Quantum Mechanics - Assetsassets.cambridge.org/97805218/69638/frontmatter/9780521869638... · Quantum Mechanics The important changes quantum mechanics has undergone in recent years

Quantum Mechanics

The important changes quantum mechanics has undergone in recent years are reflected inthis new approach for students.

A strong narrative and over 300 worked problems lead the student from experiment,through general principles of the theory, to modern applications. Stepping through resultsallows students to gain a thorough understanding. Starting with basic quantum mechan-ics, the book moves on to more advanced theory, followed by applications, perturbationmethods and special fields, and ending with new developments in the field. Historical,mathematical, and philosophical boxes guide the student through the theory. Unique tothis textbook are chapters on measurement and quantum optics, both at the forefront ofcurrent research. Advanced undergraduate and graduate students will benefit from this newperspective on the fundamental physical paradigm and its applications.

Online resources including solutions to selected problems and 200 figures, with colorversions of some figures, are available at www.cambridge.org/Auletta.

Gennaro Auletta is Scientific Director of Science and Philosophy at the Pontifical Grego-rian University, Rome. His main areas of research are quantum mechanics, logic, cognitivesciences, information theory, and applications to biological systems.

Mauro Fortunato is a Structurer at Cassa depositi e prestiti S.p.A., Rome. He is involved infinancial engineering, applying mathematical methods of quantum physics to the pricingof complex financial derivatives and the construction of structured products.

Giorgio Parisi is Professor of Quantum Theories at the University of Rome “La Sapienza.”He has won several prizes, notably the Boltzmann Medal, the Dirac Medal and Prize, andthe Daniel Heineman prize. His main research activity deals with elementary particles,theory of phase transitions and statistical mechanics, disordered systems, computers andvery large scale simulations, non-equilibrium statistical physics, optimization, and animalbehavior.

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Cambridge University Press978-0-521-86963-8 - Quantum MechanicsGennaro Auletta, Mauro Fortunato and Giorgio ParisiFrontmatterMore information

http://www.cambridge.org/9780521869638

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Quantum Mechanics

GENNARO AULETTAP o n t i f i c a l G r e g o r i a n U n i v e r s i t y , R o m e

MAURO FORTUNATOC a s s a D e p o s i t i e P r e s t i t i S . p . A . , R o m e

GIORGIO PARISI‘ ‘ L a S a p i e n z a ’ ’ U n i v e r s i t y , R o m e






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First published 2009First paperback edition 2013

Printing in the United Kingdom by TJ International Ltd. Padstow Cornwall

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Library of Congress Cataloging in Publication dataAuletta, Gennaro, 1957–

Quantum mechanics : into a modern perspective / Gennaro Auletta,Mauro Fortunato, Giorgio Parisi.

p. cm.Includes bibliographical references and index.

ISBN 978-0-521-86963-81. Quantum theory. I. Fortunato, Mauro. II. Parisi, Giorgio. III. Title.

QC174.12.A854 2009530.12–dc222009004303

ISBN 978-0-521-86963-8 HardbackISBN 978-1-107-66589-7 Paperback

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Contents

List of figures page xiList of tables xviiList of definitions, principles, etc. xviiiList of boxes xxList of symbols xxiList of abbreviations xxxiiAcknowledgements for the revised edition xxxiii

Introduction 1

Part I Basic features of quantum mechanics

1 From classical mechanics to quantum mechanics 71.1 Review of the foundations of classical mechanics 71.2 An interferometry experiment and its consequences 121.3 State as vector 201.4 Quantum probability 281.5 The historical need of a new mechanics 31Summary 40Problems 41Further reading 42

2 Quantum observables and states 432.1 Basic features of quantum observables 432.2 Wave function and basic observables 682.3 Uncertainty relation 822.4 Quantum algebra and quantum logic 92Summary 96Problems 97Further reading 99

3 Quantum dynamics 1003.1 The Schrödinger equation 1013.2 Properties of the Schrödinger equation 1073.3 Schrödinger equation and Galilei transformations 1113.4 One-dimensional free particle in a box 1133.5 Unitary transformations 117






vi Contents�

3.6 Different pictures 1253.7 Time derivatives and the Ehrenfest theorem 1293.8 Energy–time uncertainty relation 1313.9 Towards a time operator 135Summary 138Problems 139Further reading 140

4 Examples of quantum dynamics 1414.1 Finite potential wells 1414.2 Potential barrier 1454.3 Tunneling 1504.4 Harmonic oscillator 1544.5 Quantum particles in simple fields 165Summary 169Problems 170

5 Density matrix 1745.1 Basic formalism 1745.2 Expectation values and measurement outcomes 1775.3 Time evolution and density matrix 1795.4 Statistical properties of quantum mechanics 1805.5 Compound systems 1815.6 Pure- and mixed-state representation 187Summary 188Problems 189Further reading 190

Part II More advanced topics

6 Angular momentum and spin 1936.1 Orbital angular momentum 1936.2 Special examples 2076.3 Spin 2176.4 Composition of angular momenta and total angular momentum 2266.5 Angular momentum and angle 239Summary 241Problems 242Further reading 244

7 Identical particles 2457.1 Statistics and quantum mechanics 2457.2 Wave function and symmetry 2477.3 Spin and statistics 249






vii Contents�

7.4 Exchange interaction 2547.5 Two recent applications 255Summary 257Problems 257Further reading 258

8 Symmetries and conservation laws 2598.1 Quantum transformations and symmetries 2598.2 Continuous symmetries 2648.3 Discrete symmetries 2668.4 A brief introduction to group theory 267Summary 275Problems 275Further reading 276

9 The measurement problem in quantum mechanics 2779.1 Statement of the problem 2789.2 A brief history of the problem 2849.3 Schrödinger cats 2919.4 Decoherence 2979.5 Reversibility/irreversibility 3089.6 Interaction-free measurement 3159.7 Delayed-choice experiments 3209.8 Quantum Zeno effect 3229.9 Conditional measurements or postselection 3259.10 Positive operator valued measure 3279.11 Quantum non-demolition measurements 3359.12 Decision and estimation theory 341Summary 349Problems 351Further reading 353

Part III Matter and light

10 Perturbations and approximation methods 35710.1 Stationary perturbation theory 35710.2 Time-dependent perturbation theory 36610.3 Adiabatic theorem 36910.4 The variational method 37110.5 Classical limit 37210.6 Semiclassical limit and WKB approximation 37810.7 Scattering theory 38410.8 Path integrals 389Summary 398






viii Contents�

Problems 399Further reading 399

11 Hydrogen and helium atoms 40111.1 Introduction 40111.2 Quantum theory of the hydrogen atom 40311.3 Atom and magnetic field 41311.4 Relativistic corrections 42311.5 Helium atom 42611.6 Many-electron effects 431Summary 436Problems 437Further reading 438

12 Hydrogen molecular ion 43912.1 The molecular problem 43912.2 Born–Oppenheimer approximation 44012.3 Vibrational and rotational degrees of freedom 44312.4 The Morse potential 44712.5 Chemical bonds and further approximations 449Summary 453Problems 453Further reading 454

13 Quantum optics 45513.1 Quantization of the electromagnetic field 45713.2 Thermodynamic equilibrium of the radiation field 46213.3 Phase–number uncertainty relation 46313.4 Special states of the electromagnetic field 46513.5 Quasi-probability distributions 47413.6 Quantum-optical coherence 48113.7 Atom–field interaction 48413.8 Geometric phase 49713.9 The Casimir effect 501Summary 506Problems 507Further reading 509

Part IV Quantum information: state and correlations

14 Quantum theory of open systems 51314.1 General considerations 51414.2 The master equation 516






ix Contents�

14.3 A formal generalization 52314.4 Quantum jumps and quantum trajectories 52814.5 Quantum optics and Schrödinger cats 533Summary 540Problems 541Further reading 542

15 State measurement in quantum mechanics 54415.1 Protective measurement of the state 54415.2 Quantum cloning and unitarity violation 54815.3 Measurement and reversibility 55015.4 Quantum state reconstruction 55415.5 The nature of quantum states 564Summary 565Problems 565Further reading 566

16 Entanglement: non-separability 56716.1 EPR 56816.2 Bohm’s version of the EPR state 57316.3 HV theories 57716.4 Bell’s contribution 58216.5 Experimental tests 59516.6 Bell inequalities with homodyne detection 60516.7 Bell theorem without inequalities 61316.8 What is quantum non-locality? 61916.9 Further developments about inequalities 62316.10 Conclusion 625Summary 625Problems 626Further reading 627

17 Entanglement: quantum informationand computation 62817.1 Information and entropy 62817.2 Entanglement and information 63117.3 Measurement and information 63917.4 Qubits 64217.5 Teleportation 64317.6 Quantum cryptography 64617.7 Elements of quantum computation 65017.8 Quantum algorithms and error correction 659Summary 671






x Contents�

Problems 672Further reading 673

Bibliography 674Author index 710Subject index 716






Figures

1.1 Graphical representation of the Liouville theorem page 121.2 Photoelectric effect 131.3 Mach–Zehnder interferometer 151.4 The Michelson–Morley interferometer 161.5 Interferometer for detecting gravitational waves 171.6 Interference in the Mach–Zehnder interferometer 171.7 Results of the experiment performed by Grangier, Roger, and Aspect 181.8 Oscillation of electric and magnetic fields 201.9 Polarization of classical light 211.10 Decomposition of an arbitrary vector |a〉 221.11 Poincaré sphere representation of states 281.12 Mach–Zehnder interferometer with the lower path blocked by the screen S 301.13 Black-body radiation intensity corresponding to the formula of Rayleigh–

Jeans (1), Planck (2), and Wien (3) 321.14 Planck’s radiation curves in logarithmic scale for the temperatures of liquid

nitrogen, melting ice, boiling water, melting aluminium, and the solar surface 331.15 Compton effect 341.16 Dulong–Petit’s, Einstein’s, and Debye’s predictions for specific heat 361.17 Lyman series for ionized helium 371.18 The Stern–Gerlach Experiment 391.19 Momentum conservation in the Compton effect 412.1 Polarization beam splitter 442.2 Change of basis 522.3 Filters 622.4 Two sequences of two rotations of a book 652.5 Probability distributions of position and momentum for a momentum

eigenfunction 852.6 Probability distributions of position and momentum for a position

eigenfunction 862.7 Time evolution of a classical degree of freedom in phase space and graphical

representation of the uncertainty relation 882.8 Inverse proportionality between momentum and position uncertainties 892.9 Smooth complementarity between wave and particle 902.10 Illustration of the distributive law 932.11 Proposed interferometry and resulting non-Boolean algebra and Boolean

subalgebras 94






xii Figures�

2.12 Hasse diagrams of several Boolean and non-Boolean algebras 953.1 Positive potential vanishing at infinity 1093.2 Potential function tending to finite values as x →±∞ 1103.3 Potential well 1113.4 Relation between two different inertial reference frames R and R′ under

Galilei transformations 1123.5 Particle in a box of dimension a 1133.6 Energy levels of a particle in a one-dimensional box 1153.7 First three energy eigenfunctions for a one-dimensional particle confined in

a box of dimension a 1153.8 Beam Splitters as unitary operators 1193.9 Projector as a residue of the closed contour in a complex plane 1253.10 A graphical representation of the apparatus proposed by Bohr 1334.1 Schematic and asymmetric one-dimensional potential wells 1424.2 Solution of Eq. (4.8) 1434.3 Wave functions and probability densities for the first three eigenfunctions

for the symmetric finite-well potential 1444.4 Stepwise continuity 1454.5 Potential barrier 1474.6 Closed surface used to compute the flux of J 1484.7 Delta potential barrier 1494.8 Classical turning points and quantum tunneling 1514.9 Tunneling of α-particles 1524.10 Carbon atoms shown by scanning tunneling microscopy 1534.11 Potential and energy levels of the harmonic oscillator 1544.12 Eigenfunctions for the one-dimensional harmonic oscillator 1624.13 Potential energy corresponding to a particle in a uniform field 1654.14 Triangular well 1674.15 A quantum particle with energy E encounters a potential step of height

V0 < E 1714.16 Rectangular potential barrier with finite width a 1715.1 Representation of pure and mixed states on a sphere 1876.1 Angular momentum of a classical particle 1946.2 Levi–Civita tensor 1956.3 Relationship between rectangular and spherical coordinates 2006.4 s- and p-states 2066.5 Rigid rotator 2076.6 Energy levels and transition frequencies for a rigid rotator 2096.7 Cylindrical coordinates 2126.8 Energy levels of the three-dimensional harmonic oscillator 2156.9 Levels in the spectrum of hydrogen atom 2186.10 An electric dipole with charges +e and −e in an electric field gradient 2186.11 Scheme of spin superposition in single-crystal neutron interferometry 2236.12 Landé vectorial model for angular momentum 227






xiii Figures�

6.13 Graphical representation of the distribution of eigenvalues of the zcomponent of the angular momenta of two independent particles 228

6.14 Angle observable and step function 2417.1 Interferometric example of indistinguishability 2467.2 Example of counting the number of possible configurations of bosons 2537.3 Potential wells of a natural atom and of a quantum dot 2578.1 Passive and active transformations 2609.1 Representation of a measurement on the sphere of density matrices 2809.2 Two ways of tuning the coupling function 2839.3 Decohering histories 2909.4 Schrödinger cat 2929.5 Experimental realization of a Schrödinger cat with a trapped ion 2939.6 SQUID 2949.7 Wigner function of an entangled state 2969.8 Schematic representation of the experiment proposed by Scully and

co-workers 3019.9 Schematic representation of Mandel’s experiment 3099.10 Interference and visibility in Mandel’s experiment 3109.11 Scully, Englert, and Walther’s proposed experiment 3129.12 Interaction-free measurement 3169.13 Repeated interaction-free measurements 3179.14 Probability of success in repeated interaction-free measurements 3189.15 Interaction-free measurement with two cavities 3189.16 Schematic representation of Mandel’s experiment on empty waves 3199.17 Depiction of Wheeler’s experiment 3209.18 Interferometry experiment for testing delayed-choice 3219.19 Optical version of the Zeno effect 3249.20 Example of POVMs 3349.21 Plot of the estimate of the wave function 3499.22 Another example of POVM 35210.1 Stark effect 36410.2 WKB approximation: forbidden regions outside a potential well 38110.3 WKB and potential well 38210.4 WKB and potential barrier 38310.5 The different paths 39010.6 The analogy of path integral integration 39110.7 The sum over paths 39210.8 Two possible paths from i to f both passing through the same central

point c 39410.9 Path integrals and scattering 39711.1 Electron coordinates in the atomic system 40611.2 Resulting potential in the hydrogen atom 40711.3 Grotrian scheme 41111.4 Plot of the radial eigenfunctions of the hydrogenoid atom 412






xiv Figures�

11.5 Plot of the radial probability densities 41311.6 s-, p-, and d-states versus energy levels 41411.7 Landé vectorial model for the Paschen-Bach effect 41711.8 s and p levels in presence of the Paschen-Bach effect 41811.9 Paschen-Bach spectroscopical lines 41911.10 Landé vectorial model for the Zeeman effect 42011.11 Energy levels for the Zeeman effect 42211.12 Spectroscopical lines for the Zeeman effect 42212.1 Spheroidal coordinates for the H+

2 ion 44012.2 Molecular potential energy 44612.3 Vibrational and rotational levels of two electronic states I and I I in a

diatomic molecule 44712.4 Schematic diagram of the LCAO function f (E) 45012.5 LCAO energy solutions for the H+

2 molecular ion 45112.6 Symmetric and antisymmetric states of the ground level of the H+

2 molecule 45213.1 The three directions of the electromagnetic field 46013.2 Displacement operator for coherent states 46913.3 Phase-number uncertainty properties of coherent states 47113.4 Phase convention for squeezed states 47213.5 Generation of a squeezed state 47313.6 Phase-space of amplitude- and phase-squeezed states 47313.7 Representation of the Q-function of coherent, number, and squeezed states 47513.8 Representation of the W-function of coherent, number, and squeezed states 48013.9 Homodyne detection 48313.10 Jaynes–Cummings energy levels 49013.11 Rabi oscillations 49113.12 Collapse and revival 49213.13 Spontaneous and stimulated emission for a two-level atom 49513.14 Spontaneous and stimulated emission for a three-level atom 49513.15 Schematic diagram of a laser 49513.16 Parametric down conversion 49613.17 Magnetic and electric AB effects 49713.18 Parallel transport 50114.1 Bloch-sphere representation of states 52714.2 Techniques for integrating a function 53114.3 BS model for dissipation 53414.4 Interference fringes and their sensitivity to losses in the Yurke–Stoler model 53614.5 Pictorial representation of a coherent state and separation between the two

components 53714.6 Haroche’s experiment 53814.7 Interference fringes in Haroche’s experiment 54015.1 Tomographic method for reconstructing the W-function 55915.2 Tomographic measurements of the state 56016.1 Overview of the EPR – Bohm experiment 574






xv Figures�

16.2 Preparation of a singlet state 57616.3 Particle trajectories for two Gaussian slit systems after Bohm’s model, and

the corresponding quantum potential 58016.4 Trajectories for a potential barrier (E = V/2) after Bohm’s model, and the

corresponding quantum potential 58116.5 The three-dimensional Hilbert space proposed by Bell 58516.6 Scheme of the experiment proposed for proving the second Bell theorem 58916.7 Experiment proposed by CHSH 59016.8 Optimal orientation for a, a′, b, and b′ for testing the CHSH inequality 59216.9 Typical dependence of f (θ ) upon nθ for cases I-III 59516.10 Partial Grotrian diagram of atomic calcium for Freedman and Clauser’s

experiment 59916.11 Schematic diagram of apparatus and associated electronics of the

experiment by Freedman and Clauser 59916.12 Freedman–Clauser experiment and Aspect and co-workers’ experiment 60016.13 Alley-Shih and Ou-Mandel’s experiment 60216.14 Measured coincidence counting rate as a function of the polarizer angle θ1,

with θ2 fixed at 45◦ 60416.15 Experimental set-up in order to solve detection loopholes 60516.16 “Entanglement” with vacuum 60616.17 Yurke and Stoler’s experiment 60816.18 Entanglement swapping 61116.19 Variation of entanglement swapping 61216.20 Orientations for the proof of Stapp’s theorem 61416.21 The GHSZ proposed experiment 61616.22 Conditional entanglement 61916.23 Necessary criterion for separability 62317.1 Information difference in bits versus angle θ for the information-theoretic

Bell inequality 63517.2 Informational distance by quadrilateral inequality 63517.3 Schematic representation of quantum non-separability 63617.4 Diagram for entangled and disentangled states 63717.5 Representation of all density matrices 63817.6 Decompression of information 63917.7 Teleportation 64417.8 Realization of teleportation with photons 64617.9 The CNOT gate 65217.10 Implementation of a CNOT gate by means of a polarization interferometer 65317.11 The quantum computation device as an equivalent of a Mach–Zehnder

interferometer 65317.12 Generation of Bell states by means of a Hadamard gate followed by a

CNOT gate 65517.13 Preparation of a GHSZ state 65517.14 Toffoli gate 657






xvi Figures�

17.15 Representation of computational complexity 66017.16 Boolean transformation of an initial bit 66017.17 Device for solving Deutsch’s problem 66117.18 Device for solving Deutsch’s problem for n + 1 input states 66217.19 Device for solving Deutsch’s problem for n + m input states 66217.20 Representation of Shor’s theorem 66417.21 Implementation of Grover’s algorithm 66617.22 Computational steps in Grover’s algorithm 66617.23 Classical error correction 66717.24 Quantum circuit for error correction 66917.25 Environmental wave functions and their overlapping as a function

of time in quantum computation 670






Tables

2.1 Different cases and ways of expressing the basic quantum formalism page 716.1 Eigenstates and eigenvalues of lz 1996.2 Eigenvalues, for the three-dimensional harmonic oscillator 2156.3 Values of j and m and the corresponding number of possible states 2336.4 Clebsch–Gordan coefficients for j1 = j2 = 1/2 2386.5 Clebsch–Gordan coefficients for j1 = 1 and j2 = 1/2 2386.6 Clebsch–Gordan coefficients for j1 = j2 = 1 2387.1 Fermions and bosons 2507.2 Fermionic distributions 2527.3 Bosonic distributions 25411.1 Ground-state energy of helioid atoms 42913.1 Electromagnetic spectrum 45617.1 Sequence transmission 64817.2 Toffoli truth table 65817.3 Fredkin truth table 65817.4 Classical error correction 668






Definitions, principles, etc.

Corollaries

Cor. 2.1 Simultaneous measurability page 67Cor. 16.1 Bell Dispersion-free 583

Definitions

Def. 5.1 Entanglement 183Def. 9.1 Operation 328Def. 16.1 Completeness 568

Lemmas

Lemma 9.1 Elby–Bub 299Lemma 16.1 Bell I 583Lemma 16.2 Bell II 584

Principles

Pr. 1.1 Superposition 18Pr. 1.2 Complementarity 19Pr. 2.1 Quantization 44Pr. 2.2 Statistical algorithm 57Pr. 2.3 Correspondence 72Pr. 7.1 Symmetrization 248Pr. 7.2 Pauli exclusion principle 251Pr. 16.1 Separability 568Pr. 16.2 Criterion of physical reality 569

Theorems

Th. 2.1 Hermitian operators 46Th. 2.2 Finite-dimensional spectrum 47Th. 2.3 Observables’ equality 60Th. 2.4 Commuting observables 66Th. 3.1 Stone 123Th. 5.1 Schmidt decomposition 185Th. 6.1 Addition of angular momenta 229Th. 7.1 Spin and statistics 249Th. 8.1 Wigner 262






xix Pr inc iples�

Th. 8.2 Noether 264Th. 9.1 Kraus 329Th. 15.1 No-Cloning 548Th. 15.2 D’Ariano–Yuen 549Th. 15.3 Informational completeness of unsharp observables 564Th. 16.1 Bell I 584Th. 16.2 Bell II 589Th. 16.3 Stapp 613Th. 16.4 Eberhard 620Th. 16.5 Tsirelson 624Th. 17.1 Lindblad 631Th. 17.2 Holevo 642






Boxes

1.1 Interferomentry page 152.1 Hermitian and bounded operators 472.2 Example of a Hermitian operator 482.3 Unitary operators 512.4 Example of mean value 602.5 Commutation and product of Hermitian operators 642.6 Wave packet 803.1 Cyclic property of the trace 1173.2 Einstein’s box 1324.1 Relativity and tunneling time 1524.2 Scanning tunneling microscopy 1524.3 Example of a harmonic oscillator’s dynamics 1637.1 Rasetti’s discovery 2508.1 Example of passive and active transformations 2609.1 Decohering histories 2889.2 Recoil-free which path detectors? 3139.3 Complementarity 3229.4 Example of postselection 3269.5 Example of operation 327

10.1 Remarks on the Fourier transform 38811.1 Confluent hypergeometric functions 40813.1 LASER 49414.1 Coherent states and macroscopic distinguishability 53717.1 Example of factorization 663






Symbols

Latin letters

a proposition, number

a =√

m2hω

(ωx + ı ˆx

)annihilation operator

ak annihilation operator of the k-th mode of theelectromagnetic field

a† =√

m2hω

(ωx − ı ˆx

)creation operator

a†k creation operator of the k-th mode of the

electromagnetic field|a〉 (polarization) state vector (along the direction a)∣∣a j⟩

element of a discrete vector basis{∣∣a j

⟩ }A numberA(ζ ) Airy functionA vector potential

A(r, t) =∑k ck

×[akuk(r)e−ıωkt + a†

ku∗k(r)eıωkt] vector potential operator

A apparatus|A〉 ket describing a generic state of the apparatusb proposition, number|b〉 (polarization) state vector (along the direction b)∣∣b j⟩

element vector of a discrete basis{∣∣b j

⟩ }B number, intensity of the magnetic fieldB = h/8π2I rotational constant of the rigid rotatorB = ∇ × A classical magnetic field

B(r, t) = ı∑

k

(hk

2cL3ε0

) 12

×[akeı(k·r−ωkt) − a†

ke−ı(k·r−ωkt)]

bλ

magnetic field operator

c speed of light, propositionc j , c

′j generic coefficients of the j-th element of a

given discrete expansionca j coefficient of the basis element

∣∣a j⟩

cb j coefficient of the basis element∣∣b j⟩

c(0)n coefficients of the expansion of a state vector in

stationary state at an initial moment t0 = 0






xxii Symbols�

c(η), c(ξ ) coefficient of the eigenkets of continuousobservables η and ξ , respectively

|c〉 polarization state vector (along thedirection c)

C , C ′ constantsC coulomb charge unit, correlation functionC cost function

C jk cost incurred by choosing the j-th hypothesiswhen the k-th hypothesis is true

(C field of complex numbersd electric dipoled distanceD decoherence functional

D(α) = eαa†−α∗a displacement operatore exponential functione electric chargee = (ex , ey , ez) vector orthogonal to the propagation direction of

the electromagnetic field|e〉 excited state|ek〉 k-th ket of the environment’s eigenbasis

{∣∣e j⟩ }

E energyEn n-th energy level, energy eigenvalueE0 energy value of the ground state

E one-dimensional electric fieldE = −∇Ve − ∂

∂t A classical electric field

E(r, t) = ı∑

k

(hωk2ε0

) 12

×[akuk(r)e−ıωkt − a†

ku∗k(r)eıωkt] electric field operator

E environmentE effect|E〉 ket describing a generic state of the

environmentf arbitrary functionf, f′ arbitrary vectors| f 〉 final state vectorF force, arbitrary classical physical quantity

Fe classically electrical forceFm classically magnetic force

Fm(φ) eigenfunctions of lz

F(x) = ℘(ξ < x) distribution function of a random variable thatcan take values < x

g arbitrary function, gravitational acceleration|g〉 ground stateG(n) coherence of the n-th order






xxiii Symbols�

G Green functionG0(r′, t ′; r, t) =

−ı[

m2π ı h(t ′−t)

] 32

eım|r′−r|22h(t ′−t)

free Green function

G groupG generator of a a continuos transformation or of a

grouph = 6.626069 × 10−34J · s Planck constanth = h/2π|h〉 state of horizontal polarizationH Hilbert spaceHA Hilbert space of the apparatusHS Hilbert space of the system

H Hamiltonian operatorH0 unperturbed HamiltonianHA Hamiltonian of a free atomHF field HamiltonianHI interaction HamiltonianH I

I interaction Hamiltonian in Dirac pictureHJC Jaynes–Cummings HamiltonianHr planar part of the Hamiltonian

Hn(ζ ) = (−1)n eζ2 dn

dζ n e−ζ 2n-th Hermite polynomial, for all n = 0

ı imaginary unityı x Cartesian versor| i〉 initial state vectorI intensity (of radiation)I moment of inertiaI identity operator(z) = z−z∗

2ı imaginary part of a complex number zj y Cartesian versorj = J/h = (jx , jy , jz)| j〉 arbitrary ket, element of a continuous or discrete

basis| j , m〉 eigenket of Jz

J density of the probability currentJI incidental current densityJR reflected current densityJT transmitted current density

J = L + S = ( Jx , Jy , Jz) total angular momentumˆJ jump superoperator

k, k wave vectork z Cartesian versorkB Boltzmann constant|k〉 generic ket, element of a continuous or discrete

basis






xxiv Symbols�

l arbitrary lengthl = (lx , ly , lz) = L/hl± = lx ± ı ly raising and lowering operators for the levels of

the angular momentum| l〉 generic ket∣∣ l, ml

⟩eigenket of lz

L(q1, . . . , qn ; q1, . . . , qn) classical Lagrangian functionL Lagrangian multiplier operatorL = (Lx , L y , Lz) orbital angular momentumˆL Lindblad superoperatorm mass of a particle

me mass of the electronmn mass of the nucleusm p mass of the proton

ml magnetic quantum numberm j eigenvalue of jzms spin magnetic quantum number or secondary

spin quantum number|m〉 generic ket, eigenket of the energyM measure of purityM meterM arbitrary matrixn, n′ direction vectors|n〉 eigenvector of the harmonic oscillator

Hamiltonian or of the number operatorN number of elements of a given setN normalization constantN = a†a number operator

Nk = a†kak

o j j-th eigenvalue of the observable O|o〉 eigenket of the observable O∣∣o j⟩

j-th eigenket of the observable OO , O ′, O ′′ generic operators, generic observables

OH observable in the Heisenberg pictureO I observable in the Dirac pictureOS observable in the Schrödinger pictureOA apparatus’ pointerOS observable of the object systemOND non-demolition observable∣∣∣O } super-ket (or S-ket)

pk classical generalized momentum componentp = ( px , py , pz) three-dimensional momentum operator

px one-dimensional momentum operator






xxv Symbols�

ˆpx time derivative of px

pr = −ı h 1r∂∂r r radial part of the momentum operator

P(α,α∗) P-functionPj projection onto the state | j〉 or

∣∣b j⟩

P path predictabilityP path predicability operator

℘ j or ℘( j) probability of the event j℘k (D) = ℘ (D|Hk) probability density function that the particular

set D of data is observed when the system isactually in state k

℘(H j |Hk

) = Tr(ρk EH j

)conditional probability that one chooses thehypothesis H j when Hk is true

qk classical generalized position componentQ charge densityQ(α,α∗) = 1

π〈α|ρ|α〉 Q-function

Q quantum algebra(Q field of rational numbersr spherical coordinater · r′ scalar product between vectors r and r′rk k-th eigenvalue of a density matrix

r0 = h2

me2 Bohr’s radiusr = (x , y, z) three-dimensional position operatorR reflection coefficientR(r ) radial part of the eigenfunctions of lz in

speherical coordinatesR,R′ reference framesR reservoirIR field of real numbers�(z) = z+z∗

2 real part of a complex quantity zR, R(β,φ, θ ) rotation operator, generator of rotationsRO resolvent of the operator OR j =∑N

k=1 ℘Ak C jk ρk risk operator for the j-th hypothesis

| R〉 initial state of the reservoirs spin quantum numbers = (sx , sy , sz) = S/h spin vector operators± = sx ± ı sy raising and lowering spin operatorsS actionS generic quantum systemS entropyS = (Sx , Sy , Sz) spin observablet timet time operator| t〉 eigenket of the time operator






xxvi Symbols�

T transmission coefficientT temperature, classical kinetic energyT kinetic energy operatorT time reversal operatorˆT , T generic transformationu(ν, T ) energy densityuk(r) = e

L32

eık·r k-th mode function of the electromagnetic field

U scalar potentialU unitary operator

UBS beam splitting unitary operatorUPBS polarization beam-splitting unitary operatorUCNOT unitary controlled-not operatorU f Boolean unitary transformationUF Fourier unitary transformationUH unitary Hadamard operatorUp(v) unitary momentum translationUP permutation operatorUR(φ) rotation operatorUR space-reflection operatorUt time translation unitary operatorUx (a) one-dimensional space translation unitary

operatorUr(a) three-dimensional space translation unitary

operatorUθ unitary rotation operatorUφ unitary phase operator

USAτ = e

− ıh

τ∫0

dt HSA(t)unitary operator coupling system and apparatusfor time interval τ

USA,Et = e−

ıh t HSA,E unitary operator which couples the environment

E to the system and apparatus S +A at time t˜U antiunitary operator˜UT time reversal

U generic transformation that can be either unitaryor antiunitary

|v〉 state of vertical polarization|vn〉 element of a discrete basisV potential energy

Ve scalar potential of the electromagnetic field

Vc(r ) = h2l(l+1)2mr2 centrifugal-barrier potential energy

Vc classical potential energyV potential energy operator






xxvii Symbols�

V volumeV generic vectorV visibility of interference, generic vectorial spaceV visibility of interference operatorwk k-th probability weight|wn〉 element of a discrete basis vector

W (α,α∗) =1π2

∫d2αe−ηα∗+η∗αχW (η, η∗)

Wigner function

x first Cartesian axis, coordinate| x〉 eigenket of xx one-dimensional position operatorˆx time derivative of xX1 = 1√

2

(a† + a

)quadrature

X2 = ı√2

(a† − a

)quadrature

X sety second Cartesian axis, coordinateYlm(θ ,φ) spherical harmonicsz third Cartesian axis, coordinateZ atomic number

Z (β) = Tr(

e−β H)

partition function

Z parameter spaceZZ field of integer numbers

Greek letters

α angle, (complex) number

|α〉 = e−|α|2

2∑∞

n=0αn√

n!|n〉 coherent state

β angle, (complex) number, thermodynamicvariable = (kBT )−1

|β〉 coherent stateγ damping constant� Euler gamma function� phase space�k reservoir operatorδ jk Kronecker symbolδ(x) Dirac delta function

� = ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 Laplacian

�ψ uncertainty in the state |ψ〉ε small quantityε jkn Levi–Civita tensorε coupling constant

ε0 =(

ω

2ε0hl3

) 12 |d · e| vacuum Rabi frequency






xxviii Symbols�

εn = ε0√

n + 1 Rabi frequencyεSA coupling between object system and apparatusεSM coupling between object system and meter

ζ arbitrary variable, arbitrary (wave) functionζS , ζA number of possible configurations of bosons and

fermions, respectivelyη arbitrary variable, arbitrary (wave) functionη arbitrary (continuous) observable|η〉 eigenkets of ηθ angle, spherical coordinateϑ generic amplitude

ϑk(m) =⟨m∣∣∣Ut

∣∣∣ k⟩ amplitude operator connecting apremeasurement (|k〉 ), a unitary evolution (Ut ),and a measurement (|m〉 )

�lm(θ ) theta component of the spherical harmonics�(θ ) part of the spherical harmonics depending on the

polar coordinate θ

�, ˆ� arbitrary transformation (superoperator)ι constant, parameter| ι〉 internal state of a systemκ parameterλ wavelengthλc = h/mc Compton wavelength of the electronλT = h√

2mkBTthermal wavelength

� = μB Bext constant used in the Paschen–Bach effect� j Lindblad operatorμ classically magnetic dipole momentumμ

l= eh

2m l orbital magnetic momentum of a massiveparticle

μs = Q eh2m s spin magnetic momentum

μB = eh2m Bohr magneton

μ0 magnetic permeabilityν frequencyξ random variable, variableξ (r ) = R(r )r change of variable for the radial part of the wave

functionξ arbitrary (continuous) observable|ξ 〉 eigenkets of ξ�(x) Heaviside step function parity operatorρ (classical) probability densityρ density matrix (pure state)ˆρ time-evolved density matrix






xxix Symbols�

ˆρ mixed density matrixρ f density matrix for the final state of a systemρi density matrix for the initial state of a system! j reduced density matrix of the j-th subsystemρSA density matrix of the system plus apparatusρSAE density matrix of the system plus apparatus plus

environment

σ 2x =

⟨x2⟩− ⟨x ⟩2 variance of x

σx standard deviation (square root of the variance)of x

σ 2p =

⟨p2

x

⟩− ⟨ px⟩2 variance of px

σp standard deviation (square root of the variance)of px

σ+ = |e 〉〈g | raising operatorσ− = |g 〉〈e | lowering operatorσ = (σx , σy , σz) Pauli (two-dimensional) spin matricesς (s) wave component of the spin|ς〉 ket of the object systemτ time interval, interaction time between two or

more systems

τd � γ−1(λT�x

)2decoherence time

φ angle, spherical coordinateφ angle operator|φ〉 eigenket of the angle operator|ϕ〉 ,

∣∣ϕ′⟩ state vectorsϕ(ξ ) eigenfunctions of the observable with

eigenvector |ξ 〉ϕk(x) plane wavesϕk(r) spherical wavesϕp(x) momentum eigenfunctions in the position

representationϕx0 (x) position eigenfunctions in the position

representationϕp0 (px ) momentum eigenfunctions in the momentum

representationϕx (px ) position eigenfunctions in the momentum

representationϕξ (x) scalar product 〈x | ξ 〉ϕη(ξ ) scalar product 〈ξ | η〉

% flux of electric current%M magnetic flux

|%〉 generic ket for compound systemsχξ (η) = ∫ dF(x)eıηx classical characteristic function of a random

variable ξ






xxx Symbols�

χ (η, η∗) =e|η|2

∫d2αeηα

∗−αη∗Q(α,α∗)characteristic function

χW (η, η∗) = e− 12 |η|2χ (η, η∗) Wigner characteristic function

|ψ〉 ,∣∣ψ ′⟩ state vectors

|ψ(t)〉 time-evolved or time-dependent state vector∣∣ψE

⟩Eigenket of energy corresponding to eigenvalueE (in the continuous case)

|&F 〉 quantum state of the electromagnetic field|ψn〉 n-th stationary state|ψ〉H state vector in the Heisenberg picture|ψ〉 I state vector in the Dirac picture|ψ〉S state vector in the Schrödinger picture

ψ(x),ψ(r) wave functions in the position representationψ(η),ψ(ξ ) wave functions of two arbitrary continuous

observables, η and ξ , respectivelyψ(px ), ψ(p) Fourier transform of the wave functionsψ(r, s) wave function with a spinor componentψ(r , θ ,φ) eigenfunctions of lz in spherical coordinatesψp(x),ψp(r),ψk(x),ψk(r) momentum eigenfunctions in the position

representationψE (x) energy eigenfucntion in the position

representationψS ,ψA symmetric and antisymmetric wavefucntions,

respectively|&〉 ket of a compund system|&〉SA ket describing an objects system plus apparatus

compound system|&〉SM ket describing an objects system plus meter

compound system|&〉SAE ket describing an objects system plus apparatus

plus environment compound system&(x),&(r) wave function of a compound systemω = 2πν angular frequencyωB = eB

m electron cyclotron frequencyω jk ratio between energy levels Ek − E j and h' space

Other Symbols

∇ Nabla operator〈· | · ·〉 scalar product〈 j1, j2, m1, m2 | j , m〉 Clebsch–Gordan coefficient| ·〉〈· | external product〈·〉 mean value






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