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
7 - The Continuous Spectrum and Scattering States
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 Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.
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- Quantum Mechanics in Nanoscience and Engineering , pp. 61 - 73Publisher: Cambridge University PressPrint publication year: 2023