Quantum Mechanics and Quantum Information E-Book

Moses Fayngold

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1584

Veröffentlichungsjahr: 2013

Beschreibung

Alongside a thorough definition of basic concepts and their interrelations, backed by numerous examples, this textbook features a rare discussion of quantum mechanics and information 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 measurements, "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 beginning graduate quantum mechanics courses and an essential reference for physics students and physics professionals.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

Preface

Abbreviations and Notations

Chapter 1: The Failure of Classical Physics

1.1 Blackbody Radiation

1.2 Heat Capacity

1.3 The Photoelectric Effect

1.4 Atoms and Their Spectra

1.5 The Double-Slit Experiment

Problem

References

Chapter 2: The First Steps into the Unknown

2.1 The BBR and Planck's Formula

2.2 Einstein's Light Quanta and BBR

2.3 PEE Revisited

2.4 The Third Breakthrough: de Broglie Waves

2.4.1 Exercise

Problems

References

Chapter 3: Embryonic Quantum Mechanics: Basic Features

3.1 A Glimpse of the New Realm

3.2 Quantum-Mechanical Superposition of States

3.3 What Is Waving There (the Meaning of the Ψ-Function)?

3.4 Observables and Their Operators

3.5 Quantum-Mechanical Indeterminacy

3.6 Indeterminacy and the World

Exercise 3.1

3.7 Quantum Entanglement and Nonlocality

3.8 Quantum-Mechanical Phase Space

3.9 Determinism and Causality in Quantum World

Problems

References

Chapter 4: Playing with the Amplitudes

4.1 Composition of Amplitudes

4.2 Double Slit Revised I

4.3 Double Slit Revised II

4.4 Neutron Scattering in Crystals

4.5 Bosonic and Fermionic States

4.6 Path Integrals

Problems

References

Chapter 5: Basic Features and Mathematical Structure of QM

5.1 Observables: the Domain of Classical and Quantum Mechanics

5.2 Quantum-Mechanical Operators

5.3 Algebra of Operators

5.4 Eigenvalues and Eigenstates

5.5 Orthogonality of Eigenstates

5.6 The Robertson–Schrödinger Relation

5.7 The Wave Function and Measurements (Discussion)

Problems

References

Chapter 6: Representations and the Hilbert Space

6.1 Various Faces of a State Function

6.2 Unitary Transformations

6.3 Operators in the Matrix Form

6.4 The Hilbert Space

6.5 Operations in the Hilbert Space

6.6 Nonorthogonal States

Problems

References

Chapter 7: Angular Momentum

7.1 Orbital and Spin Angular Momenta

7.2 The Eigenstates and Eigenvalues of L

7.3 Operator L and Its Commutation Properties

7.4 Spin as an Intrinsic Angular Momentum

Exercise 7.1

Exercise 7.2

Exercise 7.3

Exercise 7.4

7.5 Angular Momentum of a Compound System

7.6 Spherical Harmonics

Problems

References

Chapter 8: The Schrödinger Equation

8.1 The Schrödinger Equation

8.2 State Function and the Continuity Equation

8.3 Separation of Temporal and Spatial Variables: Stationary States

8.4 The Helmholtz Equation and Dispersion Equation for a Free Particle

8.5 Separation of Spatial Variables and the Radial Schrödinger Equation

8.6 Superposition of Degenerate States

8.7 Phase Velocity and Group Velocity

8.8 de Broglie's Waves Revised

8.9 The Schrödinger Equation in an Arbitrary Basis

Problems

References

Chapter 9: Applications to Simple Systems: One Dimension

9.1 A Quasi-Free Particle

9.2 Potential Threshold

9.3 Tunneling through a Potential Barrier

9.4 Cold Emission

9.5 Potential Well

Exercise 9.1

9.6 Quantum Oscillator

9.7 Oscillator in the E-Representation

9.8 The Origin of Energy Bands

9.9 Periodic Structures

Problems

References

Chapter 10: Three-Dimensional Systems

10.1 A Particle in a 3D Box

10.2 A Free Particle in 3D (Spherical Coordinates)

10.2.1 Discussion

10.3 Some Properties of Solutions in Spherically Symmetric Potential

10.4 Spherical Potential Well

10.5 States in the Coulomb Field and a Hydrogen Atom

10.6 Atomic Currents

10.7 Periodic Table

Problems

References

Chapter 11: Evolution of Quantum States

11.1 The Time Evolution Operator

11.2 Evolution of Operators

11.3 Spreading of a Gaussian Packet

11.4 The B-Factor and Evolution of an Arbitrary State

11.5 The Fraudulent Life of an “Illegal” Spike

11.6 Jinnee Out of the Box

11.7 Inadequacy of Nonrelativistic Approximation in Description of Evolving Discontinuous States

11.7.1 Discussion

11.8 Quasi-Stationary States

11.8.1 Discussion

11.9 3D Barrier and Quasi-Stationary States

11.10 The Theory of Particle Decay

11.11 Particle–Antiparticle Oscillations

11.11.1 Discussion

11.12 A Watched Pot Never Boils (Quantum Zeno Effect)

11.13 A Watched Pot Boils Faster

Problems

References

Chapter 12: Quantum Ensembles

12.1 Pure Ensembles

12.2 Mixtures

12.3 The Density Operator

12.4 Time Evolution of the Density Operator

12.5 Composite Systems

Problems

References

Chapter 13: Indeterminacy Revisited

13.1 Indeterminacy Under Scrutiny

13.2 The Heisenberg Inequality Revised

13.3 The Indeterminacy of Angular Momentum

Exercise 13.1

13.4 The Robertson–Schrödinger Relation Revised

13.5 The N–φ Indeterminacy

13.6 Dispersed Indeterminacy

References

Chapter 14: Quantum Mechanics and Classical Mechanics

14.1 Relationship between Quantum and Classical Mechanics

14.2 QM and Optics

14.3 The Quasi-Classical State Function

14.4 The WKB Approximation

14.5 The Bohr–Sommerfeld Quantization Rules

Problems

References

Chapter 15: Two-State Systems

15.1 Double Potential Well

15.2 The Ammonium Molecule

15.3 Qubits Introduced

Problem

References

Chapter 16: Charge in Magnetic Field

16.1 A Charged Particle in EM Field

16.2 The Continuity Equation in EM Field

Exercise 16.1

16.3 Origin of the A-Momentum

16.4 Charge in Magnetic Field

16.5 Spin Precession

16.6 The Aharonov–Bohm Effect

16.7 The Zeeman Effect

Problems

References

Chapter 17: Perturbations

17.1 Stationary Perturbation Theory

17.1.1 Discussion

Exercise 17.1

17.2 Asymptotic Perturbations

17.3 Perturbations and Degeneracy

17.4 Symmetry, Degeneracy, and Perturbations

17.5 The Stark Effect

17.6 Time-Dependent Perturbations

Problems

References

Chapter 18: Light–Matter Interactions

18.1 Optical Transitions

18.2 Dipole Radiation

18.3 Selection Rules

18.3.1 Oscillator

18.3.2 Hydrogen-Like Atom

Problems

Reference

Chapter 19: Scattering

19.1 QM Description of Scattering

Exercise 19.1

19.2 Stationary Scattering

19.3 Scattering Matrix and the Optical Theorem

19.4 Diffraction Scattering

19.5 Resonant Scattering

19.6 The Born Approximation

Problems

References

Chapter 20: Submissive Quantum Mechanics

20.1 The Inverse Problem

20.2 Playing with Quantum States

20.3 Playing with Evolution: Discussion

Problems

References

Chapter 21: Quantum Statistics

21.1 Bosons and Fermions: The Exclusion Principle

21.2 Planck and Einstein Again

21.3 BBR Again

21.4 Lasers and Masers

Problems

References

Chapter 22: Second Quantization

22.1 Quantum Oscillator Revisited

22.2 Creation and Annihilation Operators: Bosons

22.3 Creation and Annihilation Operators: Fermions

Problems

References

Chapter 23: Quantum Mechanics and Measurements

23.1 Collapse or Explosion?

23.2 “Schrödinger's Cat” and Classical Limits of QM

23.3 Von Neumann's Measurement Scheme

23.4 Quantum Information and Measurements

Exercise 23.1

23.5 Interaction-Free Measurements: Quantum Seeing in the Dark

23.6 QM and the Time Arrow

Problems

References

Chapter 24: Quantum Nonlocality

24.1 Entangled Superpositions I

24.2 Entangled Superpositions II

24.3 Quantum Teleportation

24.4 The “No-Cloning” Theorem

24.5 Hidden Variables and Bell's Theorem

24.6 Bell-State Measurements

24.7 QM and the Failure of FTL Proposals

24.8 Do Lasers Violate the No-Cloning Theorem?

24.9 Imperfect Cloning

24.10 The FLASH Proposal and Quantum Compounds

Problems

References

Chapter 25: Quantum Measurements and POVMs

25.1 Projection Operator and Its Properties

25.2 Projective Measurements

25.3 POVMs

25.4 POVM as a Generalized Measurement

25.5 POVM Examples

25.6 Discrimination of Two Pure States

25.7 Neumark's Theorem

25.8 How to Implement a Given POVM

25.9 Comparison of States and Mixtures

25.10 Generalized Measurements

Problems

References

Chapter 26: Quantum Information

26.1 Deterministic Information and Shannon Entropy

26.2 von Neumann Entropy

26.3 Conditional Probability and Bayes's Theorem

26.4 KL Divergence

26.5 Mutual Information

26.6 Rényi Entropy

26.7 Joint and Conditional Renyi Entropy

26.8 Universal Hashing

26.9 The Holevo Bound

26.10 Entropy of Entanglement

Problems

References

Chapter 27: Quantum Gates

27.1 Truth Tables

27.2 Quantum Logic Gates

27.3 Shor's Algorithm

Problems

References

Chapter 28: Quantum Key Distribution

28.1 Quantum Key Distribution (QKD) with EPR

28.2 BB84 Protocol

28.3 QKD as Communication Over a Channel

28.4 Postprocessing of the Key

28.5 B92 Protocol

28.6 Experimental Implementation of QKD Schemes

28.7 Advanced Eavesdropping Strategies

Problems

References

Appendix A: Classical Oscillator

Reference

Appendix B: Delta Function

Reference

Appendix C: Representation of Observables by Operators

Appendix D: Elements of Matrix Algebra

Appendix E: Eigenfunctions and Eigenvalues of the Orbital Angular Momentum Operator

Appendix F: Hermite Polynomials

Appendix G: Solutions of the Radial Schrödinger Equation in Free Space

Appendix H: Bound States in the Coulomb Field

Reference

Index

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Preface

Quantum mechanics (QM) is one of the cornerstones of today's physics. Having emerged 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 explanatory and predictive power, it still remains the arena of lively debates about some of its basic concepts.

This book on quantum mechanics and quantum information (QI) differs from standard texts in three respects. First, it combines QM and some elements of QI in one volume. Second, it does not contain those important applications of QM (helium atom, hydrogen molecule, condensed matter) that can be found almost in 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 most current textbooks. We believe that these topics are essential for a better understanding of such fundamental concepts as quantum evolution, quantum-mechanical indeterminacy, superposition of states, quantum entanglement, and the Hilbert space and operations therein.

The book began as an attempt by one of the authors to account to himself for some aspects of QM that continued to intrigue him since his graduate school days. As his research notes grew he came up with the idea of developing them further into a textbook or monograph. The other author, who, by his own admission, used to think of 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 by his 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 explanations even in those texts he had previously considered to contain authoritative accounts on the subject.

Our original plan was to produce a short axiomatic formulation of the logical structure of quantum theory in the spirit of von Neumann's famous book [1]. It turned 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 different topics, and conversely, many apparently similar topics branched out into two or more different ones. As a result, we ended up adding lots of new examples and eventually reshaping our text into an entirely different framework. In hindsight, this evolution appears to resemble the way Ziman wrote his remarkable book on QM as a means of self-education [2].

The resulting book attempts to address the needs of students struggling to understand the weird world of quantum physics. We tried to find the optimal balance between mathematical rigor and clarity of presentation. There is always a temptation 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 the high number of equations may understandably intimidate some readers, there is still no realistic way around it. On the other hand, replacing explanations of concepts entirely by formulas, expecting them to speak for themselves, may produce only a dangerous illusion of understanding. Hence, the way we see this book: a textbook with a human face, a guidebook by necessity abundant with equations – still not an easy reading! – but also with a thorough explanation of underlying physics. This is precisely what we feel is necessary for its intended audience – mostly college students.

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 their failure in quantum domain, or using the “shock therapy” – immediately placing the reader face to face with the most paradoxical quantum phenomena and force him/her to accept from the outset the entirely new concepts and mathematical tools necessary for their description.

We do not think that this question has a unique answer: the choice of the most suitable approach depends on many factors. The “shock therapy” method is efficient, straightforward, and saves lots of time (not to mention decreasing the size of the textbook!) when aimed at the curious and bright student – but only because such a student, precisely due to his/her curiosity, has already read some material on the topic and is thus prepared to absorb new information on a higher level. We, however, aimed at a maximally broad student audience, without assuming any preliminary acquaintance with quantum ideas. The only (but necessary!) prerequisites are the knowledge of standard classical topics – mechanics, electromagnetism (EM), and thermodynamics in physics; and linear algebra with matrices and operations on them, calculus, Fourier transforms, and complex variables in mathematics. Accordingly, our book uses gradual introduction of new ideas. To make the transition from classical to quantum-mechanical concepts smoother for the first-time reader, it starts with some familiar effects whose features demonstrate the inadequacy of classical concepts for their description. The first chapter shows the failure 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 a necessary new level in the description of the world. In this respect, our book falls out of step with many modern presentations that introduce the Schrödinger equation or the Hilbert space from the very beginning. We believe that gradual approach offers a good start in the journey through the quantum world.

QM is very different from other branches of physics: it is not just formal or counterintuitive – it is, in addition, not easy to visualize. Therefore, the task of presenting QM to students poses a challenge even for the most talented pedagogues. The obstacles are very serious: apart from rather abstract mathematics involved, there is still no consensus among physicists about the exact meaning of some basic concepts such as wave function or the process of measurement [3–5]. The lack of self-explanatory models appealing to our “classical” intuition provides fertile soil for potential confusions and misunderstandings. A slightly casual use of technical language might be permissible in other areas of physics where the visual model itself provides guidance to the reader; but it often proves catastrophic for the beginning student of QM. For instance (getting back to our own experience) one of the authors, while still a student, found the representation theory almost incomprehensible. Later on, he realized the cause of his troubles: none of the many 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 that representation. The other author, on the contrary, had initially found the representation theory to be fairly easy, only to stumble upon the same block later when trying to explain the subject to his partner.

When carefully considering the logical structure of QM, one can identify a fair number of such points where it is easy to get misled or to inadvertently mislead the reader. Examples of potential snags include even such apparently well-understood concepts as the uncertainty (actually, indeterminacy) principle or the definition of quantum ensembles. In this book, we deliberately zero in on those potential pitfalls that we were able to notice by looking at them carefully and sometimes by falling into them ourselves. Sometimes we devote whole sections to the corresponding discussions, as, for instance, when probing the connection between the azimuthal angle and angular momentum (Section 13.4).

There has been a noticeable trend to present QM as an already fully accomplished construction that provides unambiguous answers to all pertinent questions. In this book, we have tried to show QM as a vibrant, still-developing science potentially capable of radically new insights and important reconstructions. For this reason, we did not shy away from the issues that up to now remain in the center of lively debates among physicists: the meaning of quantum reality, the role of the observer, the search for hidden parameters, and the increasingly important topic of quantum nonlocality [3,6–8]. We also discuss some important developments such as quantum information and its processing [9–12], interaction-free measurements [13,14], and no-cloning theorem and its implications [15–18]. We devote a special chapter to a new aspect of QM, practically unknown to the broad audience: the so-called “inverse problem” and manipulation of quantum states by local changes of interaction potential or of the environment, which is especially relevant to the physics of nanostructures [19,20]. This is so important for both practical applications and developing a new dimension in our “quantum intuition” that it should become a part 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 of quantum theory. In such cases, the reader can find the necessary information in the abundant standard texts [21,22], and the basic hope is that he/she will be already sufficiently equipped for reading them.

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

We introduce the idea of observables and their operators, eigenstates and eigenvalues, and quantum-mechanical indeterminacy with some simple illustrating examples in Chapter 3. We do not restrict to the initial concept of the wave function as a function of r and t, and show as early as possible that it could be a function of other observables. 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 forms of an operator by making it clear that, for instance, Hermitian operators are those that have a Hermitian matrix. We emphasize that going from one form to another usually means changing the representation – another potential source of confusion.

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

We did not rush to present the Schrödinger equation as soon as possible, all the more so that it is only one nonrelativistic limit of more general relativistic wave equations, which are different for particles with different spins. It expresses, among other 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). The idea 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 as traditionally 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 them in later chapters. But each such new discussion of the same topic reveals some new aspects, which helps to achieve a better understanding and gain a deeper insight into the 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 benefit to this approach, as the students who come to class after having been exposed to a different 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, and Francesco De Martini for sharing with us his latest ideas on the problem of signal exchange between entangled systems. We enjoyed working with the Project Editor Nina Stadthaus during preparation of the manuscript and are grateful to her for her patience in dealing with numerous delays at the later stages of the work.

References

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

2. Ziman, J.M. (1969) Elements of Advanced Quantum Theory, Cambridge University Press, Cambridge.

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

4. Bell, J.S. (2001) The Foundations of Quantum 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 and the Search for Reality, Little, Brown and Company, Boston, MA.

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

8. Kafatos, M. and Nadeau, R. (1990) The Conscious Universe: Part and Whole in Modern Physical Theory, Springer, New York.

9. Berman, G., Doolen, G., Mainieri, R., and Tsifrinovich, V. (1998) Introduction to Quantum Computers, World Scientific, Singapore.

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

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

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

13. Elitzur, A.C. and Vaidman, L. (1993) Quantum-mechanical interaction-free measurements. 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 – a superluminal communicator based upon a new kind of quantum measurement. Found. Phys.,12, 1171.

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

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

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

19. Agranovich, V.M. and Marchenko, V.A. (1960) Inverse Scattering Problem, Kharkov University, 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 Problem Approach, Nova Science Publishers, Inc., New York.

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

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

Abbreviations and Notations

Abbreviations

BBRblackbody radiationBSbeam splitterCMclassical mechanicsCMcenter of massCM RFreference frame of center of masseelectronEMelectromagnetismEPREinstein, Podolsky, and RosenFfermi (a unit of length in micro-world, )FTLfaster than lightIPinverse problemnneutronpa proton or positronPEEphotoelectric effectQMquantum mechanicsRFreference frameSRspecial relativityUVultraviolet (range of spectrum)

Notations

average of F1D, 2D,..., n D or NDone-dimensional, two-dimensional,..., n- dimensional or N-dimensional

1

The Failure of Classical Physics

Quantum mechanics (QM) emerged in the early twentieth century from attempts to explain some properties of blackbody radiation (BBR) and heat capacity of gases, as well as atomic spectra, light–matter interactions, and behavior of matter on the microscopic level. It soon became clear that classical physics was unable to account for 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 of classical physics. In this chapter, we briefly discuss some of the contradictions between classical concepts and observations.

1.1 Blackbody Radiation

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

A macroscopic body with absolute temperature emits radiation, which generally has a continuous spectrum. In the case of thermal equilibrium, in any frequency range the body absorbs as much radiation as it emits. We can envision such a body as the interior of an empty container whose walls are kept at a constant temperature [1, 2]. Its volume is permeated with electromagnetic (EM) waves of all frequencies and directions, so there is no overall energy transfer and no change in energy density (random fluctuations neglected). Its spectrum is independent of the material of container's walls – be it mirrors or absorbing black soot. Hence, its name – the blackbody radiation. In an experiment, we can make a small hole in the container 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 orbital motion can be represented as a combination of two mutually perpendicular oscillations in the orbital plane. An oscillating electron radiates light. Through collisions with other atoms and radiation exchange, a thermal equilibrium can be established. In equilibrium, the average kinetic energy per each degree of freedom is [1]

(1.1)

where k is the Boltzmann constant. This is known as the equipartition theorem. The same formula holds for the average potential energy of the oscillator, so the total energy (average kinetic + average potential) per degree of freedom is kT. Thus, we end up with the average total energy kT per oscillation. In an open system, such equilibrium cannot be reached because the outgoing radiation is not balanced and the energy leaks out. This is why any heated body cools down when disconnected from the source of heat.

But if the medium is sufficiently extended or contained within a cavity whose walls emit radiation toward its interior, then essentially all radiation remains confined, and thermal equilibrium can be attained. Each oscillator radiates as before, but also absorbs radiation coming from other atoms. In equilibrium, both processes balance each other. In such a case, for each temperature and each frequency there exists a certain characteristic energy density of radiation such that the rate of energy loss by atoms through emission is exactly balanced by the rate of energy gain through absorption. The quantity is called (the energy density per one unit of frequency range). In classical EM theory, it is determined by the corresponding field amplitudes and of monochromatic waves with frequency :