Introduction Why yet another book on quantum mechanics?Quantum mechanics was born in the first experimental confirmations over the years.It is considered to be the fundamental physical paradigm,and has a wide range of applications,from cosmology to chemistry,and from biology to information sciences.It is one of the greatest intellectual achievements of the past century.As an effect of its invention,the veryc oncept of physical reality was changed and"observation,""measurement,""prediction,"and"state of the system"acquired a new and deeper meaning. Probability was not unknown in physics:it was introduced by Boltzmann in order to control the behavior of a system with a very large number of particles.It was the missing concept in order to understand the thermodynamics of macroscopic bodies,but the struc ture of the physical laws remained still deterministic.The introduction of probability was needed as a consequence of our lack of knowledge of the initial conditions of the sys- tem and of our inability to solve an enormous num mber of coupled non-linear differential equations. In quantum mechanics,the tune is different:if we have 10 radioactive atoms no intrinsic unknown variables decide which of them will decay first.What we observe experimentally andom process.The original explanation of this phenom enon ir quantum mechanics was rather unexpected.All atoms have the same probability of having decayed:only when we observe the system do we select which atoms have decayed in the past.In spite of the fact that this solution seems to be in contrast with common sense.it is the only possible one in the frame rk o the conventional interpretation of quan mechanics.Heisenberg,de Broglie,Pauli,Dirac,and many others invented a formalisn that was able to explain and predict the experimental data and this formalism led,beyond the very intention of the men who constructed it,to this conceptual revolution.Then,the old problem of the relatio among the observer and the obs ed object.discussed fo e seen from a new.completely different perspective. Once established,quantum mechanics became a wonderful and extremely powerful the diffe ials,the whole chemistry,became f the first bpredicted fom the theryind not only phenmenloa le deduced from experiments.The technological discovery that shaped the second half of last century,the transistor(i.e.the basis of all the modern electronics and computers)could not have been invented without a deep command of quantum mechanics. The advances of recent years have not only concentrated on the problems of interpreta- tion that could be(wrongly)dismissed as metaphysical by some people,considering them
Introduction Why yet another book on quantum mechanics? Quantum mechanics was born in the first quarter of the twentieth century and has received an enormous number of theoretical and experimental confirmations over the years. It is considered to be the fundamental physical paradigm, and has a wide range of applications, from cosmology to chemistry, and from biology to information sciences. It is one of the greatest intellectual achievements of the past century. As an effect of its invention, the very concept of physical reality was changed, and “observation,” “measurement,” “prediction,” and “state of the system” acquired a new and deeper meaning. Probability was not unknown in physics: it was introduced by Boltzmann in order to control the behavior of a system with a very large number of particles. It was the missing concept in order to understand the thermodynamics of macroscopic bodies, but the structure of the physical laws remained still deterministic. The introduction of probability was needed as a consequence of our lack of knowledge of the initial conditions of the system and of our inability to solve an enormous number of coupled non-linear differential equations. In quantum mechanics, the tune is different: if we have 106 radioactive atoms no intrinsic unknown variables decide which of them will decay first. What we observe experimentally seems to be an irreducible random process. The original explanation of this phenomenon in quantum mechanics was rather unexpected. All atoms have the same probability of having decayed: only when we observe the system do we select which atoms have decayed in the past. In spite of the fact that this solution seems to be in contrast with common sense, it is the only possible one in the framework of the conventional interpretation of quantum mechanics. Heisenberg, de Broglie, Pauli, Dirac, and many others invented a formalism that was able to explain and predict the experimental data and this formalism led, beyond the very intention of the men who constructed it, to this conceptual revolution. Then, the old problem of the relations among the observer and the observed object, discussed for centuries by philosophers, had a unexpected evolution and now it must be seen from a new, completely different perspective. Once established, quantum mechanics became a wonderful and extremely powerful tool. The properties of the different materials, the whole chemistry, became for the first time objects that could be predicted from the theory and not only phenomenological rules deduced from experiments. The technological discovery that shaped the second half of last century, the transistor (i.e. the basis of all the modern electronics and computers) could not have been invented without a deep command of quantum mechanics. The advances of recent years have not only concentrated on the problems of interpretation that could be (wrongly) dismissed as metaphysical by some people, considering them
2 Introduction to be beyond experimental tests.In the last 30 years,the whole complex of problems con- nected to quantum mechanics and the meaning of measurements started to be studied from new perspective.Real not ony Gedanker experiments began to be done on some of the most elusive properties of quantum mechanics,i.e.the existence of correlations among spatially separated systems that could not be explained using the traditional concept of probability.The precise quantum mechanical meaning of measurements started to be ana- lyzed in a more refined way (e.g.quantum non-demolition measurements were introduced) and various concepts from statistical mechanics and other fields of physics began to be used. This is not only an academic or philosophical problem.The possibility of construct- ing a quan um computer,which would improve the speed of present day c mputers by ar incredible factor,is deeply rooted in these achievements.It is now clear that a quantum computer can solve problems,which on conventional computers take a time exploding as exponent of some parameter (e.g.the factorization into primes of a number of length N).in a time which is only a polynomial in N.The technical problems to be ove come in constructing a quantum computer are not easy to solve,but this result has a high conceptual status,telling us how deeply quantum mechanics differs from classical mechanics.Another quantum-information puzzling phenomenon,ie.teleportation,has been e proved experimemtally to eiand it isa very active area of experimenta res The arguments above explain why this new situation imposes the necessity to treat this field in a new way.The idea of writing this book came to one of us in 2000:it has taken more than eight years to accomplish this challenge Outline The book is divided into four parts: I Basic features of quantum mechanics Part I deals with the basic framework of the theory and the reasons for its birth.Fur thermore,starting from the fundamental principles,it explains the nature of quantum observables and states,and presents the dynamics of quantum systems and its main examples. ⅡMore advanced topics In Part II we introduce angular momentum,spin,identical particles,and symmetries. Moreover,we give a special emphasis to the quantum theory of measurement. IⅢMatter and light We devote Part II to some of the most important applications of quantum theory: approximation methods and perturbation theory,the hydrogen atom,simple molecules. and quantum optics. IV Quantum information:state and correlations Finally,we deal with the most recent topics:the quantum theory of open systems,state
2 Introduction to be beyond experimental tests. In the last 30 years, the whole complex of problems connected to quantum mechanics and the meaning of measurements started to be studied from a new perspective. Real, not only Gedanken experiments began to be done on some of the most elusive properties of quantum mechanics, i.e. the existence of correlations among spatially separated systems that could not be explained using the traditional concept of probability. The precise quantum mechanical meaning of measurements started to be analyzed in a more refined way (e.g. quantum non-demolition measurements were introduced) and various concepts from statistical mechanics and other fields of physics began to be used. This is not only an academic or philosophical problem. The possibility of constructing a quantum computer, which would improve the speed of present day computers by an incredible factor, is deeply rooted in these achievements. It is now clear that a quantum computer can solve problems, which on conventional computers take a time exploding as exponent of some parameter (e.g. the factorization into primes of a number of length N), in a time which is only a polynomial in N. The technical problems to be overcome in constructing a quantum computer are not easy to solve, but this result has a high conceptual status, telling us how deeply quantum mechanics differs from classical mechanics. Another quantum-information puzzling phenomenon, i.e. teleportation, has been recently proved experimentally to exist and it is a very active area of experimental research. The arguments above explain why this new situation imposes the necessity to treat this field in a new way. The idea of writing this book came to one of us in 2000; it has taken more than eight years to accomplish this challenge. Outline The book is divided into four parts: I Basic features of quantum mechanics Part I deals with the basic framework of the theory and the reasons for its birth. Furthermore, starting from the fundamental principles, it explains the nature of quantum observables and states, and presents the dynamics of quantum systems and its main examples. II More advanced topics In Part II we introduce angular momentum, spin, identical particles, and symmetries. Moreover, we give a special emphasis to the quantum theory of measurement. III Matter and light We devote Part III to some of the most important applications of quantum theory: approximation methods and perturbation theory, the hydrogen atom, simple molecules, and quantum optics. IV Quantum information: state and correlations Finally, we deal with the most recent topics: the quantum theory of open systems, state
3 Methodology measurement,quantum correlations and non-locality,and quantum information and computation. In this book there is material for four one-semester courses.It may also serve as a guide for short course or tuto rials on specific and more advanced topics Methodology (1)In our exposition we have tried to follow a "logical"order,starting from the principles of classical mechanics,the need of quantum mechanics with its fun- Then, ward to the dynamics and to more sophisticated stuff.applications,and special areas. (2)We have made an effort to use a pedagogical style.In particular: (i)We prove or let the reader prove (through problems that are solved on the book' website)practically all our results:we try to lead the reader to reach them step by step from previous ones. (ii)We have made the choice to present Dirac algebra and operatorial formalism from the very beginning.instead of starting with the wave-function formalism.The lat ter is obtained naturally as a particular representation of the former.This approach has the advantage that we are not obliged to repeat the fundamental mathematical tools of the theory. (iii)We present our main principles and results in a pragm tic way.trying to intro- duce new concepts on the basis of experimental evidence,rather than in an axiomatic way.which may result cumbersome for readers who are learning quantum mechanics. (iv)We have made an effort to pay particular attention to cross-references in order to help the (inexpert)reader to quickly find the necessary background and related problems. (3)We have taken into account some of the most recent developments at theoretical and experimental level,as well as with respect t to technological applicat ns:quantun optics,quantum information,quantum non-locality,state measurement,etc (4)We believe that measurement theory constitutes a fundamental part of quan- tum mechanics.As a consequence we have devoted an entire chapter to this sue. (5)When necessary.we have emphasized interpretational as well as historical issues.such as complementarity,measurement,nature of quantum states,and so on. (6)We propose to the reader a large number of problems(more than 300).and the less nes (about half of the m)are solved in a cal way (7)From time to time.we have chosen to treat special topics in"boxes
3 Methodology measurement, quantum correlations and non-locality, and quantum information and computation. In this book there is material for four one-semester courses. It may also serve as a guide for short courses or tutorials on specific and more advanced topics. Methodology (1) In our exposition we have tried to follow a “logical” order, starting from the principles of classical mechanics, the need of quantum mechanics with its fundamental assumptions (superposition, complementarity, and uncertainty principles). Then, we present the main features of observables and states, before going forward to the dynamics and to more sophisticated stuff, applications, and special areas. (2) We have made an effort to use a pedagogical style. In particular: (i) We prove or let the reader prove (through problems that are solved on the book’s website) practically all our results: we try to lead the reader to reach them step by step from previous ones. (ii) We have made the choice to present Dirac algebra and operatorial formalism from the very beginning, instead of starting with the wave-function formalism. The latter is obtained naturally as a particular representation of the former. This approach has the advantage that we are not obliged to repeat the fundamental mathematical tools of the theory. (iii) We present our main principles and results in a pragmatic way, trying to introduce new concepts on the basis of experimental evidence, rather than in an axiomatic way, which may result cumbersome for readers who are learning quantum mechanics. (iv) We have made an effort to pay particular attention to cross-references in order to help the (inexpert) reader to quickly find the necessary background and related problems. (3) We have taken into account some of the most recent developments at theoretical and experimental level, as well as with respect to technological applications: quantum optics, quantum information, quantum non-locality, state measurement, etc. (4) We believe that measurement theory constitutes a fundamental part of quantum mechanics. As a consequence we have devoted an entire chapter to this issue. (5) When necessary, we have emphasized interpretational as well as historical issues, such as complementarity, measurement, nature of quantum states, and so on. (6) We propose to the reader a large number of problems (more than 300), and the less trivial ones (about half of them) are solved in a pedagogical way. (7) From time to time, we have chosen to treat special topics in “boxes.”
Introduction Apparatus Besides a large number of cross-references,we also list the following tools: (1)The book contains 200 figures among drawings,photographs,and graphs,distributed in all chapters (a sample of color figures can be found on the book's website).We graphic very imporant aspect of figure captions are particularly accurate and often self-contained (2)The book contains an extensive bibliography (almost 600 entries,most of which are papers,and other publications. (3)The book contains full,accurate,and comprehensive indices(table of contents,subject index,author index,list of figures,list of tables,list of abbreviations,list of symbols list of boxes.list of theorems.)toether withasummary of the main concepts at the end of each chapter. Readers This book is addressed to people who want to learn quantum mechanics or deepen their knowledge of the subject.The requirement for understanding the book is a knowledge of calculus ectorial a mechanics The book isprimarily intended for third-and fourth-year physics.However,it may also be used for other curricula(such as mathematics,engineer- ing.chemistry.computer sciences.etc.)Furthermore.it may well be used as a reference book for graduate students,researchers.and practitioners who want a rapid ac cess to spe cific topics.To this purpose the extensive indices and lists are of great help.It may even serve as an introduction to specific areas(quantum optics,entanglement,quantum informa- tion,measurement theory)for experienced professionals from different fields of physics Finally,the book may prove useful for scientists of other disciplines who want to learr something about quantum mechanics Acknowledgements We would like to thank Enrico Beltrametti,Michael Heller,Artur Ekert,and Willem de Muynck for a critical reading of part of the manuscript.We would also like to warmly thank all those persons-friends,colleagues,teachers,students-who helped and influenced us in the writing of the book Finally,we dedicate this book to our families for their continual support and love,and for tolerating our many absences during the completion of the book
4 Introduction Apparatus Besides a large number of cross-references, we also list the following tools: (1) The book contains 200 figures among drawings, photographs, and graphs, distributed in all chapters (a sample of color figures can be found on the book’s website). We consider this graphic support a very important aspect of our exposition. In this context, figure captions are particularly accurate and often self-contained. (2) The book contains an extensive bibliography (almost 600 entries, most of which are quoted in the text) and a “Further reading” section at the end of each chapter. Name of authors in italics in citations refer to books, those in roman text refer to journals, papers, and other publications. (3) The book contains full, accurate, and comprehensive indices (table of contents, subject index, author index, list of figures, list of tables, list of abbreviations, list of symbols, list of boxes, list of theorems, definitions, and so on) together with a summary of the main concepts at the end of each chapter. Readers This book is addressed to people who want to learn quantum mechanics or deepen their knowledge of the subject. The requirement for understanding the book is a knowledge of calculus, vectorial analysis, operator algebra, and classical mechanics. The book is primarily intended for third- and fourth-year undergraduate students in physics. However, it may also be used for other curricula (such as mathematics, engineering, chemistry, computer sciences, etc.). Furthermore, it may well be used as a reference book for graduate students, researchers, and practitioners, who want a rapid access to specific topics. To this purpose the extensive indices and lists are of great help. It may even serve as an introduction to specific areas (quantum optics, entanglement, quantum information, measurement theory) for experienced professionals from different fields of physics. Finally, the book may prove useful for scientists of other disciplines who want to learn something about quantum mechanics. Acknowledgements We would like to thank Enrico Beltrametti, Michael Heller, Artur Ekert, and Willem de Muynck for a critical reading of part of the manuscript. We would also like to warmly thank all those persons – friends, colleagues, teachers, students – who helped and influenced us in the writing of the book. Finally, we dedicate this book to our families for their continual support and love, and for tolerating our many absences during the completion of the book.
PARTI BASIC FEATURES OF QUANTUM MECHANICS
PARTI BASIC FEATURES OF QUANTUM MECHANICS