3.5 Eigenfunctions and Eigenvalues 3.6 The Particle in a One-Dimensional Box 3.7 The Indeterminacy or Uncertainty Principle 3.8 Expectation Values Summary of Postulates 3.9 Particles in Two- and Three-Dimensional Boxes 3.10 Particles in Circular Boxes 3.11 Particles in Spherical Boxes 3.12 The Rigid Rotor Appendix 3A: More on Circular Coordinates and the Circular Box 4. PARTICLES IN VARYING POTENTIAL FIELDS TRANSITIONS 4.1 Finite Potential Barriers 4.2 The Quantum Mechanical Harmonic Oscillator 4.3 The Hydrogen Atom 4.4 The Shapes of Orbitals 4.5 Transitions between Energy Levels 5. THE STRUCTURE OF ATOMS 5.1 Electron Spin Magnetic Phenomena 5.2 The Pauli Exclusion Principle the Aufbau Principle 5.3 Electronic Configurations of Atoms 5.4 Calculation of Atomic Structures 5.5 Atomic Structure and Periodic Behavior 5.6 Term Splitting and the Vector Model 5.7 Fine Structure and Spin-Orbit Interactions Appendix 5A: The Stern-Gerlach Experiment 6. THE CHEMICAL BOND IN THE SIMPLEST MOLECULES: H[2+ AND H[2 6.1 Bonding Forces between Atoms 6.2 The Simplest Molecule: The Hydrogen Molecule-Ion, H[2+ 6.3 H[2+: Molecular Orbitals and the LCAO Approximation 6.4 H[2+: Obtaining the Energy Curve 6.5 H[2+: Correlation of Orbitals Excited States 6.6 The H[2 Molecule: Simple MO Description 6.7 Symmetry Properties of Identical Particles 6.8 H[2: The Valence Bond Representation 6.9 H[2: Beyond the Simple MO and VB Approximations 6.10 H[2: Excited Electronic States Appendix 6A: Orthogonality Appendix 6B: Hermitian Operators 7. MORE ABOUT DIATOMIC MOLECULES 7.1 Vibrations of Diatomic Molecules 7.2 Rotations of Diatomic Molecules 7.3 Spectra of Diatomic Molecules 7.4 The Ionic Bond 7.5 Homonuclear Diatomic Molecules: Molecular Orbitals and Orbital Correlation 7.6 Homonuclear Diatomic Molecules: Aufbau Principle and Structure of First-Row Molecules 7.7 Introduction to Heteronuclear Diatomic Molecules: Electronegativity 7.8 Bonding in LiH: Crossing and Noncrossing Potential Curves 7.9 Other First-Row Diatomic Hydrides 7.10 Isoelectronic and Other Series Appendix 7A: Perturbation Theory 8. TRIATOMIC MOLECULES 8.1 Electronic Structure and Geometry in the Simplest Cases: H[3 and H[3+ 8.2 Dihydrides: Introduction to the Water Molecule 8.3 Hybrid Orbitals 8.4 Delocalized Orbitals in H[2O: The General MO Method 8.5 Bonding in More Complex Triatomic Molecules 8.6 Normal Coordinates and Modes of Vibration 8.7 A Solvable Example: The Vibrational Modes of CO[2 8.8 Transitions and Spectra of Polyatomic Molecules: Rotations and Vibrations 8.9 Transitions and Spectra of Polyatomic Molecules: Magnetic Transitions 8.10 Transitions and Spectra of Polyatomic Molecules: Electronic Transitions 9. LARGER POLYATOMIC MOLECULES 9.1 Small Molecules 9.2 Catenated Carbon Compounds Transferability 9.3 Other Extended Structures 9.4 Some Steric Effects 9.5 Complex Ions and Other Coordination Compounds: Simple Polyhedra 9.6 Chirality and Optical Rotation 9.7 Chiral and Other Complex Ions 9.8 Magnetic Properties of Complexes 9.9 Electronic Structure of Complexes Appendix 9A: Schmidt Orthogonalization 10. INTERMOLECULAR FORCES 10.1 Long-Range Forces: Interactions between Charge Distributions 10.2 Empirical Intermolecular Potentials 10.3 Weakly Associated Molecules 11. THE STRUCTURE OF SOLIDS 11.1 Some General Properties of Solids 11.2 Space Lattices and Crystal Symmetry 11.3 X-ray Diffraction from Crystals: The Bragg Model 11.4 The Laue Model 11.5 Determination of Crystal Structures 11.6 Techniques of Diffraction 11.7 Molecular Crystals 11.8 Structures of Ionic Crystals 11.9 Binding Energy of Ionic Crystals 11.10 Covalent Solids 11.11 The Free-Electron Theory of Metals 11.12 The Band Theory of Solids 11.13 Conductors, Insulators, and Semiconductors 11.14 Other Forms of Condensed Matter APPENDICES I. Systems of Units II. Partial Derivatives III. Glossary of Symbols IV. Searching the Scientific Literature INDEX
In this thesis we study the role of nonradiative degrees of freedom on quantum optical properties of mesoscopic quantum dots placed in the structured electromagnetic reservoir of a photonic crystal. We derive a quantum theory of the role of acoustic and optical phonons in modifying the optical absorption lineshape, polarization dynamics, and population dynamics of a two-level atom (quantum dot) in the ``colored" electromagnetic vacuum of a photonic band gap (PBG) material. This is based on a microscopic Hamiltonian describing both radiative and vibrational processes quantum mechanically. Phonon sidebands in an ordinary electromagnetic reservoir are recaptured in a simple model of optical phonons using a mean-field factorization of the atomic and lattice displacement operators. Our formalism is then used to treat the non-Markovian dynamics of the same system within the structured electromagnetic density of states of a photonic crystal. We elucidate the extent to which phonon-assisted decay limits the lifetime of a single photon-atom bound state and derive the modified spontaneous emission dynamics due to coupling to various phonon baths. We demonstrate that coherent interaction with undamped phonons can lead to enhanced lifetime of a photon-atom bound state in a PBG by (i) dephasing and reducing the transition electric dipole moment of the atom and (ii) reducing the quantum mechanical overlap of the state vectors of the excited and ground state (polaronic shift). This results in reduction of the steady-state atomic polarization but an increase in the fractionalized upper state population in the photon-atom bound state. We demonstrate, on the other hand, that the lifetime of the photon-atom bound state in a PBG is limited by the lifetime of phonons due to lattice anharmonicities (break-up of phonons into lower energy phonons) and purely nonradiative decay. We demonstrate how these additional damping effects limit the extent of the polaronic (Franck-Condon) shift of the atomic excited state. We also derive the modified polarization decay and dephasing rates in the presence of such damping. This leads to a microscopic, quantum theory of the optical absorption lineshapes. Our model and formalism provide a starting point for describing dephasing and relaxation in the presence of external coherent fields and multiple quantum dot interactions in electromagnetic reservoirs with radiative memory effects.
The optically generated excitations in semiconductors constitute a genuine many-body system. To describe its quantum-optical features, we have to expand significantly the theoretical models used so far. However, the important insights of Chapters 16–23 are already presented in a form in which most of them can directly be used and generalized to analyze central properties of the optical excitations in solids. As for atoms, the optical transitions in semiconductors are induced via dipole interaction between photons and electrons. We can thus efficiently construct a systematic quantum-optical theory for semiconductors by following the cluster-expansion approach. One of the main differences from atoms is that the electronic excitations in semiconductors form a strongly interacting many-body system. Thus, we must systematically treat the arising Coulomb-induced hierarchy problem together with the quantum-optical one. Moreover, the coupling of electrons to lattice vibrations, i.e., the phonons, produces yet another hierarchy problem. In addition, in solid-state spectroscopy one often uses multimode light fields such that one cannot rely on the single-mode simplifications to study semiconductor quantum optics. As shown in Chapter 15, the Coulomb-, phonon-, and photon-induced hierarchy problems have formally an identical structure. Thus, we start the analysis by investigating how semiconductor quantum optics emerges from the dynamics of correlated clusters. We first focus on the basic properties of the optical transitions in the classical regime. This means investigating the fundamental optical phenomena resulting from the singlets. The full singlet–doublet approach is presented in Chapters 28–30.