Graduate Texts in Physics For further volumes: http://www.com/series/8431 www.com Graduate Texts in Physics Graduate Texts in Physics publishes core learning/teaching material for graduate- and advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS- or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, ex- periments and applications (as relevant) for their mastery and teaching, respectively. In- ternational in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions or references for scien- tists entering, or requiring timely knowledge of, a research field.
Series Editors Professor Richard Needs Cavendish Laboratory JJ Thomson Avenue Cambridge CB3 0HE, UK E-mail: rn11@cam.uk Professor William T. Rhodes Florida Atlantic University Imaging Technology Center Department of Electrical Engineering 777 Glades Road SE, Room 456 Boca Raton, FL 33431, USA E-mail: wrhodes@fau. Eugene Stanley Boston University Center for Polymer Studies Department of Physics 590 Commonwealth Avenue, Room 204B Boston, MA 02215, USA E-mail: hes@bu.com Rainer Dick Advanced Quantum Mechanics Materials and Photons With 62 Figures 123 www.com Rainer Dick University of Saskatchewan Saskatoon, Saskatchewan S7N5E2, Canada rainer.ca ISSN 1868-4513 e-ISSN 1868-4521 ISBN 978-1-4419-8076-2 e-ISBN 978-1-4419-8077-9 DOI 10.1007/978-1-4419-8077-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011943751 c Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly anal- ysis.
Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.com Preface Quantum mechanics was invented in an era of intense and seminal scientific re- search between 1900 and 1928 (and in many regards continues to be developed and expanded) because neither the properties of atoms and electrons, nor the spectrum of radiation from heat sources could be explained by the classical theories of mechanics, electrodynamics and thermodynamics. It was a major intellectual achievement and a breakthrough of curiosity driven fundamental research which formed quantum theory into one of the pillars of our present understanding of the fundamental laws of nature.
The properties and behav- ior of every elementary particle is governed by the laws of quantum theory. However, the rule of quantum mechanics is not limited to atomic and sub- atomic scales, but also affects macroscopic systems in a direct and profound manner. The electric and thermal conductivity properties of materials are de- termined by quantum effects, and the electromagnetic spectrum emitted by a star is primarily determined by the quantum properties of photons. It is there- fore not surprising that quantum mechanics permeates all areas of research in advanced modern physics and materials science, and training in quantum mechanics plays a prominent role in the curriculum of every major physics or chemistry department.
The ubiquity of quantum effects in materials implies that quantum mechanics also evolved into a major tool for advanced technological research. The con- struction of the first nuclear reactor in Chicago in 1942 and the development of nuclear technology could not have happened without a proper understanding of the quantum properties of particles and nuclei. However, the real breakthrough for a wide recognition of the relevance of quantum effects in technology occured with the invention of the transistor in 1948 and the ensuing rapid development of semiconductor electronics. This proved once and for all the importance of quantum mechanics for the applied sciences and engineering, only 22 years after publication of the Schrödinger equation! Electronic devices like transis- tors rely heavily on the quantum mechanical emergence of energy bands in materials, which can be considered as a consequence of combination of many atomic orbitals or as a consequence of delocalized electron states probing a lattice structure.
Today the rapid developments of spintronics, photonics and nanotechnology provide continuing testimony to the technological relevance of quantum mechanics. As a consequence, every physicist, chemist and electrical engineer nowadays has to learn aspects of quantum mechanics, and we are witnessing a time v www.com vi Preface when also mechanical and aerospace engineers are advised to take at least a 2nd year course, due to the importance of quantum mechanics for elasticity and stability properties of materials. Furthermore, quantum information appears to become inceasingly relevant for computer science and information technology, and a whole new area of quantum technology will likely follow in the wake of this development. Therefore it seems safe to posit that within the next two generations, 2nd and 3rd year quantum mechanics courses will become as abundant and important in the curricula of science and engineering colleges as first and second year calculus courses.
Quantum mechanics continues to play a dominant role in particle physics and atomic physics - after all, the Standard Model of particle physics is a quantum theory, and the spectra and stability of atoms cannot be explained without quantum mechanics. However, most scientists and engineers use quantum me- chanics in advanced materials research. Furthermore, the dominant interaction mechanisms in materials (beyond the nuclear level) are electromagnetic, and many experimental techniques in materials science are based on photon probes. The introduction to quantum mechanics in the present book takes this into account by including aspects of condensed matter theory and the theory of photons at earlier stages and to a larger extent than other quantum mechanics texts.
Quantum properties of materials provide neat and very interesting il- lustrations of time-independent and time-dependent perturbation theory, and many students are better motivated to master the concepts of quantum me- chanics when they are aware of the direct relevance for modern technology. A focus on the quantum mechanics of photons and materials is also perfectly suited to prepare students for future developments in quantum information technology, where entanglement of photons or spins, decoherence, and time evolution operators will be key concepts. Other novel features of the discussion of quantum mechanics in this book concern attention to relevant mathematical aspects which otherwise can only be found in journal articles or mathematical monographs. Special appendices include a mathematically rigorous discussion of the completeness of Sturm- Liouville eigenfunctions in one spatial dimension, an evaluation of the Baker- Campbell-Hausdorff formula to higher orders, and a discussion of logarithms of matrices.
Quantum mechanics has an extremely rich and beautiful mathemat- ical structure. The growing prominence of quantum mechanics in the applied sciences and engineering has already reinvigorated increased research efforts on its mathematical aspects. Both students who study quantum mechanics for the sake of its numerous applications, as well as mathematically inclined students with a primary interest in the formal structure of the theory should therefore find this book interesting. This book emerged from a quantum mechanics course which I had introduced at the University of Saskatchewan in 2001.
It should be suitable both for advanced undergraduate and introductory graduate courses on the subject. To make advanced quantum mechanics accessible to wider audiences which might not have been exposed to standard second and third year courses on www.com Preface vii atomic physics, analytical mechanics, and electrodynamics, important aspects of these topics are briefly, but concisely introduced in special chapters and appendices. The success and relevance of quantum mechanics has reached far beyond the realms of physics research, and physicists have a duty to dissemi- nate the knowledge of quantum mechanics as widely as possible. Saskatoon, Saskatchewan, Canada Rainer Dick www.com To the Students Congratulations! You have reached a stage in your studies where the topics of your inquiry become ever more interesting and more relevant for modern research in basic science and technology.
Together with your professors, I will have the privilege to accompany you along the exciting road of your own discovery of the bizarre and beautiful world of quantum mechanics. I will aspire to share my own excitement that I continue to feel for the subject and for science in general. You will be introduced to many analytical and technical skills that are used in everyday applications of quantum mechanics. These skills are essential in virtually every aspect of modern research.
A proper understanding of a mate- rials science measurement at a synchrotron requires a proper understanding of photons and quantum mechanical scattering, just like manipulation of qubits in quantum information research requires a proper understanding of spin and photons and entangled quantum states. Quantum mechanics is ubiquitous in modern research. It governs the formation of microfractures in materials, the conversion of light into chemical energy in chlorophyll or into electric impulses in our eyes, and the creation of particles at the Large Hadron Collider. Technical mastery of the subject is of utmost importance for understanding quantum mechanics.
Trying to decipher or apply quantum mechanics without knowing how it really works in the calculation of wave functions, energy levels, and cross sections is just idle talk, and always prone for misconceptions. There- fore we will go through a great many technicalities and calculations, because you and I (and your professor!) have a common goal: You should become an expert in quantum mechanics. However, there is also another message in this book. The apparently exotic world of quantum mechanics is our world.
Our bodies and all the world around us is built on quantum effects and ruled by quantum mechanics. It is not apparent and only visible to the cognoscenti. Therefore we have developed a mode of thought and explanation of the world that is based on classical pictures – mostly waves and particles in mechanical interaction. This mode of thought was sufficient for survivial of our species so far, and it culminated in a powerful tool called classical physics.
However, by 1900 those who were paying attention had caught enough glimpses of the underlying non-classical world to embark on the exciting journey of discovering quantum mechanics. Indeed, every single atom in your body is ruled by the laws of quantum mechanics, and could not even exist as a classical particle. The electrons that provide the light for your ix www.com x To the Students long nights of studying generate this light in stochastic quantum leaps from a state of a single electron to a state of an electron and a photon. And maybe the most striking example of all: There is absolutely nothing classical in the sunlight that provides the energy for all life on Earth.
Quantum theory is not a young theory any more. The scientific foundations of the subject were developed over half a century between 1900 and 1949, and many of the mathematical foundations were even developed in the 19th century. The steepest ascent in the development of quantum theory appeared between 1924 and 1928, when matrix mechanics, Schrödinger’s equation, the Dirac equation and field quantization were invented. I have included numerous references to original papers from this period, not to ask you to read all those papers – after all, the primary purpose of a textbook is to put major achieve- ments into context, provide an introductory overview at an appropriate level, and replace often indirect and circuitous original derivations with simpler ex- planations – but to honour the people who brought the then nascent theory to maturity.
Quantum theory is an extremely well established and developed theory now, which has proven itself on numerous occasions. However, we still continue to improve our collective understanding of the theory and its wide ranging applications, and we test its predicitions and its probabilistic inter- pretation with ever increasing accuracy.