This page intentionally left blank 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.cambridge.org/9780521860963. A. F. J. 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&TBellLaboratories,NewJersey.Heinventedhotelectronspectroscopy,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. Applied Quantum Mechanics Second Edition A. F. J. Levi cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521860963 © Cambridge University Press 2006 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Firstpublishedinprintformat 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 forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Dass ich erkenne, was die Welt Im Innersten zusammenhält Goethe (Faust, I.382–3) Contents Preface to the first edition page xiii Preface to the second edition xv MATLAB® programs xvi 1 Introduction 1 1.1 Motivation 1 1.2 Classical mechanics 4 1.2.1 Introduction 4 1.2.2 The one-dimensional simple harmonic oscillator 7 1.2.3 Harmonic oscillation of a diatomic molecule 10 1.2.4 The monatomic linear chain 13 1.2.5 The diatomic linear chain 15 1.3 Classical electromagnetism 18 1.3.1 Electrostatics 18 1.3.2 Electrodynamics 24 1.4 Example exercises 39 1.5 Problems 53 2 Toward quantum mechanics 57 2.1 Introduction 57 2.1.1 Diffraction and interference of light 58 2.1.2 Black-body radiation and evidence for quantization of light 62 2.1.3 Photoelectric effect and the photon particle 64 2.1.4 Secure quantum communication 66 2.1.5 The link between quantization of photons and other particles 70 2.1.6 Diffraction and interference of electrons 71 2.1.7 When is a particle a wave? 72 2.2 The Schrödinger wave equation 73 2.2.1 The wave function description of an electron in free space 79 2.2.2 The electron wave packet and dispersion 80 2.2.3 The hydrogen atom 83 2.2.4 Periodic table of elements 89 2.2.5 Crystal structure 93 2.2.6 Electronicpropertiesofbulksemiconductorsandheterostructures 96 vii CONTENTS 2.3 Example exercises 103 2.4 Problems 114 3 Using the Schrödinger wave equation 117 3.1 Introduction 117 3.1.1 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 3.3.1 One-dimensional rectangular potential well with infinite barrier energy 123 3.4 Numerical solution of the Schrödinger equation 126 3.5 Current flow 128 3.5.1 Current in a rectangular potential well with infinite barrier energy 129 3.5.2 Current flow due to a traveling wave 131 3.6 Degeneracy as a consequence of symmetry 131 3.6.1 Boundstatesinthreedimensionsanddegeneracyofeigenvalues 131 3.7 Symmetric finite-barrier potential 133 3.7.1 Calculation of bound states in a symmetric finite-barrier potential 135 3.8 Transmission and reflection of unbound states 137 3.8.1 Scattering from a potential step when m =m 138 1 2 3.8.2 Scattering from a potential step when m (cid:1)=m 140 1 2 3.8.3 Probability current density for scattering at a step 141 3.8.4 Impedance matching for unity transmission across a potential step 142 3.9 Particle tunneling 145 3.9.1 Electron tunneling limit to reduction in size of CMOS transistors 149 3.10 The nonequilibrium electron transistor 150 3.11 Example exercises 155 3.12 Problems 168 4 Electron propagation 171 4.1 Introduction 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 4.6.1 Transmission probability for a rectangular potential barrier 182 4.6.2 Transmission as a function of energy 185 4.6.3 Transmission resonances 186 4.7 Resonant tunneling 188 4.7.1 Heterostructure bipolar transistor with resonant tunnel-barrier 190 4.7.2 Resonant tunneling between two quantum wells 192 viii