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Moses Fayngold, Vadim Fayngold

Quantum Mechanics and Quantum Information

PHYSICS TEXTBOOK

Quantum

Mechanics

and Quantum

Information

ISBN 978-3-527-40647-0

Fayngold • Fayngold

www.wiley-vch.de

Moses Fayngold graduated from the State University of Samarkand and got his PhD at the Nuclear Research Institute of Academy of Science in Uzbekistan (former USSR). He has combined teaching and research in colleges of USSR and USA, most recently as a Senior University Lectur-er at the Physics Department of the New Jersey Institute of Technology. He has lectured on Quantum Mech-anics and Special Relativity to both undergraduate and graduate students. His research interests and areas of activity include Special and General Relativity, Quantum Mechanics, Optics and optical imaging, particle scattering and propagation in peri-odic structures.

Vadim Fayngold holds two degrees – M.S. in Physics and B.S. in Computer Science. While working as a research assistant at the Department of Com-puter Engineering (Polytechnic Uni-versity, New York), he focused on computer simulation of complex processes in fluid dynamics. The combined expertise he developed there has spurred his interest in the Quantum Information theory. Vadim came to the idea of writing this book while working on computer animations of various relativistic and quantum-mechanical phenomena.

Alongside a thorough definition of basic concepts and their interrelations, backed by numerous examples, this textbook features a rare discussion of quantum mechanics and informa-tion theory combined in one text. It deals with important topics hardly found in regular textbooks, including the Robertson-Schrödinger relation, incompatibility between angle and angular momentum, “dispersed indeterminacy”, interaction-free mea-surements, “submissive quantum mechanics”, and many others. With its in-depth discussion of key concepts complete with problems and exercises, this book is poised to become the standard textbook for advanced undergraduate and begin-ning graduate quantum mechanics courses and an essential reference for physics students and physics professionals.

From the contents:

• Embryonic Quantum Mechanics: Basic Features • Playing with the Amplitudes • Representations and the Hilbert Space• Angular Momentum• Evolution of Quantum States• Indeterminacy Revisited • “Submissive” Quantum Mechanics• Quantum Statistics• Second Quantization• Quantum Mechanics and Measurements• Quantum Non-Locality• Quantum Measurements and POVMs• Quantum Information• Quantum Gates• Quantum Key Distribution

Moses Fayngold and Vadim

Fayngold

Quantum Mechanics andQuantum Information

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Moses Fayngold and Vadim Fayngold

Quantum Mechanics and QuantumInformation

A Guide through the Quantum World

The Authors

Moses FayngoldNJITDept. of Physi csNewark, NJ 07102-1982fayngold@mai laps.org

Vadim Fayngoldvadim.resear [email protected]

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Contents

Preface XIIIAbbreviations and Notations XIX

1 The Failure of Classical Physics 11.1 Blackbody Radiation 11.2 Heat Capacity 41.3 The Photoelectric Effect 91.4 Atoms and Their Spectra 121.5 The Double-Slit Experiment 14

Problem 19References 19

2 The First Steps into the Unknown 212.1 The BBR and Planck’s Formula 212.2 Einstein’s Light Quanta and BBR 242.2.1 Discussion 272.3 PEE Revisited 302.4 The Third Breakthrough: de Broglie Waves 312.4.1 Exercise 33

Problems 35References 35

3 Embryonic Quantum Mechanics: Basic Features 373.1 A Glimpse of the New Realm 373.2 Quantum-Mechanical Superposition of States 393.3 What Is Waving There (the Meaning of the C-Function)? 423.4 Observables and Their Operators 473.5 Quantum-Mechanical Indeterminacy 493.6 Indeterminacy and the World 533.7 Quantum Entanglement and Nonlocality 583.8 Quantum-Mechanical Phase Space 623.9 Determinism and Causality in Quantum World 63

jV

3.9.1 Discussion 63Problems 66References 66

4 Playing with the Amplitudes 694.1 Composition of Amplitudes 694.2 Double Slit Revised I 744.3 Double Slit Revised II 774.4 Neutron Scattering in Crystals 784.5 Bosonic and Fermionic States 814.6 Path Integrals 89


5 Basic Features and Mathematical Structure of QM 955.1 Observables: the Domain of Classical and Quantum

Mechanics 955.2 Quantum-Mechanical Operators 975.3 Algebra of Operators 1005.4 Eigenvalues and Eigenstates 1025.5 Orthogonality of Eigenstates 1075.6 The Robertson–Schr€odinger Relation 1105.7 The Wave Function and Measurements (Discussion) 112


6 Representations and the Hilbert Space 1196.1 Various Faces of a State Function 1196.2 Unitary Transformations 1216.3 Operators in the Matrix Form 1256.4 The Hilbert Space 1296.5 Operations in the Hilbert Space 1356.6 Nonorthogonal States 142


7 Angular Momentum 1497.1 Orbital and Spin Angular Momenta 1497.2 The Eigenstates and Eigenvalues of L̂ 1517.3 Operator L̂ and Its Commutation Properties 1547.4 Spin as an Intrinsic Angular Momentum 1647.5 Angular Momentum of a Compound System 1837.6 Spherical Harmonics 188


VIj Contents

8 The Schr€odinger Equation 1998.1 The Schr€odinger Equation 1998.2 State Function and the Continuity Equation 2008.3 Separation of Temporal and Spatial Variables: Stationary

States 2038.4 The Helmholtz Equation and Dispersion Equation for a Free

Particle 2058.5 Separation of Spatial Variables and the Radial Schr€odinger

Equation 2078.6 Superposition of Degenerate States 2098.7 Phase Velocity and Group Velocity 2128.8 de Broglie’s Waves Revised 2188.9 The Schr€odinger Equation in an Arbitrary Basis 222


9 Applications to Simple Systems: One Dimension 2279.1 A Quasi-Free Particle 2279.2 Potential Threshold 2329.3 Tunneling through a Potential Barrier 2369.4 Cold Emission 2419.5 Potential Well 2449.6 Quantum Oscillator 2499.7 Oscillator in the E-Representation 2549.8 The Origin of Energy Bands 2579.9 Periodic Structures 260


10 Three-Dimensional Systems 27310.1 A Particle in a 3D Box 27310.2 A Free Particle in 3D (Spherical Coordinates) 27410.2.1 Discussion 27710.3 Some Properties of Solutions in Spherically Symmetric

Potential 27710.4 Spherical Potential Well 27810.5 States in the Coulomb Field and a Hydrogen Atom 28110.6 Atomic Currents 28710.7 Periodic Table 290


11 Evolution of Quantum States 29511.1 The Time Evolution Operator 29511.2 Evolution of Operators 299

Contents jVII

11.3 Spreading of a Gaussian Packet 30111.4 The B-Factor and Evolution of an Arbitrary State 30311.5 The Fraudulent Life of an “Illegal” Spike 30611.6 Jinnee Out of the Box 31111.7 Inadequacy of Nonrelativistic Approximation in Description

of Evolving Discontinuous States 31511.7.1 Discussion 31611.8 Quasi-Stationary States 31711.8.1 Discussion 32311.9 3D Barrier and Quasi-Stationary States 32411.10 The Theory of Particle Decay 32711.11 Particle–Antiparticle Oscillations 33111.11.1 Discussion 33711.12 A Watched Pot Never Boils (Quantum Zeno Effect) 33911.13 A Watched Pot Boils Faster 344


12 Quantum Ensembles 35512.1 Pure Ensembles 35512.2 Mixtures 35612.3 The Density Operator 35812.4 Time Evolution of the Density Operator 36612.5 Composite Systems 368


13 Indeterminacy Revisited 37713.1 Indeterminacy Under Scrutiny 37713.2 The Heisenberg Inequality Revised 38013.3 The Indeterminacy of Angular Momentum 38213.4 The Robertson–Schr€odinger Relation Revised 38413.5 The N–f Indeterminacy 38813.6 Dispersed Indeterminacy 390


14 Quantum Mechanics and Classical Mechanics 39714.1 Relationship between Quantum and Classical Mechanics 39714.2 QM and Optics 40014.3 The Quasi-Classical State Function 40114.4 The WKB Approximation 40414.5 The Bohr–Sommerfeld Quantization Rules 406


VIIIj Contents

15 Two-State Systems 41115.1 Double Potential Well 41115.2 The Ammonium Molecule 41515.3 Qubits Introduced 419

Problem 422References 422

16 Charge in Magnetic Field 42316.1 A Charged Particle in EM Field 42316.2 The Continuity Equation in EM Field 42516.3 Origin of the A-Momentum 42716.4 Charge in Magnetic Field 42916.5 Spin Precession 43216.6 The Aharonov–Bohm Effect 43716.6.1 Discussion 44116.7 The Zeeman Effect 442


17 Perturbations 44717.1 Stationary Perturbation Theory 44717.1.1 Discussion 45017.2 Asymptotic Perturbations 45517.3 Perturbations and Degeneracy 45717.4 Symmetry, Degeneracy, and Perturbations 46017.5 The Stark Effect 46217.6 Time-Dependent Perturbations 465


18 Light–Matter Interactions 47318.1 Optical Transitions 47318.2 Dipole Radiation 47418.3 Selection Rules 47718.3.1 Oscillator 47818.3.2 Hydrogen-Like Atom 478

Problems 480Reference 480

19 Scattering 48119.1 QM Description of Scattering 48119.2 Stationary Scattering 48719.3 Scattering Matrix and the Optical Theorem 49019.4 Diffraction Scattering 49419.5 Resonant Scattering 498

Contents jIX

19.6 The Born Approximation 501Problems 504References 505

20 Submissive Quantum Mechanics 50720.1 The Inverse Problem 50720.2 Playing with Quantum States 50920.3 Playing with Evolution: Discussion 514


21 Quantum Statistics 52521.1 Bosons and Fermions: The Exclusion Principle 52521.1.1 Discussion 53121.2 Planck and Einstein Again 54021.3 BBR Again 54221.4 Lasers and Masers 543


22 Second Quantization 54722.1 Quantum Oscillator Revisited 54722.2 Creation and Annihilation Operators: Bosons 54822.3 Creation and Annihilation Operators: Fermions 552


23 Quantum Mechanics and Measurements 55723.1 Collapse or Explosion? 55723.2 “Schr€odinger’s Cat” and Classical Limits of QM 56323.3 Von Neumann’s Measurement Scheme 57123.3.1 Discussion 57523.4 Quantum Information and Measurements 57823.5 Interaction-Free Measurements: Quantum Seeing in the Dark 58623.6 QM and the Time Arrow 593


24 Quantum Nonlocality 59924.1 Entangled Superpositions I 59924.2 Entangled Superpositions II 60124.2.1 Discussion 60424.3 Quantum Teleportation 60424.4 The “No-Cloning” Theorem 607

Xj Contents

24.5 Hidden Variables and Bell’s Theorem 61324.6 Bell-State Measurements 61924.7 QM and the Failure of FTL Proposals 62724.8 Do Lasers Violate the No-Cloning Theorem? 62824.9 Imperfect Cloning 63624.10 The FLASH Proposal and Quantum Compounds 643


25 Quantum Measurements and POVMs 65125.1 Projection Operator and Its Properties 65125.2 Projective Measurements 65525.3 POVMs 65825.4 POVM as a Generalized Measurement 66425.5 POVM Examples 66625.6 Discrimination of Two Pure States 67025.7 Neumark’s Theorem 68125.8 How to Implement a Given POVM 68625.9 Comparison of States and Mixtures 69525.10 Generalized Measurements 697


26 Quantum Information 70326.1 Deterministic Information and Shannon Entropy 70326.2 von Neumann Entropy 70926.3 Conditional Probability and Bayes’s Theorem 71126.4 KL Divergence 71626.5 Mutual Information 71726.6 R�enyi Entropy 71926.7 Joint and Conditional Renyi Entropy 72126.8 Universal Hashing 72626.9 The Holevo Bound 73126.10 Entropy of Entanglement 733


27 Quantum Gates 73727.1 Truth Tables 73727.2 Quantum Logic Gates 74127.3 Shor’s Algorithm 746


Contents jXI

28 Quantum Key Distribution 75328.1 Quantum Key Distribution (QKD) with EPR 75328.2 BB84 Protocol 75828.3 QKD as Communication Over a Channel 76628.4 Postprocessing of the Key 76928.5 B92 Protocol 77628.6 Experimental Implementation of QKD Schemes 77928.7 Advanced Eavesdropping Strategies 788


Appendix A: Classical Oscillator 795Reference 799

Appendix B: Delta Function 801Reference 807

Appendix C: Representation of Observables by Operators 809Appendix D: Elements of Matrix Algebra 813Appendix E: Eigenfunctions and Eigenvalues of the Orbital Angular

Momentum Operator 817Appendix F: Hermite Polynomials 821Appendix G: Solutions of the Radial Schr€odinger Equation in Free

Space 825Appendix H: Bound States in the Coulomb Field 827Reference 829

Index 831

XIIj Contents

Preface

Quantum mechanics (QM) is one of the cornerstones of today’s physics. Havingemerged more than a century ago, it now forms the basis of all modern technology.And yet, while being long established as a science with astounding explanatoryand predictive power, it still remains the arena of lively debates about some of itsbasic concepts.

This book on quantum mechanics and quantum information (QI) differs fromstandard texts in three respects. First, it combines QM and some elements of QI inone volume. Second, it does not contain those important applications of QM(helium atom, hydrogen molecule, condensed matter) that can be found almostin any other textbook. Third, it contains important topics (Chapters 13 and 20,Sections 6.6, 11.4–11.7, 24.1–24.2, 24.6, among others) that are not covered in mostcurrent textbooks. We believe that these topics are essential for a better under-standing of such fundamental concepts as quantum evolution, quantum-mechani-cal indeterminacy, superposition of states, quantum entanglement, and the Hilbertspace and operations therein.

The book began as an attempt by one of the authors to account to himself for someaspects of QM that continued to intrigue him since his graduate school days. As hisresearch notes grew he came up with the idea of developing them further into atextbook or monograph. The other author, who, by his own admission, used to thinkof himself as something of an expert in QM, was not initially impressed by the idea,citing a huge number of excellent contemporary presentations of the subject.Gradually, however, as he grew involved in discussing the issues brought up byhis younger colleague, he found it hard to explain some of them even to himself.Moreover, to his surprise, in many instances he could not find satisfactory explan-ations even in those texts he had previously considered to contain authoritativeaccounts on the subject.

Our original plan was to produce a short axiomatic formulation of the logicalstructure of quantum theory in the spirit of von Neumann’s famous book [1]. Itturned out, however, that the key ideas of QM are not so easy to compartmentalize.There were, on the one hand, too many interrelations between apparently differenttopics, and conversely, many apparently similar topics branched out into two ormore different ones. As a result, we ended up adding lots of new examples andeventually reshaping our text into an entirely different framework. In hindsight, this

jXIII

evolution appears to resemble the way Ziman wrote his remarkable book on QM as ameans of self-education [2].

The resulting book attempts to address the needs of students struggling tounderstand the weird world of quantum physics. We tried to find the optimalbalance between mathematical rigor and clarity of presentation. There is always atemptation to “go easy on math”, to “get to the subject matter quicker.” We, however,feel that mathematical formalism must be treated with proper respect. While thehigh number of equations may understandably intimidate some readers, there isstill no realistic way around it. On the other hand, replacing explanations of conceptsentirely by formulas, expecting them to speak for themselves, may produce only adangerous illusion of understanding. Hence, the way we see this book: a textbookwith a human face, a guidebook by necessity abundant with equations – still not aneasy reading! – but also with a thorough explanation of underlying physics. This isprecisely what we feel is necessary for its intended audience – mostly collegestudents.

There is an ongoing discussion of what would be the best way to introduce QM:doing it gradually, starting from some familiar classical concepts and showing theirfailure in quantum domain, or using the “shock therapy” – immediately placing thereader face to face with the most paradoxical quantum phenomena and force him/her to accept from the outset the entirely new concepts and mathematical toolsnecessary for their description.

We do not think that this question has a unique answer: the choice of the mostsuitable approach depends on many factors. The “shock therapy” method isefficient, straightforward, and saves lots of time (not to mention decreasing thesize of the textbook!) when aimed at the curious and bright student – but onlybecause such a student, precisely due to his/her curiosity, has already read somematerial on the topic and is thus prepared to absorb new information on a higherlevel. We, however, aimed at a maximally broad student audience, without assumingany preliminary acquaintance with quantum ideas. The only (but necessary!)prerequisites are the knowledge of standard classical topics – mechanics, electro-magnetism (EM), and thermodynamics in physics; and linear algebra with matricesand operations on them, calculus, Fourier transforms, and complex variables inmathematics. Accordingly, our book uses gradual introduction of new ideas. Tomake the transition from classical to quantum-mechanical concepts smoother forthe first-time reader, it starts with some familiar effects whose features demonstratethe inadequacy of classical concepts for their description. The first chapter shows thefailure of classical physics in explaining such phenomena as the blackbody radiation(BBR), heat capacity, and photoeffect. We tried to do it by way of pointing to QM as anecessary new level in the description of the world. In this respect, our book falls outof step with many modern presentations that introduce the Schr€odinger equation orthe Hilbert space from the very beginning. We believe that gradual approach offers agood start in the journey through the quantum world.

QM is very different from other branches of physics: it is not just formal orcounterintuitive – it is, in addition, not easy to visualize. Therefore, the task ofpresenting QM to students poses a challenge even for the most talented pedagogues.

XIVj Preface

The obstacles are very serious: apart from rather abstract mathematics involved,there is still no consensus among physicists about the exact meaning of some basicconcepts such as wave function or the process of measurement [3–5]. The lack ofself-explanatory models appealing to our “classical” intuition provides fertile soil forpotential confusions and misunderstandings. A slightly casual use of technicallanguage might be permissible in other areas of physics where the visual modelitself provides guidance to the reader; but it often proves catastrophic for thebeginning student of QM. For instance (getting back to our own experience) oneof the authors, while still a student, found the representation theory almostincomprehensible. Later on, he realized the cause of his troubles: none of themany textbooks he used ever explicitly mentioned one seemingly humble detail,namely, that the eigenstates used as the basis may themselves be given in this or thatrepresentation. The other author, on the contrary, had initially found the represen-tation theory to be fairly easy, only to stumble upon the same block later when tryingto explain the subject to his partner.

When carefully considering the logical structure of QM, one can identify a fairnumber of such points where it is easy to get misled or to inadvertently mislead thereader. Examples of potential snags include even such apparently well-understoodconcepts as the uncertainty (actually, indeterminacy) principle or the definition ofquantum ensembles. In this book, we deliberately zero in on those potential pitfallsthat we were able to notice by looking at them carefully and sometimes by fallinginto them ourselves. Sometimes we devote whole sections to the correspondingdiscussions, as, for instance, when probing the connection between the azimuthalangle and angular momentum (Section 13.4).

There has been a noticeable trend to present QM as an already fully accomplishedconstruction that provides unambiguous answers to all pertinent questions. In thisbook, we have tried to show QM as a vibrant, still-developing science potentiallycapable of radically new insights and important reconstructions. For this reason, wedid not shy away from the issues that up to now remain in the center of lively debatesamong physicists: the meaning of quantum reality, the role of the observer, thesearch for hidden parameters, and the increasingly important topic of quantumnonlocality [3,6–8]. We also discuss some important developments such as quantuminformation and its processing [9–12], interaction-free measurements [13,14], andno-cloning theorem and its implications [15–18]. We devote a special chapter to anew aspect of QM, practically unknown to the broad audience: the so-called “inverseproblem” and manipulation of quantum states by local changes of interactionpotential or of the environment, which is especially relevant to the physics ofnanostructures [19,20]. This is so important for both practical applications anddeveloping a new dimension in our “quantum intuition” that it should become apart of all standard quantum textbooks.

Unfortunately, everything comes at a price. As mentioned in the beginning,focusing on some conceptual issues has left little room for some applications ofquantum theory. In such cases, the reader can find the necessary information in theabundant standard texts [21,22], and the basic hope is that he/she will be alreadysufficiently equipped for reading them.

Preface jXV

This book describes mostly nonrelativistic QM and has a clear structure. Aftershowing the limitations of classical physics, we discuss the new concepts necessaryto explain the observed phenomena. Then we formulate de Broglie’s hypothesis,followed by two postulates: the probabilistic nature of the wave function and thesuperposition principle. It is shown then that the whole mathematical framework ofthe theory follows from these two postulates.

We introduce the idea of observables and their operators, eigenstates and eigen-values, and quantum-mechanical indeterminacy with some simple illustrating exam-ples in Chapter 3. We do not restrict to the initial concept of the wave function as afunction of r and t, and show as early as possible that it could be a function of otherobservables. The ideas of representation theory are introduced at the first opportunity,and it is emphasized from the beginning that the wave function “has many faces.”

We made every effort to bridge the gulf between the differential and matrix formsof an operator by making it clear that, for instance, Hermitian operators are thosethat have a Hermitian matrix. We emphasize that going from one form to anotherusually means changing the representation – another potential source of confusion.

We discuss the superposition principle in more detail than other textbooks. Inparticular, we emphasize that the sum of squares of probability amplitudes gives 1only if the corresponding eigenfunctions are square integrable and the basis wasorthogonalized. We introduce nonorthogonal bases as early as in Chapter 6 andderive the normalization condition for this more general case.

We did not rush to present the Schr€odinger equation as soon as possible, all themore so that it is only one nonrelativistic limit of more general relativistic waveequations, which are different for particles with different spins. It expresses, amongother things, the law of conservation of energy, and it follows naturally (Chapter 8)once the energy, momentum, and the angular momentum operators are introduced.

We feel that we’ve paid proper attention to quantum ensembles (Chapter 12). Theidea of the pure and mixed ensembles is essential for the theory of measurements.On the other hand, we make it explicitly clear that not every collection of particles“in the same physical situation” forms a statistical mixture of pure states astraditionally defined, due to the possibilities of local interaction and entanglement.

In many cases, the sequence of topics reminds a helix: we return to some of themin later chapters. But each such new discussion of the same topic reveals some newaspects, which helps to achieve a better understanding and gain a deeper insight intothe problem.

We are not aware of other textbooks that use both the “old” and Dirac’s notation.We cover both notations and use them interchangeably. There is a potential benefitto this approach, as the students who come to class after having been exposed to adifferent notation in an introductory course are often put at a relative disadvantage.

We want to thank Nick Herbert and Vladimir Tsifrinovich for valuable discussions,Boris Zakhariev for acquainting us with the new aspects of the inverse problem, andFrancesco De Martini for sharing with us his latest ideas on the problem of signalexchange between entangled systems. We enjoyed working with the Project EditorNina Stadthaus during preparation of the manuscript and are grateful to her for herpatience in dealing with numerous delays at the later stages of the work.

XVIj Preface

References

1 von Neumann, J. (1955) MathematicalFoundations of Quantum Mechanics,Princeton University Press, Princeton, NJ.

2 Ziman, J.M. (1969) Elements of AdvancedQuantum Theory, Cambridge UniversityPress, Cambridge.

3 Herbert, N. (1987) Quantum Reality:Beyond New Physics, Anchor Books,New York.

4 Bell, J.S. (2001) The Foundations ofQuantum Mechanics, World Scientific,Singapore.

5 Bell, J.S. (1990) Against measurement,in Sixty Two Years of Uncertainty,(ed A. Miller), Plenum Press, New York.

6 Gribbin, J. (1995) Schrodinger’s Kittens andthe Search for Reality, Little, Brown andCompany, Boston, MA.

7 Greenstein, G. and Zajonc, A.G. (1997)The Quantum Challenge: Modern Researchon the Foundations of Quantum Mechanics,Jones & Bartlett Publishers, Boston, MA.

8 Kafatos, M. and Nadeau, R. (1990) TheConscious Universe: Part and Whole inModern Physical Theory, Springer,New York.

9 Berman, G., Doolen, G., Mainieri, R., andTsifrinovich, V. (1998) Introduction toQuantum Computers, World Scientific,Singapore.

10 Benenti, G., Casati, G., and Strini, G.(2004) Principles of Quantum Computationand Information, World Scientific,Singapore.

11 Bouwmeester, D., Ekert, A., and Zeilinger,A. (2000) The Physics of QuantumInformation: Quantum Cryptography,Quantum Teleportation, QuantumComputation, Springer, Berlin.

12 Nielsen, M.A. and Chuang, I.L. (2000)Quantum Computation and QuantumInformation, Cambridge University Press,Cambridge.

13 Elitzur, A.C. and Vaidman, L. (1993)Quantum-mechanical interaction-freemeasurements. Found. Phys., 23 (7),987–997.

14 Kwiat, P., Weinfurter, H., and Zeilinger,A. (1996) Quantum seeing in the dark. Sci.Am., 275, 72–78.

15 Herbert, N. (1982) FLASH – asuperluminal communicator based upona new kind of quantum measurement.Found. Phys., 12, 1171.

16 Wooters, W.K. and Zurek, W.H. (1982) Asingle quantum cannot be cloned. Nature,299, 802.

17 Dieks, D. (1982) Communication by EPRdevices. Phys. Lett. A, 92 (6), 271.

18 van Enk, S.J. (1998) No-cloning andsuperluminal signaling. arXiv: quant-ph/9803030v1.

19 Agranovich, V.M. and Marchenko, V.A.(1960) Inverse Scattering Problem, KharkovUniversity, Kharkov (English edition:Gordon and Breach, New York, 1963).

20 Zakhariev, B.N. and Chabanov, V.M.(2007) Submissive Quantum Mechanics:New Status of the Theory in Inverse ProblemApproach, Nova Science Publishers, Inc.,New York.

21 Blokhintsev, D.I. (1964) Principles ofQuantum Mechanics, Allyn & Bacon,Boston, MA.

22 Landau, L. and Lifshits, E. (1965)Quantum Mechanics: Non-RelativisticTheory, 2nd edn, Pergamon Press,Oxford.

References jXVII

Abbreviations and Notations

Abbreviations

BBR blackbody radiationBS beam splitterCM classical mechanicsCM center of massCM RF reference frame of center of masse electronEM electromagnetismEPR Einstein, Podolsky, and RosenF fermi (a unit of length in micro-world, 1 F ¼ 10�15 m)FTL faster than lightIP inverse problemn neutronp a proton or positronPEE photoelectric effectQM quantum mechanicsRF reference frameSR special relativityUV ultraviolet (range of spectrum)

Notations

�F or Fh i average of F

1D, 2D, . . . , n D or ND one-dimensional, two-dimensional, . . . , n- dimensionalor N-dimensional

jXIX

Det Mð Þ ¼ DDD Mð Þ ¼a11 a12 � � � a1n

a21 a22 � � � a2n

..

. ...

} ...

an1 an2 � � � ann

��

��

determinant of matrix

a11 a12 � � � a1n

a21 a22 � � � a2n

..

. ...

} ...

an1 an2 � � � ann

0BBB@

1CCCA

XXj Abbreviations and Notations

1The Failure of Classical Physics

Quantum mechanics (QM) emerged in the early twentieth century from attempts toexplain some properties of blackbody radiation (BBR) and heat capacity of gases, aswell as atomic spectra, light–matter interactions, and behavior of matter on themicroscopic level. It soon became clear that classical physics was unable to accountfor these phenomena. Not only did classical predictions disagree with experiments,but even the mere existence of atoms seemed to be a miracle in the framework ofclassical physics. In this chapter, we briefly discuss some of the contradictionsbetween classical concepts and observations.

1.1Blackbody Radiation

First, we outline the failure of classical physics to describe some properties ofradiation.

A macroscopic body with absolute temperature T > 0 emits radiation, whichgenerally has a continuous spectrum. In the case of thermal equilibrium, in anyfrequency range the body absorbs as much radiation as it emits. We can envisionsuch a body as the interior of an empty container whose walls are kept at a constanttemperature [1,2]. Its volume is permeated with electromagnetic (EM) waves of allfrequencies and directions, so there is no overall energy transfer and no change inenergy density (random fluctuations neglected). Its spectrum is independent ofthe material of container’s walls – be it mirrors or absorbing black soot. Hence, itsname – the blackbody radiation. In an experiment, we can make a small hole in thecontainer and record the radiation leaking out.

There is an alternative way [3] to think of BBR. Consider an atom in a medium.According to classical physics, its electrons orbit the atomic nucleus. Each orbitalmotion can be represented as a combination of two mutually perpendicularoscillations in the orbital plane. An oscillating electron radiates light. Throughcollisions with other atoms and radiation exchange, a thermal equilibrium can be

Quantum Mechanics and Quantum Information: A Guide through the Quantum World,First Edition. Moses Fayngold and Vadim Fayngold.� 2013 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2013 by Wiley-VCH Verlag GmbH & Co. KGaA.

j1

established. In equilibrium, the average kinetic energy per each degree of freedomis [1]

Ei ¼ 12kT ; ð1:1Þ

where k is the Boltzmann constant. This is known as the equipartition theorem. Thesame formula holds for the average potential energy of the oscillator, so the totalenergy (average kinetic þ average potential) per degree of freedom is kT. Thus, weend up with the average total energy kT per oscillation. In an open system, suchequilibrium cannot be reached because the outgoing radiation is not balanced andthe energy leaks out. This is why any heated body cools down when disconnectedfrom the source of heat.

But if the medium is sufficiently extended or contained within a cavity whose wallsemit radiation toward its interior, then essentially all radiation remains confined, andthermal equilibrium can be attained. Each oscillator radiates as before, but alsoabsorbs radiation coming from other atoms. In equilibrium, both processes balanceeach other. In such a case, for each temperature Tand each frequency v there exists acertain characteristic energy density rðv;TÞ of radiation such that the rate of energyloss by atoms through emission is exactly balanced by the rate of energy gainthrough absorption. The quantity rðv;TÞ is called spectral energy density (theenergy density per one unit of frequency range). In classical EM theory, it isdetermined by the corresponding field amplitudes EEðvÞ and BðvÞ of mono-chromatic waves with frequency v:

rðv;TÞ ¼ 14

e0jEEðvÞj2 þ jBðvÞj2m0

!¼ 1

2e0jEEðvÞj2 ¼ jBðvÞj2

2m0: ð1:2Þ

The last two expressions in (1.2) are obtained in view of the relationB ¼ ffiffiffiffiffiffiffiffiffi

m0e0p

n̂� EEÞð , where n̂ is the unit vector along the wave propagation. Notethat v is the angular frequency,1) and all quantities involved are measured in the restframe of the given medium.2)

Under the described conditions, rðv;TÞ is a universal function of v and T.According to thermodynamics (Kirchhoff’s law of thermal radiation), it must be theproduct of v3 and another universal function of v/T [2,3]:

rðv;TÞ ¼ av3fv

T

� �: ð1:3Þ

Using (1.3), one can show that the total energy density gðTÞ of BBR is

gðTÞ ¼ð1

0

rðv;TÞdv ¼ sT4; ð1:4Þ

1) We will use throughout the book the angular frequency v, which is ordinary frequency f(number of cycles/s) multiplied by 2p. In the physicists’ jargon, the word “angular” is usuallydropped.

2) Rest frame of an object is the frame of reference where the object’s center of mass is at rest.

2j 1 The Failure of Classical Physics

where

s � a

ð1

0

j3f ðjÞdj; j � v

T: ð1:5Þ

The relation (1.4), known as the Stefan–Boltzmann law, is exact and has beenexperimentally confirmed. Figure 1.1 shows a few graphs of rðv;TÞ obtainedfrom experiments. But all attempts to derive the pivotal function f ðv=TÞ determin-ing rðv;TÞ and s have failed.

By 1900 there were two half-successful attempts to derive rðv;TÞ. Their resultswere different due to the different models chosen to represent radiation.

The first model considered radiation as EM waves. In this model, the moleculesinteracting with radiation were represented as harmonic oscillators; similarly,each monochromatic component of radiation can also be considered as anoscillator with the corresponding frequency. Then, the total energy density couldbe evaluated as a product of the average energy hEi¼ kT per one EM oscillator andthe number N of oscillators occupying all states with frequency v [4]. Such anapproach results in

rðv;TÞ ¼ NðvÞhEi ¼ v2

p2c3kT : ð1:6Þ

This expression is known as the Rayleigh–Jeans formula. Note that it does have theform (1.3). But, while matching the data at low frequencies, it diverges at highfrequencies (Figure 1.2), predicting the infinite spectral density rðv;TÞ and infinitetotal energy density gðTÞ at v ! 1; even at low temperatures! This conclusion ofclassical theory was dubbed “the UV catastrophe.” Something was wrong with theclassical notion of energy exchange between matter and radiation as a continuousprocess, especially when applied to the high-frequency range!

Figure 1.1 The BBR spectrum at various temperatures.

1.1 Blackbody Radiation j3

The second model suggested by W. Wien used the Newtonian view of radiation asa flux of particles. Applying to such particles Boltzmann’s statistical treatment, heobtained the expression

rðv;TÞ ¼ constv3 e�cðv=TÞ; ð1:7Þwhere c is another constant. This expression also satisfies the requirement (1.3),and, in contrast with (1.6), it describes accurately the experimental data for highfrequencies. However, it does not match the data at low frequencies (Figure 1.2).Something was wrong with the notion of radiation as classical particles, especially inthe low-frequency range!

Thus, regardless of whether we view radiation as purely classical waves (Maxwell)or purely classical particles (Newton), either view only partially succeeds. The wavepicture works well in describing low frequencies, and the particle picture works forhigh frequencies, but both fail to describe all available data. That was the firstindication that the EM radiation is neither exactly waves nor exactly particles.

1.2Heat Capacity

Heat capacity is the amount of heat dQ required to change a body’s temperature T by1 K: C ¼ dQ=dT : We model the body as an ideal gas whose molecules do not interactwith each other. The analysis for an ideal gas hinges on the number of degrees offreedom. For an atom considered as a point-like object, three mutually independentdirections of its motion (or three components of its position vector) form three degreesof freedom. A diatomic molecule presents a more complex case. If it is a rigid pair of twopoint masses, then it has five degrees of freedom – three coordinates of its center ofmass and two angular coordinates specifying the orientation of its axis. If the separations between the two masses can change (e.g., two masses connected by a spring), then itbecomes a variable s, and the total number j of degrees of freedom jumps from 5 to 6.This is the maximal number for a diatomic molecule formed from two point-like atoms.The number j here can also be determined as 3� 2 (three degrees of freedom perparticle times the number of particles).

Figure 1.2 The BBR spectrum according to different approaches treating radiation as classicalwaves or particles, respectively. (a) BBR spectrum; (b) the Rayleigh–Jeans curve; (c) the Wien curve.


But as stressed in comment to Eq. (1.1), the vibrational degree of freedom“absorbs” the energy kT, where the additional amount (1/2)kT is due to the averagepotential energy of vibration. This can be formally described by adding and extradegree of freedom for each vibration, so that for a classical diatomic molecule we canwrite j0 ¼ jþ 1¼ 7.

Suppose we have a system of particles (e.g., a container with gas) in a state ofthermodynamic equilibrium. The gas in this case is described by the ideal gasequation PV ¼ NkT [3,5,6], where P and V are the gas pressure and volume,respectively, and N is the number of molecules. For one mole of gas, that is,N ¼ NA, where NA is the Avogadro number, we have

PV ¼ RT ; ð1:8Þwhere R ¼ NAk is the universal gas constant.

Let us now recall the relationship between pressure P and the internal energy U ofthe gas, P ¼ ð2=j0ÞðU=VÞ. Combining this with (1.8) gives

U ¼ j0

2RT : ð1:9Þ

There are two different types of heat capacity depending on two possible ways oftransferring heat to a system. We can heat a gas keeping it either at fixed volume or atfixed pressure. The corresponding molar heat capacities will be denoted as cP and cV ,respectively. To find them, recall the first law of thermodynamics [2,3],dQ ¼ dU þ dW ¼ dU þ P dV , where dW is an incremental work done by thesystem against external forces while changing its volume by an incremental amountdV. Applying the basic definition C ¼ dQ=dT , we have for the case of fixed volumedV ¼ 0:

C ! cV ¼ @U@T

¼ j0

2R: ð1:10Þ

When P¼ const, we obtain

C ! cP ¼ @U@T

þ P@V@T

¼ cV þ P@V@T

: ð1:11Þ

By virtue of (1.8) taken at P¼ const, this gives

cP ¼ cV þ R ¼ j0

2þ 1

� �R: ð1:12Þ

The ratio

c � cPcV

¼ 1 þ 2j0

ð1:13Þ

gives us direct information about the number j0.In the outlined classical picture, the number j0 and thereby cP; cV, and c are all

independent of T. But this contradicts the experiments. Heat capacities of all sub-stances at low temperatures turn out to be noticeably less than predicted and go to zeroin the limitT ! 0. Shown in Table 1.1 are the classically predicted values of cV ; cP; andc for a few different substances and their experimental values at room temperature [6].

1.2 Heat Capacity j5

The table shows a very interesting (and mixed) picture. The measurement resultsalmost exactly confirm theoretical predictions for monatomic gases such as helium. Thecalculated and measured values of c and c are in excellent agreement for j ¼ 3.

For diatomic gases (j¼ 6, j0 ¼ 7), however, the results are more complicated.Consider, for instance, hydrogen. Its measured value of cV ¼ 20:36 is significantlylower than the expected value 24.42. A similar discrepancy is observed for c: themeasured value is 1.41 instead of predicted 1.33. A closer look at these numbersreveals something very strange: they are still in excellent agreement with Equa-tions 1.10–1.13, but for j0¼ 5 instead of 7. Thus, the agreement can be recovered,but only at the cost of decreasing the number j0 ascribed to a diatomic molecule. Itlooks like two “effective” degrees of freedom “freezes” when the particles get boundto one another. Which kind of motion could possibly undergo “freezing”? Certainlynot 2 out of 3 translational motions: there is nothing in the isotropic space that couldsingle out one remaining motion. It could be either 2 rotational motions, or 1vibrational motion. Running ahead of ourselves, we will say here that it is vibrationalmotion that freezes first as the gas is cooled down. Already at room temperatures, wecannot pump energy into molecular vibrations. At these temperatures, the connec-tion between the two atoms in an H2 molecule is effectively absolutely rigid.

Thus, already the experimental results for different gases at room temperature showthat something is wrong with the classical picture. But the situation becomes evenworse if we carry out experiments for the same gas at widely different temperatures.As an example, consider the data for molecular hydrogen (Table 1.2).

At sufficiently low temperatures, experimental values become lower thanthe classical prediction even after we ascribe to the diatomic molecules onlyfive degrees of freedom instead of effective seven. For instance, the experimental

Table 1.1 Predicted versus observed heat capacities at T¼ 293 K (in J/(mol K))

Gas j cV cP c

Theory Exp Theory Exp Theory Exp

Helium (He) 3 12.47 12.46 20.79 20.90 1.67 1.67Hydrogen (H2) 6 24.42 20.36 32.73 28.72 1.33 1.41Water vapor (H2O) 9 37.41 27.80 45.73 36.16 1.22 1.31Methane (CH4) 15 62.36 27.21 70.67 35.57 1.13 1.30

Table 1.2 Constant-volume heat capacity of hydrogen asa function of temperature (in degrees Kelvin)

T cV

197 18.3290 13.6040 12.46


value cV ¼ 12:46 at 40 K is significantly less than cV ¼ 20:36 measured at roomtemperature. It could still fall within the classical prediction, but only at the cost ofreducing the number of degrees of freedom from 5 to 3. It looks as if more and moredegrees of freedom become frozen as the substance is cooled down. This time wecan, by the same argument as before, assume the freezing of two rotational degreesof freedom associated with the spatial orientation of the molecule.

The same tendency is observed in the measurements of c (Figure 1.3). Contrary tothe classical predictions, experiments show that c increases with T, and thisbehavior, in view of Equation 1.13, can be attributed to the same mysteriousmechanism of “freezing.” We can thus say that decrease of cV and increase of cat low temperatures represent two sides of the same coin.

The general feature can be illustrated by Figure 1.4. It shows that heat capacitiesfall off with temperature in a step-like fashion. Each step is associated with thefreezing of one or two degrees of freedom. At the end of this road, all of the initialdegrees of freedom are frozen and accordingly heat capacities approach zero.

If we now increase the temperature, starting from the absolute zero, we observethe same phenomenon in reverse. As we heat a body, it effectively regains its degreesof freedom all the way back to their normal number at sufficiently hightemperatures.

What causes these strange effects? And what is the “normal” number for c tobegin with? Classical physics cannot answer these questions even in the simplestcase of a monatomic gas. Let us, for instance, get back to helium. We started with theapparently innocuous statement that a helium atom at room temperature has j ¼ 3.But after a second thought we can ask: “Why is j equal to 3, in the first place?” Afterall, the helium atom consists of the nucleus and two electrons, so there are threeparticles in it, and therefore there must be j ¼ 3 � 3 ¼ 9. Further, the He nucleusconsists of two protons and two neutrons, so altogether we have six particles in a

01.0

1.2

1.4

1.6H2

O2

γ

500

1.286

1000 1500

T (K)

2000

Figure 1.3 Experimental values of c as functions of T for H2 and O2. The dashed horizontal lineis the classical prediction for a diatomic molecule with vibrational degree of freedom ( j0 ¼ 7).(Reproduced from Refs [3,6].)

1.2 Heat Capacity j7

helium atom and accordingly the number j must be j ¼ 6 � 3 ¼ 18 instead of 3. Andif we also realize that each nucleon, in turn, consists of three quarks, which makesthe total number of particles in the helium atom equal to 14, then the number j mustbe j ¼ 14 � 3 ¼ 42. Accordingly, the theoretical prediction for the molar heatcapacities for He must be cV ¼ 174:6 and cP ¼ 182:9 instead of 12.46 and 20.8,respectively. In other words, already at room temperature and in the simplest case ofa monatomic gas, there is a wide (by about one order of magnitude!) discrepancybetween theory and experiment. Experiment shows that nearly all degrees offreedom of the subatomic particles are frozen so fundamentally that they are asgood as nonexisting, at least at room temperatures, and only the remaining threedegrees determining the motion of the atom as a whole survive. Why is this so?

One could try to explain this by the fact that the binding forces between theelectrons and the atomic nucleus are so strong that they practically stop anyrelative motion within an atom; this is true even more so for the protons andneutrons within a nucleus, and so on. As we go farther down the subatomicscale, the interaction forces increase enormously, thus “turning off” the corre-sponding degrees of freedom. But this argument does not hold. The notion aboutthe forces is true, but the conclusion that it must “turn off” the correspondingmotions is wrong.

The equipartition theorem is a very general statement that applies to anyconservative (i.e., described by potential) forces, regardless of their physical natureor magnitude. As a simple example, consider two different types of diatomicmolecules in thermodynamic equilibrium. Let each molecule be represented bya system of two masses connected by a spring, but the spring constant is muchhigher for one molecule than for the other. Equilibrium is established in the processof collisions between the molecules, in which they can exchange their energy.Eventually, molecules of both types will have, on average, an equal amount ofvibration energy. The total mechanical energy of vibration for a spring described by

CV

Low T Medium T High T T

Figure 1.4 Graph of the heat capacity versus temperature for a diatomic ideal gas.
