Book contents
- Quantum Mechanics in Nanoscience and Engineering
- Additional material
- Quantum Mechanics in Nanoscience and Engineering
- Copyright page
- Contents
- Preface: Who Can Benefit from Reading This Book?
- 1 Motivation
- 2 The State of a System
- 3 Observables and Operators
- 4 The Schrödinger Equation
- 5 Energy Quantization
- 6 Wave Function Penetration, Tunneling, and Quantum Wells
- 7 The Continuous Spectrum and Scattering States
- 8 Mechanical Vibrations and the Harmonic Oscillator Model
- 9 Two-Body Rotation and Angular Momentum
- 10 The Hydrogen-Like Atom
- 11 The Postulates of Quantum Mechanics
- 12 Approximation Methods
- 13 Many-Electron Systems
- 14 Many-Atom Systems
- 15 Quantum Dynamics
- 16 Incoherent States
- 17 Quantum Rate Processes
- 18 Thermal Rates in a Bosonic Environment
- 19 Open Quantum Systems
- 20 Open Many-Fermion Systems
- Index
- References
13 - Many-Electron Systems
Published online by Cambridge University Press: 11 May 2023
- Quantum Mechanics in Nanoscience and Engineering
- Additional material
- Quantum Mechanics in Nanoscience and Engineering
- Copyright page
- Contents
- Preface: Who Can Benefit from Reading This Book?
- 1 Motivation
- 2 The State of a System
- 3 Observables and Operators
- 4 The Schrödinger Equation
- 5 Energy Quantization
- 6 Wave Function Penetration, Tunneling, and Quantum Wells
- 7 The Continuous Spectrum and Scattering States
- 8 Mechanical Vibrations and the Harmonic Oscillator Model
- 9 Two-Body Rotation and Angular Momentum
- 10 The Hydrogen-Like Atom
- 11 The Postulates of Quantum Mechanics
- 12 Approximation Methods
- 13 Many-Electron Systems
- 14 Many-Atom Systems
- 15 Quantum Dynamics
- 16 Incoherent States
- 17 Quantum Rate Processes
- 18 Thermal Rates in a Bosonic Environment
- 19 Open Quantum Systems
- 20 Open Many-Fermion Systems
- Index
- References
Summary
The foundations for understanding the electronic structure of many-electron atoms are introduced. We start from the discovery of the spin and introduce spin operators. The spin existence is shown to “upgrade” the state of single particles into a product space with the spin subspace, and to impose constraints on states of identical particles, which must be symmetric (bosons) or antisymmetric (fermions) under particle transpositions. The many-electron state in the atom is therefore approximated as an antisymmetrized products (Slater determinant) of single-electron states (spin-orbitals). The variationally optimal orbitals are shown to be solutions to the Hartree–Fock equations, and the assignment of electrons to these orbitals in the atomic ground state reflects the Pauli exclusion and Aufbau principles, thus explaining the trends in the periodic table of the elements in terms of their electronic configurations. Special attention is given to two-electron systems, demonstrating the exchange stabilization of triplet versus singlet states (Hund’s rule).
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- Quantum Mechanics in Nanoscience and Engineering , pp. 200 - 236Publisher: Cambridge University PressPrint publication year: 2023