This page intentionally left blank www.com Applied Quantum Mechanics, Second Edition Electrical and mechanical engineers, materials scientists and applied physicists will find Levi’s uniquely practical explanation of quantum mechanics invaluable. This updated and expanded edition of the bestselling original text now covers quantization of angular momentum and quantum communication, and problems and additional references are included. Using real-world engineering examples to engage the reader, the author makes quantum mechanics accessible and relevant to the engineering student. Numerous illustrations, exercises, worked examples and problems are included; MATLAB® source code to support the text is available from www.
Levi is Professor of Electrical Engineering and of Physics and Astronomy at the University of Southern California. He joined USC in 1993 after working for 10 years at AT & T Bell Laboratories, New Jersey. He invented hot electron spectroscopy, discovered ballistic electron transport in transistors, created the first microdisk laser, and carried out groundbreaking work in parallel fiber optic interconnect components in computer and switching systems. His current research interests include scaling of ultra-fast electronic and photonic devices, system-level integration of advanced optoelectronic technologies, manufacturing at the nanoscale, and the subject of Adaptive Quantum Design.com Applied Quantum Mechanics Second Edition A.com cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www.org/9780521860963 © Cambridge University Press 2006 This publication is in copyright.
Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 isbn-13 978-0-511-19111-4 eBook (EBL) isbn-10 0-511-19111-1 eBook (EBL) isbn-13 978-0-521-86096-3 hardback isbn-10 0-521-86096-2 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com Dass ich erkenne, was die Welt Im Innersten zusammenhält Goethe (Faust, I.com Contents Preface to the first edition page xiii Preface to the second edition xv MATLAB® programs xvi 1 Introduction 1 1.2 The one-dimensional simple harmonic oscillator 7 1.3 Harmonic oscillation of a diatomic molecule 10 1.4 The monatomic linear chain 13 1.5 The diatomic linear chain 15 1.5 Problems 53 2 Toward quantum mechanics 57 2.1 Diffraction and interference of light 58 2.2 Black-body radiation and evidence for quantization of light 62 2.3 Photoelectric effect and the photon particle 64 2.4 Secure quantum communication 66 2.5 The link between quantization of photons and other particles 70 2.6 Diffraction and interference of electrons 71 2.7 When is a particle a wave? 72 2.2 The Schrödinger wave equation 73 2.1 The wave function description of an electron in free space 79 2.2 The electron wave packet and dispersion 80 2.3 The hydrogen atom 83 2.4 Periodic table of elements 89 2.6 Electronic properties of bulk semiconductors and heterostructures 96 vii www.4 Problems 114 3 Using the Schrödinger wave equation 117 3.1 The effect of discontinuity in the wave function and its slope 118 3.2 Wave function normalization and completeness 121 3.3 Inversion symmetry in the potential 122 3.1 One-dimensional rectangular potential well with infinite barrier energy 123 3.4 Numerical solution of the Schrödinger equation 126 3.1 Current in a rectangular potential well with infinite barrier energy 129 3.2 Current flow due to a traveling wave 131 3.6 Degeneracy as a consequence of symmetry 131 3.1 Bound states in three dimensions and degeneracy of eigenvalues 131 3.7 Symmetric finite-barrier potential 133 3.1 Calculation of bound states in a symmetric finite-barrier potential 135 3.8 Transmission and reflection of unbound states 137 3.1 Scattering from a potential step when m1 = m2 138 3.2 Scattering from a potential step when m1 = m2 140 3.3 Probability current density for scattering at a step 141 3.4 Impedance matching for unity transmission across a potential step 142 3.1 Electron tunneling limit to reduction in size of CMOS transistors 149 3.10 The nonequilibrium electron transistor 150 3.12 Problems 168 4 Electron propagation 171 4.2 The propagation matrix method 172 4.3 Program to calculate transmission probability 177 4.4 Time-reversal symmetry 178 4.5 Current conservation and the propagation matrix 180 4.6 The rectangular potential barrier 182 4.1 Transmission probability for a rectangular potential barrier 182 4.2 Transmission as a function of energy 185 4.1 Heterostructure bipolar transistor with resonant tunnel-barrier 190 4.2 Resonant tunneling between two quantum wells 192 viii www.8 The potential barrier in the delta function limit 197 4.9 Energy bands in a periodic potential 199 4.2 The propagation matrix applied to a periodic potential 201 4.3 The tight binding approximation 207 4.4 Crystal momentum and effective electron mass 209 4.10 Other engineering applications 213 4.11 The WKB approximation 214 4.1 Tunneling through a high-energy barrier of finite width 215 4.13 Problems 234 5 Eigenstates and operators 238 5.1 The postulates of quantum mechanics 238 5.2 One-particle wave function space 239 5.3 Properties of linear operators 240 5.1 Product of operators 241 5.2 Properties of Hermitian operators 241 5.3 Normalization of eigenfunctions 243 5.4 Completeness of eigenfunctions 243 5.5 Measurement of real numbers 246 5.1 Expectation value of an operator 247 5.2 Time dependence of expectation value 248 5.3 Uncertainty of expectation value 249 5.4 The generalized uncertainty relation 253 5.6 The no cloning theorem 255 5.7 Density of states 256 5.1 Density of electron states 256 5.2 Calculating density of states from a dispersion relation 263 5.3 Density of photon states 264 5.9 Problems 277 6 The harmonic oscillator 280 6.1 The harmonic oscillator potential 280 6.2 Creation and annihilation operators 282 6.1 The ground state of the harmonic oscillator 284 6.2 Excited states of the harmonic oscillator and normalization of eigenstates 287 6.3 The harmonic oscillator wave functions 291 6.1 The classical turning point of the harmonic oscillator 295 6.1 The superposition operator 300 ix www.2 Measurement of a superposition state 300 6.3 Time dependence of creation and annihilation operators 301 6.5 Quantization of electromagnetic fields 305 6.2 Quantization of an electrical resonator 306 6.6 Quantization of lattice vibrations 307 6.7 Quantization of mechanical vibrations 308 6.9 Problems 323 7 Fermions and bosons 326 7.1 The symmetry of indistinguishable particles 327 7.2 Fermi–Dirac distribution and chemical potential 334 7.1 Writing a computer program to calculate the chemical potential 337 7.2 Writing a computer program to plot the Fermi–Dirac distribution 338 7.3 Fermi–Dirac distribution function and thermal equilibrium statistics 339 7.3 The Bose–Einstein distribution function 342 7.5 Problems 351 8 Time-dependent perturbation 353 8.1 An abrupt change in potential 354 8.2 Time-dependent change in potential 356 8.2 First-order time-dependent perturbation 359 8.1 Charged particle in a harmonic potential 360 8.3 Fermi’s golden rule 363 8.4 Elastic scattering from ionized impurities 366 8.1 The coulomb potential 369 8.2 Linear screening of the coulomb potential 375 8.5 Photon emission due to electronic transitions 384 8.1 Density of optical modes in three-dimensions 384 8.3 Background photon energy density at thermal equilibrium 385 8.4 Fermi’s golden rule for stimulated optical transitions 385 8.5 The Einstein and coefficients 387 8.7 Problems 407 9 The semiconductor laser 412 9.2 Spontaneous and stimulated emission 413 9.1 Absorption and its relation to spontaneous emission 416 x www.3 Optical transitions using Fermi’s golden rule 419 9.1 Optical gain in the presence of electron scattering 420 9.4 Designing a laser diode 422 9.1 The optical cavity 422 9.2 Mirror loss and photon lifetime 428 9.3 The Fabry–Perot laser diode 429 9.4 Semiconductor laser diode rate equations 430 9.5 Numerical method of solving rate equations 434 9.1 The Runge–Kutta method 435 9.2 Large-signal transient response 437 9.6 Noise in laser diode light emission 440 9.7 Why our model works 443 9.9 Problems 449 10 Time-independent perturbation 450 10.2 Time-independent nondegenerate perturbation 451 10.1 The first-order correction 452 10.2 The second-order correction 453 10.3 Harmonic oscillator subject to perturbing potential in x 456 10.4 Harmonic oscillator subject to perturbing potential in x2 458 10.5 Harmonic oscillator subject to perturbing potential in x3 459 10.3 Time-independent degenerate perturbation 461 10.1 A two-fold degeneracy split by time-independent perturbation 462 10.3 The two-dimensional harmonic oscillator subject to perturbation in xy 465 10.4 Perturbation of two-dimensional potential with infinite barrier energy 467 10.5 Problems 482 11 Angular momentum and the hydrogenic atom 485 11.1 Classical angular momentum 485 11.2 The angular momentum operator 487 11.1 Eigenvalues of angular momentum operators L̂z and L̂2 489 11.3 Spherical coordinates and spherical harmonics 492 11.4 The rigid rotator 498 11.3 The hydrogen atom 499 11.1 Eigenstates and eigenvalues of the hydrogen atom 500 11.2 Hydrogenic atom wave functions 508 xi www.4 Fine structure of the hydrogen atom and electron spin 515 11.6 Problems 529 Appendix A Physical values 532 Appendix B Coordinates, trigonometry, and mensuration 537 Appendix C Expansions, differentiation, integrals, and mathematical relations 540 Appendix D Matrices and determinants 546 Appendix E Vector calculus and Maxwell’s equations 548 Appendix F The Greek alphabet 551 Index 552 xii www.com Preface to the first edition The theory of quantum mechanics forms the basis for our present understanding of physical phenomena on an atomic and sometimes macroscopic scale. Today, quantum mechanics can be applied to most fields of science. Within engineering, important subjects of practical significance include semiconductor transistors, lasers, quantum optics, and molecular devices.
As technology advances, an increasing number of new electronic and opto-electronic devices will operate in ways which can only be understood using quantum mechanics. Over the next thirty years, fundamentally quantum devices such as single-electron memory cells and photonic signal processing systems may well become commonplace. Applications will emerge in any discipline that has a need to understand, control, and modify entities on an atomic scale. As nano- and atomic-scale structures become easier to manufacture, increasing numbers of individuals will need to understand quantum mechanics in order to be able to exploit these new fabrication capabilities.
Hence, one intent of this book is to provide the reader with a level of understanding and insight that will enable him or her to make contributions to such future applications, whatever they may be. The book is intended for use in a one-semester introductory course in applied quantum mechanics for engineers, material scientists, and others interested in understanding the critical role of quantum mechanics in determining the behavior of practical devices. To help maintain interest in this subject, I felt it was important to encourage the reader to solve problems and to explore the possibilities of the Schrödinger equation. To ease the way, solutions to example exercises are provided in the text, and the enclosed CD- ROM contains computer programs written in the MATLAB language that illustrate these solutions.
The computer programs may be usefully exploited to explore the effects of changing parameters such as temperature, particle mass, and potential within a given problem. In addition, they may be used as a starting point in the development of designs for quantum mechanical devices. The structure and content of this book are influenced by experience teaching the subject. Surprisingly, existing texts do not seem to address the interests or build on the computing skills of today’s students.
This book is designed to better match such student needs. Some material in the book is of a review nature, and some material is merely an introduction to subjects that will undoubtedly be explored in depth by those interested in pursuing more advanced topics. The majority of the text, however, is an essentially self-contained study of quantum mechanics for electronic and opto-electronic applications.com PREFACE TO THE FIRST EDITION There are many important connections between quantum mechanics and classical mechanics and electromagnetism. For this and other reasons, Chapter 1 is devoted to a review of classical concepts.
This establishes a point of view with which the predictions of quantum mechanics can be compared.