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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This topical and timely textbook is a collection of problems for students, researchers, and practitioners interested in state-of-the-art material and device applications in quantum mechanics. Most problem are relevant either to a new device or a device concept or to current research topics which could spawn new technology. It deals with the practical aspects of the field, presenting a broad range of essential topics currently at the leading edge of technological innovation.
Includes discussion on:
Properties of Schroedinger Equation
Operators
Bound States in Nanostructures
Current and Energy Flux Densities in Nanostructures
Density of States
Transfer and Scattering Matrix Formalisms for Modelling Diffusive Quantum Transport
Perturbation Theory, Variational Approach and their Applications to Device Problems
Electrons in a Magnetic or Electromagnetic Field and Associated Phenomena
Time-dependent Perturbation Theory and its Applications
Optical Properties of Nanostructures
Problems in Quantum Mechanics: For Material Scientists, Applied Physicists and Device Engineers is an ideal companion to engineering, condensed matter physics or materials science curricula. It appeals to future and present engineers, physicists, and materials scientists, as well as professionals in these fields needing more in-depth understanding of nanotechnology and nanoscience.
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Seitenzahl: 422
Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title Page
Copyright
Dedication
About the Authors
Preface
References
Suggested Reading
Chapter 1: General Properties of the Schrödinger Equation
References
Suggested Reading
Chapter 2: Operators
References
Suggested Reading
Chapter 3: Bound States
References
Suggested Reading
Chapter 4: Heisenberg Principle
References
Suggested Reading
Chapter 5: Current and Energy Flux Densities
References
Suggested Reading
Chapter 6: Density of States
References
Suggested Reading
Chapter 7: Transfer Matrix
References
Suggested Reading
Chapter 8: Scattering Matrix
References
Chapter 9: Perturbation Theory
References
Suggested Reading
Chapter 10: Variational Approach
References
Suggested Reading
Chapter 11: Electron in a Magnetic Field
Suggested Reading
Chapter 12: Electron in an Electromagnetic Field and Optical Properties of Nanostructures
Preliminary
Absorption in a quantum well
References
Chapter 13: Time-Dependent Schrödinger Equation
References
Suggested Reading
Appendix A: Postulates of Quantum Mechanics
References
Appendix B: Useful Relations for the One-Dimensional Harmonic Oscillator
References
Appendix C: Properties of Operators [1–5]
References
Appendix D: The Pauli Matrices and their Properties [1–5]
References
Appendix E: Threshold Voltage in a High Electron Mobility Transistor Device
References
Suggested Reading
Appendix F: Peierls's Transformation [1, 2]
References
Appendix G: Matlab Code
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: General Properties of the Schrödinger Equation
Figure 1.1 Illustration of electron impinging on the left on an arbitrary conduction band energy profile under bias. is the potential difference between the two contacts.
Chapter 3: Bound States
Figure 3.1 Illustration of the bound state energy of a one-dimensional delta scatterer located at a heterointerface. This energy depends on the magnitude of the potential jump .
Figure 3.2 The one-dimensional particle in a box problem. inside the box in the range and is otherwise.
Figure 3.3 Bound state problem for a one-dimensional attractive delta scatterer in the middle of a box of size with infinite barriers, i.e., outside the box.
Figure 3.4 A one-dimensional potential well with equal to zero in the interval and outside.
Figure 3.5 Erroneously calculated spatial variation of the wave function of an excited state in a one-dimensional heterostructure.
Figure 3.6 A one-dimensional potential well in a semi-infinite space.
Figure 3.7 Two coupled finite square wells, each of width .
Figure 3.8 A triangular quantum well formed by a constant electric field in the region . for negative . The quantity is the magnitude of the charge of the electron.
Figure 3.9 The two-dimensional particle in a box. in a box of size and equal to outside the box.
Figure 3.10 The bound state energy of a one-dimensional delta scatter located at a heterointerface depends on the magnitude of the potential jump .
Chapter 4: Heisenberg Principle
Figure 4.1 Diffraction pattern observed on a screen at a distance from a parallel screen with an opening due to a particle incident from the left with momentum perpendicular to the plane of both screens.
Chapter 5: Current and Energy Flux Densities
Figure 5.1 Illustration of electron impinging from the left on a potential step with height . The effective mass is assumed to be and on the left and right side of the step, respectively.
Figure 5.2 Plot of the transmission probability , reflection probability , and absorption probability as a function of electron incident kinetic energy for a repulsive delta scatterer with strength equal to 0.1 eV-Å and equal to 0 (no absorption). The effective mass is assumed to be .
Figure 5.3 Plot of the transmission probability , reflection probability , and absorption probability as a function of electron incident kinetic energy for a delta scatterer with strength equal to 0.1 eV-Å and equal to 0.2 eV-Å. The effective mass is assumed to be .
Figure 5.4 Illustration of an electron impinging from the left on an absorbing well. The effective mass is assumed to be the same throughout.
Figure 5.5 Plot of the transmission probability , reflection probability , and absorption probability as a function of the electron incident kinetic energy for a potential well of width of 50 Å, depth equal to 0.3 eV, and absorbing potential equal to 0.1 eV. The effective mass is assumed to be .
Figure 5.6 Illustration of an electron impinging from the left on a potential well. The effective mass is assumed to be the same throughout.
Chapter 6: Density of States
Figure 6.1 Illustration of the formation of a quantum dot (bottom right) through the gradual squeezing of a bulk piece of semiconductor (upper left). When the dimension of the bulk structure is reduced in one direction to a size comparable to the de Broglie wavelength, the resulting electron gas is referred to as a two-dimensional electron gas (2DEG) because the carriers are free to move in two directions only. If quantum confinement occurs in two directions, as illustrated in the bottom left figure, the resulting electron gas is referred to as a one-dimensional electron gas (1DEG) since an electron in this structure is free to move in one direction only. If confinement is imposed in all three directions (bottom right frame), we get a quantum dot (0DEG).
Figure 6.2 (Left) Parabolic energy dispersion relation close to the bottom of the conduction band () of a typical semiconductor. (Right) Corresponding energy dependence of the three-dimensional DOS in a bulk semiconductor.
Figure 6.3 Density of states of electrons and holes as a function of energy in a two-dimensional electron and hole gas. Here, is the first subband energy for electrons and is that for holes.
Figure 6.4 Spatial dependence of the total charge concentration, (in C/m), in the QW (see Equation (6.69)).
Figure 6.6 Spatial dependence of the electrostatic potential (in mV) across the QW (see Equation (6.73)). The dielectric constant of GaAs was set equal to 12.9.
Figure 6.7 Energy band profile under the gate of a HEMT device consisting of an AlGaAs/GaAs heterostructure. The substrate is held at ground and a voltage is applied to the gate.
Figure 6.8 Self-consistent scheme to calculate the current–voltage characteristics in a nanoscale device under the approximation of ballistic transport.
Figure 6.9 Reflection from a infinite potential wall. If for , the reflection coefficient for all values of , the wavevector of the electron incident from the left.
Figure 6.10 Plot of the normalized electron density for three different temperatures for an electron with an effective mass . From bottom to top, the temperature is set equal to 4.2 K, 77 K, and 300 K, respectively.
Figure 6.11 Richardson–Dushman thermionic current across a metal/vacuum interface modeled as a potential step of height . and are the Fermi energy and work function of the metal, respectively.
Figure 6.12 Only one-eighth of the spherical shell between the two spheres of radii and must be taken into account to calculate the density of distinct phonon modes present in the blackbody cavity within the corresponding energy range.
Figure 6.13 Plot of the normalized spectral energy distribution versus wavelength in Equation (6.161) for three different temperatures. From left to right, the curves correspond to , 1000, and 300 K.
Chapter 7: Transfer Matrix
Figure 7.1 Approximation of an arbitrary conduction band energy profile as a series of steps. The effective mass may be assumed to be different in each interval. There is no bias applied between the two contacts, i.e., .
Figure 7.2 Scattering problem for an electron incident from the left on a potential energy step of height . The electron effective mass is assumed to be different on both sides of the step.
Figure 7.3 Plot of the transmission () and reflection () probabilities given by Equations (7.70) and (7.71), respectively, as a function of the reduced wavevector . Notice that when .
Figure 7.4 Scattering problem for an electron incident from the left on a periodic potential energy profile composed of identical unit cells.
Figure 7.5 Basic unit cell used to calculate the energy dispersion relation of an infinite periodic lattice. The effective mass is assumed to be the same throughout.
Figure 7.6 Schematic of a quantum well (dashed line) of width with an arbitrary conduction band energy profile and maximum depth . The zero of energy is selected to coincide with the bottom of the well. Also shown are the locations of the two lowest bound states and in the well. The latter coincide with the energies for unit transmission probability for an electron incident from the left barrier region. The quantities and are the reflection and transmission amplitudes, respectively, of the incident electron [13].
Figure 7.7 Connections between the incoming and outgoing wave amplitudes across a region where the potential energy profile is approximated by a constant. The effective mass is assumed constant throughout.
Chapter 8: Scattering Matrix
Figure 8.1 The scattering matrix relates the current amplitudes of incoming [] to outgoing [] waves on both sides of a region of width containing an arbitrary potential energy profile. The scattering matrix is defined such that Equation (8.1) is satisfied.
Figure 8.2 The only non-zero elements of the scattering matrix describing a free propagating region are simple phase shifts of the waves incident from the left and right contacts. The effective mass is assumed to be constant throughout.
Figure 8.3 Summing the current transmission amplitudes of multiple Feynman paths to calculate the total current amplitude
transmitted
through two successive sections. The electron is incident from the left.
Figure 8.6 Summing the current reflection amplitudes of multiple Feynman paths to calculate the total current amplitude
reflected
by two successive sections. The electron is incident from the right.
Figure 8.4 Summing the current reflection amplitudes of multiple Feynman paths to calculate the total current amplitude
reflected
by two successive sections. The electron is incident from the left.
Figure 8.5 Summing the current transmission amplitudes of multiple Feynman paths to calculate the total current amplitude
transmitted
through two successive sections. The electron is incident from the right.
Figure 8.7 A quasi two-dimensional electron waveguide of finite width containing randomly placed scatterers that cause elastic scattering.
Chapter 9: Perturbation Theory
Figure 9.1 The quantum-confined DC Stark effect. Application of an electric field transverse to the heterointerfaces of a quantum well skews the electron and hole wave functions in opposite directions. That decreases the overlap between them and partially quenches photoluminescence intensity. The band bending within the quantum well due to the electric field also reduces the effective bandgap and causes a red-shift of the suppressed peak in the photoluminescence spectrum.
Figure 9.2 The Lennard-Jones potential plotted in one dimension.
Chapter 11: Electron in a Magnetic Field
Figure 11.1 Illustration of a 2DEG with a uniform magnetic field applied perpendicular to it (pointing in the -direction).
Chapter 12: Electron in an Electromagnetic Field and Optical Properties of Nanostructures
Figure 12.1 Quantum well in the – plane.
Chapter 13: Time-Dependent Schrödinger Equation
Figure 13.1 Linear approximation to the parabolic energy dispersion relation for a free particle over the energy range describing the spectral content of the initial wave packet.
Appendix E: Threshold Voltage in a High Electron Mobility Transistor Device
Figure E.1 A HEMT is implemented with a heterostructure comprising a narrow bandgap semiconductor and a wide bandgap semiconductor. The wide gap semiconductor is doped with donor atoms and the resulting free electrons transfer to the narrow bandgap semiconductor to minimize their potential energies. This results in spatial separation of the electrons from their parent donors, which causes the electron mobility to be high because of suppression of scattering due to ionized donor atoms. A quasi two-dimensional layer (2DEG) of electrons forms at the heterointerface. The energy band diagram along the direction perpendicular to the heterointerface is shown. There is a gate bias applied to the top metallic gate leading to a difference between the Fermi level in the bulk narrow gap semiconductor and the Fermi level in the gate metal . The quantity is the work function of the metal. The wide bandgap layer is doped uniformly with donors for and undoped in the region .
List of Tables
Chapter 12: Electron in an Electromagnetic Field and Optical Properties of Nanostructures
Table 12.1 The quantity in a quantum well. Note that the quantity is electron spin dependent
Chapter 13: Time-Dependent Schrödinger Equation
Table 13.1 First and last row values of and for the case of 100% reflecting boundaries at both ends of the solution domain
Problem Solving in Quantum Mechanics
From Basics to Real-World Applications for Materials Scientists, Applied Physicists, and Devices Engineers
Marc Cahay
Spintronics and Vacuum Nanoelectronics Laboratory, University of Cincinnati, USA
Supriyo Bandyopadhyay
School of Engineering, Virginia Commonwealth University, USA
This edition first published 2017
© 2017 John Wiley & Sons Ltd
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M. Cahay would like to thank the students who took his course on Quantum Systems taught at the University of Cincinnati over the last four years and who carefully checked many of the problems in this book, including (in alphabetic order): Joshua Alexander, Kelsey Baum, Graham Beach, Jordan Bishop, Ryan Blanford, Mike Bosken, Joseph Buschur, James Charles, Sumeet Chaudhary, Triet Dao, Aaron Diebold, Chelsey Duran, Adam Fornalczyk, Sthitodhi Ghosh, Adam Hauke, Erik Henderson, Phillip Horn, Alexander Jones, Henry Jentz, Jesse Kreuzmann, Alex Lambert, Craig Mackson, Ashley Mattson, James McNay, Thinh Nguyen, Francois Nyamsi, Jesse Owens, Nandhakishore Perulalan, Kaleb Posey, Logan Reid, Charles Skipper, Adam Steller, Matthew Strzok, William Teleger, Nicole Wallenhorst, Brian Waring, Aaron Welton, Samuel Wenke, and Robert Wolf.
M. Cahay would like to thank his wife, Janie, for her support and patience during this time-consuming project.
Supriyo Bandyopadhyay dedicates this book to his family and students.
About the Authors
Marc Cahayis a professor in the Department of Electrical Engineering and Computing Systems at the University of Cincinnati. His research interests include modeling of carrier transport in semiconductors, quantum mechanical effects in heterostructures, heterojunction bipolar transistors, spintronics, and cold cathodes. He has also been involved in experimental investigations of cold cathodes, and more recently organic light-emitting diodes based on rare-earth monosulfide materials. He has published over 200 papers in journals and conference proceedings in these areas. He also has organized many national and international symposia and conferences on his areas of expertise. Together with Supriyo Bandyopadhyay, he has authored a book titled Introduction to Spintronics. A second edition of this book was released by CRC Press in 2015.
Supriyo Bandyopadhyay is a professor in the Department of Electrical and Computer Engineering at Virginia Commonwealth University. His research interests include spintronics, nanomagnetism, straintronics, self-assembly of nanostructures, quantum dot devices, carrier transport in nanostructures, quantum networks, and quantum computing. He has published over 300 journal articles and conference papers in these fields, serves on the editorial boards of nine journals devoted to these fields, and has served on the organizing and program committees of many international conferences in these areas of research. He has authored a book titled Physics of Nanostructured Solid State Devices published by Springer in 2012.
The authors encourage comments about the book contents via email to [email protected] and [email protected].
Preface
Over the last two decades, there has been a dramatic increase in the study of physical and biological systems at the nanoscale. In fact, this millenium has been referred to as the “nanomillenium.” The fields of nanoscience and nanoengineering have been fuelled by recent spectacular discoveries in mesoscopic physics, a new understanding of DNA sequencing, the advent of the field of quantum computing, tremendous progress in molecular biology, and other related fields. A fundamental understanding of physical phenomena at the nanoscale level will require future generations of engineers and scientists to grasp the intricacies of the quantum world and master the fundamentals of quantum mechanics developed by many pioneers since the 1920s. For electrical engineers, condensed matter physicists, and materials scientists who are involved with electronic and optical device research, quantum mechanics will assume a special significance. For instance, progress in the semiconductor industry has tracked Gordon Moore's prediction in 1965 regarding continued downscaling of electronic devices on a chip [1]. The density of transistors in a semiconductor chip has increased ever since in a geometric progression, roughly doubling every 18 months. In state-of-the-art semiconductor chips, the separation between the source and drain in currently used fin field effect transistors (FinFETs) is below 10 nm. All future devices for semiconductor chip applications are likely to be strongly affected by the laws of quantum mechanics, and an understanding of these laws and tenets must be added to the repertoire of a device engineer and scientist [2].
Another challenge is to understand the quantum mechanical laws that will govern device operation when the projected density of 1013 transistors per cm2, anticipated by 2017, is finally reached. Density increase, however, comes with a cost: if energy dissipation does not scale down concomitantly with device dimensions there will be thermal runaway, resulting in chip meltdown. This doomsday scenario has been dubbed the “red brick wall” by the International Technology Roadmap for Semiconductors [3]. The foremost challenge is to find alternatives to the current semiconductor technology that would lead to a drastic reduction in energy dissipation during device operation. Such a technology, if and when it emerges, will very likely draw heavily on quantum mechanics as opposed to classical physics. Alternatives based on semiconductor heterostructures employing AlGaAs/GaAs or other III–V or II–VI materials have been investigated for several decades and have led to myriad quantum mechanical devices and architectures exploiting the special properties of quantum wells, wires, and dots [4–7]. Future device engineers, applied physicists, and material scientists will therefore need to be extremely adept at quantum mechanics.
The need for reform in the teaching of quantum mechanics at both the undergraduate and graduate levels is now evident [8], and has been discussed in many articles over the last few years [9–16] and at dedicated conferences on the subject, including many recent Gordon Research conferences. There are already some efforts under way at academic institutions to better train undergraduate students in this area. Many curricula have been modified to include more advanced classes in quantum mechanics for students in the engineering disciplines [10, 11]. This initiative has been catalyzed by the recent enthusiasm generated by the prospects of quantum computing and quantum communication [17]. This is a discipline that embraces knowledge in four different fields: electrical engineering, physics, materials science, and computer science.
Many textbooks have been written on quantum mechanics [18–30]. Only a few have dealt with practical aspects in the field suitable for a wide audience comprised of device engineers, applied physicists, and materials scientists [31–44]. In fact, quantum mechanics is taught very differently by high energy physicists and electrical engineers. In order for the subject to be entertaining and understandable to either discipline, they must be taught by their own kind to avoid a culture shock for the uninitiated students. Carr and McKagan have recently discussed the significant problems with graduate quantum mechanics education [13]. Typically, most textbooks are inadequate or devote too little time to exploring topics of current exciting new research and development that would prepare graduate students for the rapidly growing fields of nanoscience, nanoengineering and nanotechnology. As pointed out by Carr and McKagan, from a purely theoretical point of view, the history of quantum mechanics can be divided into four periods. In the first ten years following the 1926 formulation of the famous equation by Schrödinger, the early pioneers in the field developed the formalism taught in many undergraduate classes, including wave mechanics, its matrix formulations, and an early version of its interpretation with the work of Bohm and Bohr, among others. Then, until the mid 1960s, new concepts were developed, mostly addressing many-body aspects, with landmark achievements such as a formulation of density functional theory. This was accompanied by quantum electrodynamics and a successful explanation of low temperature superconductivity by Bardeen, Cooper, and Schrieffer. The third period began in 1964 with the pioneering work of Bell. The question of interpretation of quantum mechanics reached a deeper level with many theoretical advances, which eventually led to the fourth period in the field starting with the pioneering work of Aspect et al. in 1982 and the first successful experimental proof of Bell's inequality. Fundamental research in quantum mechanics now includes the fields of quantum computing and quantum communication, which have progressed in large strides helped by the rapid technological advances in non-linear optics, spintronic devices, and other systems fabricated with sophisticated techniques such as molecular beam epitaxy, metal organic chemical vapor deposition, atomic layer epitaxy, and various self-assembly techniques. The tremendous progress in the field has also been accelerated with the development of new characterization techniques including scanning and tunneling electron microscopy, atomic force microscopy, near field scanning optical microscopy, single photon detection, single electron detection, and others.
Many books dedicated to problems in quantum mechanics have appeared over the years. Most of them concentrate on exercises to help readers master the principles and fundamentals of the theory. In contrast, this work is a collection of problems for students, researchers, and practitioners interested in state-of-the-art material and device applications. It is not a textbook filled with precepts. Since examples are always better than precepts, this book is a collection of practical problems in quantum mechanics with solutions. Every problem is relevant either to a new device or a device concept, or to topics of current material relevant to the most recent research and development in practical quantum mechanics that could lead to new technological developments. The collection of problems covered in this book addresses topics that are covered in quantum mechanics textbooks but whose practical applications are often limited to a few end-of-the-book problems, if even that.
The present book should therefore be an ideal companion to a graduate-level textbook (or the instructor's personal lecture notes) in an engineering, condensed matter physics, or materials science curriculum. This book can not only be used by graduate students preparing for qualifying exams but is an ideal resource for the training of professional engineers in the fast-growing field of nanoscience. As such, it is appealing to a wide audience comprised not only of future generations of engineers, physicists, and material scientists but also of professionals in need of refocusing their areas of expertise toward the rapidly burgeoning areas of nanotechnology in our everyday life. The student is expected to have some elementary knowledge of quantum mechanics gleaned from modern physics classes. This includes a basic exposure to Planck's pioneering work, Bohr's concept of the atom, the meaning of the de Broglie wavelength, a first exposure to Heisenberg's uncertainty principle, and an introduction to the Schrödinger equation, including its solution for simple problems such as the particle in a box and the analysis of tunneling through a simple rectangular barrier. The authors have either organized or served on panels of many international conferences dedicated to the field of nanoscience and nanotechnology over the last 25 years. They have given or organized many short courses in these areas and given many invited talks in their field of expertise spanning nanoelectronics, nano-optoelectronics, nanoscale device simulations, spintronics, and vacuum nanoelectronics, among others. They also routinely teach graduate classes centered on quantum mechanical precepts, and therefore have first-hand experience of student needs and where their understanding can fall short.
The problems in this book are grouped by theme in 13 different chapters. At the beginning of each chapter, we briefly describe the theme behind the set of problems and refer the reader to specific sections of existing books that offer some of the clearest exposures to the material needed to tackle the problems. The level of difficulty of each problem is indicated by an increasing number of asterisks. Most solutions are typically sketched with an outline of the major steps. Intermediate and lengthy algebra steps are kept to a minimum to keep the size of the book reasonable. Additional problems are suggested at the end of each chapter and are extensions of or similar to those solved explicitly.
Each chapter contains a section on further reading containing references to articles where some of the problems treated in this book were used to investigate specific practical applications. There are several appendices to complement the set of problems. Appendix A reviews the postulates of quantum mechanics. Appendix B reviews some basic properties of the one-dimensional harmonic oscillator. Appendix C reviews some basic definitions and properties of quantum mechanical operators. Appendix D reviews the concept of Pauli matrices and their basic properties. Appendix E is a derivation of an analytical expression for the threshold voltage of a high electron mobility transistor. Appendix F is a derivation of Peierl's transformation, which is crucial to the study of the properties of a particle in an external electromagnetic field. Finally, Appendix G contains some of the Matlab code necessary to solve some of the problems and generate figures throughout the book.
The problems in this book have been collected by the authors over a period of 25 years while teaching different classes dealing with the physics and engineering of devices at the submicron and nanoscale levels. These problems were solved by the authors as part of several classes taught at the undergraduate and graduate levels at their respective institutions. For instance, some of the exercises have been assigned as homework or exam questions as part of first-year graduate courses on High-Speed Electronic Devices and Quantum Systems taught by M. Cahay at the University of Cincinnati. Since 2003, M. Cahay has also taught a class on Introduction to Quantum Computing with his colleagues in the Physics Department at the University of Cincinnati. S. Bandyopadhyay has taught a multi-semester graduate level course in Quantum Theory of Solid State Devices in three different institutions: University of Notre Dame, University of Nebraska, and Virginia Commonwealth University.
Should this edition be a success, we intend to upgrade future editions of this book with solutions to all the suggested problems. This book could not obviously cover all aspects of current research. For instance, topics left out are quantization of phonon modes, Coulomb and spin blockaded transport in nanoscale devices, and carrier transport in carbon nanotubes and graphene, among others. Future editions will include new sets of problems on these topics as well as others based on suggestions by readers, keeping pace with the most recent topics which will, without a doubt, bloom in the exciting fields of nanoscience, nanoengineering, and nanotechnology.
The contents of this book are as follows:
Chapter 1: General Properties of the Schrödinger Equation
This chapter describes some general properties of the time-independent effective mass Schrödinger equation (EMSE), which governs the steady-state behavior of an electron in a solid with spatially varying potential profile. The solid may consist of one or more materials (e.g., a heterostructure or superlattice); hence the effective mass of the electron may vary in space. The EMSE is widely used in studying the electronic and optical properties of solids. This chapter also discusses some general properties of the EMSE, including the concepts of linearly independent solutions and their Wronskian. It is shown that in the presence of a spatially varying effective mass, the Ben Daniel–Duke boundary conditions must be satisfied. The concept of quantum mechanical wave impedance is introduced to point out the similarity between solutions to the time-independent Schrödinger equation and transmission line theory in classical electrodynamics and microwave theory.
Chapter 2: Operators
All quantum mechanical operators describing physical variables are Hermitian. This chapter derives several useful identities involving operators. This includes derivations of the shift operator, the Glauber identity, the Baker–Hausdorff formula, the hypervirial theorem, Ehrenfest's theorem, and various quantum mechanical sum rules. The concept of unitary transformation is also introduced and illustrated through a calculation of the polarizability of the one-dimensional harmonic oscillator. Usage of the operator identities and theorems derived in this chapter is illustrated in other chapters. Some general definitions and properties of operators are reviewed in Appendix C, which the reader should consult before trying out the problems in this chapter.
Chapter 3: Bound States
The problems in this chapter deal with one-dimensional bound state calculations, which can be performed analytically or via the numerical solution of a transcendental equation. These problems give some insight into more complicated three-dimensional bound state problems whose solutions typically require numerically intensive approaches.
Chapter 4: Heisenberg Principle
This chapter starts with three different proofs of the generalized Heisenberg uncertainty relations followed by illustrations of their application to the study of some bound state and scattering problems, including diffraction from a slit in a screen and quantum mechanical tunneling through a potential barrier.
Chapter 5: Current and Energy Flux Densities
This set of problems introduces the current density operator, which is applied to the study of various tunneling problems, including the case of a general one-dimensional heterostructure under bias (i.e., subjected to an electric field), the tunneling of an electron through an absorbing one-dimensional delta scatterer and potential well, and the calculation of the dwell time above a quantum well (QW). The dwell time is the time that an electron traversing a QW potential, with energy above the well's barrier, lingers within the well region. This chapter also includes an introduction to a quantum mechanical version of the energy conservation law based on the concept of quantum mechanical energy flux derived from the Schrödinger equation. Some basic tunneling problems are revisited using the conservation of energy flux principle.
Chapter 6: Density of States
This chapter introduces the important concept of density of states (DOS) going from bulk to quantum confined structures. Applications of DOS expressions are illustrated by studying the onset of degeneracy in quantum confined structures, by calculating the intrinsic carrier concentration in a two-dimensional electron gas, and by establishing the relation between the three- and two-dimensional DOS. We also illustrate the use of the DOS concept to calculate the electron density due to reflection from an infinite potential wall, the electron charge concentration in a QW in the presence of carrier freeze-out, and the threshold voltage, gate capacitance, and current–voltage characteristics of a high electron mobility transistor. Finally, the DOS concept is applied to study the properties of the blackbody radiation in three- and one-dimensional cavities.
Chapter 7: Transfer Matrix
In this chapter, the use of the transfer matrix approach to solving the time-independent Schrödinger equation is illustrated for simple examples such as tunneling through a one-dimensional delta scatterer and through a square potential barrier. Using the cascading rule for transfer matrices, a general expression for the reflection and tunneling coefficient through an arbitrary potential energy profile is then derived in the presence of an applied bias across the structures. A derivation of the Floquet and Bloch theorems pertaining to an infinite repeated structure is then given based on the transfer matrix technique. The approach is also used to develop the Kroenig–Penney model for an infinite lattice with an arbitrary unit cell potential. Properties of the tunneling coefficient through finite repeated structures are then discussed, as well as their connection to the energy band structure of the infinite periodic lattice. The chapter concludes with the connection between the bound state and the tunneling problem for an arbitrary one-dimensional potential energy profile and a calculation of the dwell time above an arbitrary potential well.
Chapter 8: Scattering Matrix
The concept of a scattering matrix to solve tunneling problems is first described, including their cascading rule. Explicit analytical expression of the scattering matrix through a one-dimensional delta scatterer, two delta scatterers in series separated by a distance L, a simple potential step, a square barrier, and a double barrier resonant tunneling diode are then derived. The connection between transfer and scattering matrices is then discussed, as well as applications of these formalisms to the study of electron wave propagation through an arbitrary one-dimensional energy profile.
Chapter 9: Perturbation Theory
This chapter starts with a brief introduction to first-order time-independent perturbation theory and applies it to the study of an electro-optic modulator and calculation of band structure in a crystal. It then introduces Fermi's Golden Rule, which is a well-known result of time-dependent perturbation theory, and applies it to calculate the scattering rate of electrons interacting with impurities in a solid. Such rates determine the carrier mobility in a solid at low temperatures when impurity scattering dominates over phonon scattering. Fermi's Golden Rule is also applied to calculate the electron–photon interaction rate in a solid, and the absorption coefficient quantifying absorption of light as a function of light frequency.
Chapter 10: Variational Approach
Another important approach to finding approximate solutions to the Schrödinger equation is based on the Rayleigh–Ritz variational principle. For a specific problem, if the wave function associated with the ground or first excited states of a Hamiltonian cannot be calculated exactly, a suitable guess for the general shape of the wave functions associated with these states can be inferred using some symmetry properties of the system and the general properties of the Schrödinger equation as studied in Chapter 1. In this chapter, we first briefly describe the Rayleigh–Ritz variational procedure and apply it to the calculation of the energy of the ground and first excited states of problems for which an exact solution is known. Next, some general criteria for the existence of a bound state in a one-dimensional potential with finite range are derived.
Chapter 11: Electron in a Magnetic Field
Many important phenomena in condensed matter physics, such as the quantum Hall effect, require an understanding of the quantum mechanical behavior of electrons in a magnetic field. In this chapter, we introduce the concept of a vector potential and gauge to incorporate magnetic fields in the Hamiltonian of an electron. We then study quantum confined systems and derive the eigenstates of an electron in such systems subjected to a magnetic field, an example being the formation of Landau levels in a two-dimensional electron gas with a magnetic field directed perpendicular to the plane of the electron gas. The effect of a magnetic field (other than lifting spin degeneracy via the Zeeman effect) is to modify the momentum operator through the introduction of a magnetic vector potential. We study properties of the transformed momentum operator and conclude by deriving the polarizability of a harmonic oscillator in a magnetic field.
Chapter 12: Electron in an Electromagnetic Field and Optical Properties of Nanostructures
This chapter deals with the interaction between an electron and an electromagnetic field. We derive the electron–photon interaction Hamiltonian and apply it to calculate absorption coefficients. Some problems dealing with emission of light are also examined, concluding with the derivation of the Schrödinger equation for an electron in an intense laser field.
Chapter 13: Time-Dependent Schrödinger Equation
This chapter examines several properties of one-dimensional Gaussian wave packets, including a calculation of the spatio-temporal dependence of their probability current and energy flux densities and a proof that their average kinetic energy is a constant of motion. An algorithm to study the time evolution of wave packets based on the Crank–Nicholson scheme is discussed for the cases of totally reflecting and absorbing boundary conditions at the ends of the simulation domain.
This book should be of interest to any reader with a preliminary knowledge of quantum mechanics as taught in a typical modern physics class in undergraduate curricula. It should be a strong asset to professionals refocusing their expertise on many different areas of nanotechnology that affect our daily life. If successful, future editions of the book will be geared toward practical aspects of quantum mechanics useful to chemists, chemical engineers, and researchers and practitioners in the field of nanobiotechnology.
This book will be an ideal companion to a graduate-level textbook (or the instructor's personal lecture notes) in an engineering, physics, or materials science curriculum. It can not only be used by graduate students eager to better grasp the field of quantum mechanics and its applications, but should also help faculty develop teaching materials. Moreover, it will be an ideal resource for the training of professional engineers in the fast-growing fields of nanoscience, nanoengineering, and nanotechnology. As such it should be appealing to a wide audience of future generations of engineers, physicists, and material scientists.
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Suggested Reading
• Kogan, V. I. and Galitsiv, V. M. (1963)
Problems in Quantum Mechanics
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• ter Haar, D. (1964)
Selected Problems in Quantum Mechanics
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• Constantinesu, F. and Magyari, E. (1976)
Problems in Quantum Mechanics
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• Peleg, Y., Pnini, R., and Zaarir, Z. (1998)
Schaum's Outline of Theory and Problems of Quantum Mechanics
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• Mavromatis, H. A. (1992)
Exercises in Quantum Mechanics: A Collection of Illustrative Problems and Their Solutions
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• Squires, G. L. (1995)
Problems in Quantum Mechanics with Solutions
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Problems and Solutions on Quantum Mechanics Compiled by the Physics Coaching Class, University of Science and Technology of China
, World Scientific, Singapore.
• Basdevant, J.-L. and Dalibard, J. (2000)
The Quantum Mechanics Solver: How to Apply Quantum Theory to Modern Physics
, Springer, New York.Capri, A. Z. (2002)
Problems and Solutions in Non-Relativistic Quantum Mechanics
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• Tamvakis, K. (2005)
Problems and Solutions in Quantum Mechanics
, Cambridge University Press, Cambridge.
• Gold'man, I. I. and Krivchenkov, V. D. (2006)
Problems in Quantum Mechanics
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• d'Emilio, E. and Picasso, L. E. (2011)
Problems in Quantum Mechanics with Solutions
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• Cini, M., Fucito, R., and Sbragaglia, M. (2012)
Solved Problems in Quantum and Statistical Mechanics
, Springer, New York.
• Flügge, S. (1999)
Practical Quantum Mechanics
, Springer, New York.
• Harrison, W. A. (2000)
Applied Quantum Mechanics
, World Scientific, River Edge, NJ.
• Marchildon, L. (2002)
Quantum Mechanics: From Basic Principles to Numerical Methods and Applications
, Springer, New York.
• Blinder, S. M. (2004)
Introduction to Quantum Mechanics: In Chemistry, Materials Science, and Biology
, Elsevier, Boston.
• Tang, C. L. (2005),
Fundamentals of Quantum Mechanics: For Solid State Electronics and Optics
, Cambridge University Press, New York.
• Harrison, P. (2005)
Quantum Wells, Wires, and Dots
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• Robinett, R. W. (2006)
Quantum Mechanics, Classical Results, Modern Systems, and Visualized Examples
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Quantum Mechanics for Scientists and Engineers
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• Zettili, N. (2009)
Quantum Mechanics: Concepts and Application
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• Kim, D. M. (2010)
Introductory Quantum Mechanics for Semiconductor Nanotechnology
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• Ahn, D. and Park, S.-H. (2011)
Engineering Quantum Mechanics
, Wiley-IEEE, Hoboken, NJ.
• Sullivan, D. M. (2012)
Quantum Mechanics for Electrical Engineers
, Wiley-Blackwell, Oxford.
• Band, Y. B. and Avishai, Y. (2013)
Quantum Mechanics with Applications to Nanotechnology and Information Science
, Academic Press, New York.
