# Quantum Mechanics For Applied Physics And Engineering

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Quantum Mechanics for Applied Physics and Engineering is devoted to the use of quantum mechanics in applied physics and engineering. Topics covered include elementary quantum theory, quantum statistics and many-particle systems, and energy bands in crystals. Approximation techniques for the Schrödinger equation are also described. Comprised of seven chapters, this book opens with an overview of basic quantum mechanics and includes a discussion on wave-particle duality, probability current density, and periodic boundary conditions. Quantum statistics is then considered as a prelude to the free-electron theory of metals, along with the use of perturbation theory to evaluate modifications in free-electron theory. The following chapters explore the use of WKB approximation to deduce the transmission coefficient for electron tunneling in solids; the theory of electronic energy bands; and the application of the Schrödinger equation to the problem of the periodic potential of a crystalline solid. Examples from solid-state physics are employed to illustrate specific applications and to demonstrate the principal results that can be deduced by means of quantum theory. This monograph is written primarily for engineers and applied physicists.

## Table of Contents

Preface

Acknowledgments

Part I Elementary Quantum Theory

Chapter 1 An Introduction to Quantum Mechanics

1 Wave-Particle Duality

2 Classical Wave Motion

3 Periodic Boundary Conditions and Complex Fourier Components

4 Fourier Series and Fourier Integrals

5 Wave Nature of Particles

6 Development of the Time-Dependent and Time-Independent Schrödinger Wave Equations

7 Wave-Packet Solutions and the Uncertainty Relation

8 Expectation Values for Quantum-Mechanical Operators

9 Probability Current Density

10 Energy Levels and Density of States

11 Reflection and Transmission Coefficients for a Particle Beam at a Potential-Energy Step Discontinuity and at a Rectangular Barrier

12 Bound-State Problems

Problems

Answers to Multiple Choice Problems

Part II Quantum Statistics of Many-Particle Systems; Formulation of the Free-Electron Model for Metals

Chapter 2 Many-Particle Systems and Quantum Statistics

1 Wave Functions for a Many-Particle System

2 Statistics for a Many-Particle System

Problems

Chapter 3 Free-Electron Model and the Boltzmann Equation

1 Free-Electron Gas in Three Dimensions

2 Electronic Specific Heat

3 Electrical Conductivity and the Derivation of Ohm's Law

4 Thermal Electron Emission from Metals

5 General Method for Evaluating Statistical Quantities Involving Fermi-Dirac Statistics

6 The Temperature Dependence of the Fermi Energy and Other Applications of the General Approximation Technique

7 The Boltzmann Equation

Problems

Part III Approximation Techniques for the Schrödinger Equation

Chapter 4 The WKB Approximation and Electron Tunneling

1 Development of the WKB Approximation

2 Application of the WKB Technique to Barrier Penetration

3 Tunneling in Metal-Insulator-Metal Structures

4 Tunnel Current at 0°K between Two Metals Separated by a Rectangular Barrier

5 Tunnel Current at 0°K for Barriers of Arbitrary Shape

6 Temperature Dependence of the Electron Tunnel Current

7 Applications of Electron Tunneling

Chapter 5 Perturbation Theory, Diffraction of Valence Electrons, and the Nearly-Free-Electron Model

1 Stationary-State Perturbation Theory

2 Elementary Treatment of Diagonalization

3 Higher-Order Perturbations and Applications

4 Degenerate Case for Second-Order Treatment

5 Removal of Degeneracy in Second Order

6 Time-Dependent Perturbation Theory

7 Example: Harmonic Perturbation

8 Example: Constant Perturbation in First Order

9 Example: Constant Perturbation in Second Order

10 Transition Probability and Fermi's Golden Rule

11 Differential Cross Section for Scattering

12 Diffraction of Electrons by the Periodic Potential of a Crystal

13 Diffraction of Conduction Electrons and the Nearly-Free-Electron Model

14 Differential Scattering Cross Section for Plane-Wave States and a Coulomb Potential

Problems

Part IV Energy Bands in Crystals

Chapter 6 The Periodicity of Crystalline Solids

1 Generalities

2 Unit Cells and Bravais Lattices

3 Miller Indices and Crystal Directions

4 Some Specific Crystal Structures

5 Crystal Bonding

6 The Reciprocal Lattice: Fourier Space for Arbitrary Functions That Have the Lattice Periodicity

7 Wigner-Seitz Cell

8 First Brillouin Zone

9 Higher Brillouin Zones

Problems

Chapter 7 Bloch's Theorem and Energy Bands for a Periodic Potential

1 Fourier Series Expansions for Arbitrary Functions of Position within the Crystal

2 The Periodic Potential Characteristic of the Perfect Monocrystal

3 The Hamiltonian for an Electron in a Periodic Potential

4 Fourier Series Derivation of Bloch's Theorem

5 Properties of Bloch Functions

6 Correspondence with the Free-Electron Model

7 Additional Properties of Bloch Functions

8 Energy Bands from the Viewpoint of the One-Electron Atomic Levels

9 Energy Gaps and Energy Bands: Insulators, Semiconductors, and Metals

Problems

Appendix Physical Constants: Symbols, Units, and Values

References

Index

## Product details

- No. of pages: 444
- Language: English
- Copyright: © Academic Press 1981
- Published: January 1, 1981
- Imprint: Academic Press
- eBook ISBN: 9780323150095

## About the Author

### Albert T. Jr. Fromhold

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