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3 Quantum Mechanics The important changes quantum mechanics has undergone in recent years are reflected in this 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 results allows students to gain a thorough understanding. Starting with basic quantum mechanics, the book moves on to more advanced theory, followed by applications, perturbation methods and special fields, and ending with new developments in the field. Historical, mathematical, and philosophical boxes guide the student through the theory. Unique to this textbook are chapters on measurement and quantum optics, both at the forefront of current research. Advanced undergraduate and graduate students will benefit from this new perspective on the fundamental physical paradigm and its applications. Online resources including solutions to selected problems and 200 figures, with color versions of some figures, are available at Gennaro Auletta is Scientific Director of Science and Philosophy at the Pontifical Gregorian University, Rome. His main areas of research are quantum mechanics, logic, cognitive sciences, information theory, and applications to biological systems. Mauro Fortunato is a Structurer at Cassa depositi e prestiti S.p.A., Rome. He is involved in financial engineering, applying mathematical methods of quantum physics to the pricing of 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, and the Daniel Heineman prize. His main research activity deals with elementary particles, theory of phase transitions and statistical mechanics, disordered systems, computers and very large scale simulations, non-equilibrium statistical physics, optimization, and animal behavior.

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5 Quantum Mechanics GENNARO AULETTA Pontifical Gregorian University, Rome MAURO FORTUNATO Cassa Depositi e Prestiti S.p.A., Rome GIORGIO PARISI La Sapienza University, Rome

6 CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York Information on this title: G. Auletta, M. Fortunato and G. Parisi 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN ISBN ebook (EBL) hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

7 Contents List of figures List of tables List of definitions, principles, etc. List of boxes List of symbols List of abbreviations page xi xvii xviii xx xxi xxxii Introduction 1 Part I Basic features of quantum mechanics 1 From classical mechanics to quantum mechanics Review of the foundations of classical mechanics An interferometry experiment and its consequences State as vector Quantum probability The historical need of a new mechanics 31 Summary 40 Problems 41 Further reading 42 2 Quantum observables and states Basic features of quantum observables Wave function and basic observables Uncertainty relation Quantum algebra and quantum logic 92 Summary 96 Problems 97 Further reading 99 3 Quantum dynamics The Schrödinger equation Properties of the Schrödinger equation Schrödinger equation and Galilei transformations One-dimensional free particle in a box Unitary transformations 117

8 vi Contents 3.6 Different pictures Time derivatives and the Ehrenfest theorem Energy time uncertainty relation Towards a time operator 135 Summary 138 Problems 139 Further reading Examples of quantum dynamics Finite potential wells Potential barrier Tunneling Harmonic oscillator Quantum particles in simple fields 165 Summary 169 Problems Density matrix Basic formalism Expectation values and measurement outcomes Time evolution and density matrix Statistical properties of quantum mechanics Compound systems Pure- and mixed-state representation 187 Summary 188 Problems 189 Further reading 190 Part II More advanced topics 6 Angular momentum and spin Orbital angular momentum Special examples Spin Composition of angular momenta and total angular momentum Angular momentum and angle 239 Summary 241 Problems 242 Further reading Identical particles Statistics and quantum mechanics Wave function and symmetry Spin and statistics 249

9 vii Contents 7.4 Exchange interaction Two recent applications 255 Summary 257 Problems 257 Further reading Symmetries and conservation laws Quantum transformations and symmetries Continuous symmetries Discrete symmetries A brief introduction to group theory 267 Summary 275 Problems 275 Further reading The measurement problem in quantum mechanics Statement of the problem A brief history of the problem Schrödinger cats Decoherence Reversibility/irreversibility Interaction-free measurement Delayed-choice experiments Quantum Zeno effect Conditional measurements or postselection Positive operator valued measure Quantum non-demolition measurements Decision and estimation theory 341 Summary 349 Problems 351 Further reading 353 Part III Matter and light 10 Perturbations and approximation methods Stationary perturbation theory Time-dependent perturbation theory Adiabatic theorem The variational method Classical limit Semiclassical limit and WKB approximation Scattering theory Path integrals 389 Summary 398

10 viii Contents Problems 399 Further reading Hydrogen and helium atoms Introduction Quantum theory of the hydrogen atom Atom and magnetic field Relativistic corrections Helium atom Many-electron effects 431 Summary 436 Problems 437 Further reading Hydrogen molecular ion The molecular problem Born Oppenheimer approximation Vibrational and rotational degrees of freedom The Morse potential Chemical bonds and further approximations 449 Summary 453 Problems 453 Further reading Quantum optics Quantization of the electromagnetic field Thermodynamic equilibrium of the radiation field Phase number uncertainty relation Special states of the electromagnetic field Quasi-probability distributions Quantum-optical coherence Atom field interaction Geometric phase The Casimir effect 501 Summary 506 Problems 507 Further reading 509 Part IV Quantum information: state and correlations 14 Quantum theory of open systems General considerations The master equation 516

11 ix Contents 14.3 A formal generalization Quantum jumps and quantum trajectories Quantum optics and Schrödinger cats 533 Summary 540 Problems 541 Further reading State measurement in quantum mechanics Protective measurement of the state Quantum cloning and unitarity violation Measurement and reversibility Quantum state reconstruction The nature of quantum states 564 Summary 565 Problems 565 Further reading Entanglement: non-separability EPR Bohm s version of the EPR state HV theories Bell s contribution Experimental tests Bell inequalities with homodyne detection Bell theorem without inequalities What is quantum non-locality? Further developments about inequalities Conclusion 625 Summary 625 Problems 626 Further reading Entanglement: quantum information and computation Information and entropy Entanglement and information Measurement and information Qubits Teleportation Quantum cryptography Elements of quantum computation Quantum algorithms and error correction 659 Summary 671

12 x Contents Problems 672 Further reading 673 Bibliography 674 Author index 710 Subject index 716

13 Figures 1.1 Graphical representation of the Liouville theorem page Photoelectric effect Mach Zender interferometer The Michelson Morley interferometer Interferometer for detecting gravitational waves Interference in the Mach Zender interferometer Results of the experiment performed by Grangier, Roger, and Aspect Oscillation of electric and magnetic fields Polarization of classical light Decomposition of an arbitrary vector a Poincaré sphere representation of states Mach Zender interferometer with the lower path blocked by the screen S Black-body radiation intensity corresponding to the formula of Rayleigh Jeans (1), Planck (2), and Wien (3) Planck s radiation curves in logarithmic scale for the temperatures of liquid nitrogen, melting ice, boiling water, melting aluminium, and the solar surface Compton effect Dulong Petit s, Einstein s, and Debye s predictions for specific heat Lyman series for ionized helium The Stern Gerlach Experiment Momentum conservation in the Compton effect Polarization beam splitter Change of basis Filters Two sequences of two rotations of a book Probability distributions of position and momentum for a momentum eigenfunction Probability distributions of position and momentum for a position eigenfunction Time evolution of a classical degree of freedom in phase space and graphical representation of the uncertainty relation Inverse proportionality between momentum and position uncertainties Smooth complementarity between wave and particle Illustration of the distributive law Proposed interferometry and resulting non-boolean algebra and Boolean subalgebras 94

14 xii Figures 2.12 Hasse diagrams of several Boolean and non-boolean algebras Positive potential vanishing at infinity Potential function tending to finite values as x ± Potential well Relation between two different inertial reference frames R and R under Galilei transformations Particle in a box of dimension a Energy levels of a particle in a one-dimensional box First three energy eigenfunctions for a one-dimensional particle confined in a box of dimension a Beam Splitters as unitary operators Projector as a residue of the closed contour in a complex plane A graphical representation of the apparatus proposed by Bohr Schematic and asymmetric one-dimensional potential wells Solution of Eq. (4.8) Wave functions and probability densities for the first three eigenfunctions for the symmetric finite-well potential Stepwise continuity Potential barrier Closed surface used to compute the flux of J Delta potential barrier Classical turning points and quantum tunneling Tunneling of α-particles Carbon atoms shown by scanning tunneling microscopy Potential and energy levels of the harmonic oscillator Eigenfunctions for the one-dimensional harmonic oscillator Potential energy corresponding to a particle in a uniform field Triangular well A quantum particle with energy E encounters a potential step of height V 0 < E Rectangular potential barrier with finite width a Representation of pure and mixed states on a sphere Angular momentum of a classical particle Levi Civita tensor Relationship between rectangular and spherical coordinates s- and p-states Rigid rotator Energy levels and transition frequencies for a rigid rotator Cylindrical coordinates Energy levels of the three-dimensional harmonic oscillator Levels in the spectrum of hydrogen atom An electric dipole with charges +e and e in an electric field gradient Scheme of spin superposition in single-crystal neutron interferometry Landé vectorial model for angular momentum 227

15 xiii Figures 6.13 Graphical representation of the distribution of eigenvalues of the z component of the angular momenta of two independent particles Angle observable and step function Interferometric example of indistinguishability Example of counting the number of possible configurations of bosons Potential wells of a natural atom and of a quantum dot Passive and active transformations Representation of a measurement on the sphere of density matrices Two ways of tuning the coupling function Decohering histories Schrödinger cat Experimental realization of a Schrödinger cat with a trapped ion SQUID Wigner function of an entangled state Schematic representation of the experiment proposed by Scully and co-workers Schematic representation of Mandel s experiment Interference and visibility in Mandel s experiment Scully, Englert, and Walther s proposed experiment Interaction-free measurement Repeated interaction-free measurements Probability of success in repeated interaction-free measurements Interaction-free measurement with two cavities Schematic representation of Mandel s experiment on empty waves Depiction of Wheeler s experiment Interferometry experiment for testing delayed-choice Optical version of the Zeno effect Example of POVMs Plot of the estimate of the wave function Another example of POVM Stark effect WKB approximation: forbidden regions outside a potential well WKB and potential well WKB and potential barrier The different paths The analogy of path integral integration The sum over paths Two possible paths from i to f both passing through the same central point c Path integrals and scattering Electron coordinates in the atomic system Resulting potential in the hydrogen atom Grotrian scheme Plot of the radial eigenfunctions of the hydrogenoid atom 412

16 xiv Figures 11.5 Plot of the radial probability densities s-, p-, and d-states versus energy levels Landé vectorial model for the Paschen-Bach effect s and p levels in presence of the Paschen-Bach effect Paschen-Bach spectroscopical lines Landé vectorial model for the Zeeman effect Energy levels for the Zeeman effect Spectroscopical lines for the Zeeman effect Spheroidal coordinates for the H + 2 ion Molecular potential energy Vibrational and rotational levels of two electronic states I and II in a diatomic molecule Schematic diagram of the LCAO function f (E) LCAO energy solutions for the H + 2 molecular ion Symmetric and antisymmetric states of the ground level of the H + 2 molecule The three directions of the electromagnetic field Displacement operator for coherent states Phase-number uncertainty properties of coherent states Phase convention for squeezed states Generation of a squeezed state Phase-space of amplitude- and phase-squeezed states Representation of the Q-function of coherent, number, and squeezed states Representation of the W-function of coherent, number, and squeezed states Homodyne detection Jaynes Cummings energy levels Rabi oscillations Collapse and revival Spontaneous and stimulated emission for a two-level atom Spontaneous and stimulated emission for a three-level atom Schematic diagram of a laser Parametric down conversion Magnetic and electric AB effects Parallel transport Bloch-sphere representation of states Techniques for integrating a function BS model for dissipation Interference fringes and their sensitivity to losses in the Yurke Stoler model Pictorial representation of a coherent state and separation between the two components Haroche s experiment Interference fringes in Haroche s experiment Tomographic method for reconstructing the W-function Tomographic measurements of the state Overview of the EPR Bohm experiment 574

17 xv Figures 16.2 Preparation of a singlet state Particle trajectories for two Gaussian slit systems after Bohm s model, and the corresponding quantum potential Trajectories for a potential barrier (E = V/2) after Bohm s model, and the corresponding quantum potential The three-dimensional Hilbert space proposed by Bell Scheme of the experiment proposed for proving the second Bell theorem Experiment proposed by CHSH Optimal orientation for a, a,b,and b for testing the CHSH inequality Typical dependence of f (θ) upon nθ for cases I-III Partial Grotrian diagram of atomic calcium for Freedman and Clauser s experiment Schematic diagram of apparatus and associated electronics of the experiment by Freedman and Clauser Freedman Clauser experiment and Aspect and co-workers experiment Alley-Shih and Ou-Mandel s experiment Measured coincidence counting rate as a function of the polarizer angle θ 1, with θ 2 fixed at Experimental set-up in order to solve detection loopholes Entanglement with vacuum Yurke and Stoler s experiment Entanglement swapping Variation of entanglement swapping Orientations for the proof of Stapp s theorem The GHSZ proposed experiment Conditional entanglement Necessary criterion for separability Information difference in bits versus angle θ for the information-theoretic Bell inequality Informational distance by quadrilateral inequality Schematic representation of quantum non-separability Diagram for entangled and disentangled states Representation of all density matrices Decompression of information Teleportation Realization of teleportation with photons The CNOT gate Implementation of a CNOT gate by means of a polarization interferometer The quantum computation device as an equivalent of a Mach Zender interferometer Generation of Bell states by means of a Hadamard gate followed by a CNOT gate Preparation of a GHSZ state Toffoli gate 657

18 xvi Figures Representation of computational complexity Boolean transformation of an initial bit Device for solving Deutsch s problem Device for solving Deutsch s problem for n + 1 input states Device for solving Deutsch s problem for n + m input states Representation of Shor s theorem Implementation of Grover s algorithm Computational steps in Grover s algorithm Classical error correction Quantum circuit for error correction Environmental wave functions and their overlapping as a function of time in quantum computation 670

19 Tables 2.1 Different cases and ways of expressing the basic quantum formalism page Eigenstates and eigenvalues of ˆl z Eigenvalues, for the three-dimensional harmonic oscillator Values of j and m and the corresponding number of possible states Clebsch Gordan coefficients for j 1 = j 2 = 1/ Clebsch Gordan coefficients for j 1 = 1 and j 2 = 1/ Clebsch Gordan coefficients for j 1 = j 2 = Fermions and bosons Fermionic distributions Bosonic distributions Ground-state energy of helioid atoms Electromagnetic spectrum Sequence transmission Toffoli truth table Fredkin truth table Classical error correction 668

20 Definitions, principles, etc. Corollaries Cor. 2.1 Simultaneous measurability page 67 Cor Bell Dispersion-free 583 Definitions Def. 5.1 Entanglement 183 Def. 9.1 Operation 328 Def Completeness 568 Lemmas Lemma 9.1 Elby Bub 299 Lemma 16.1 Bell I 583 Lemma 16.2 Bell II 584 Principles Pr. 1.1 Superposition 18 Pr. 1.2 Complementarity 19 Pr. 2.1 Quantization 44 Pr. 2.2 Statistical algorithm 57 Pr. 2.3 Correspondence 72 Pr. 7.1 Symmetrization 248 Pr. 7.2 Pauli exclusion principle 251 Pr Separability 568 Pr Criterion of physical reality 569 Theorems Th. 2.1 Hermitian operators 46 Th. 2.2 Finite-dimensional spectrum 47 Th. 2.3 Observables equality 60 Th. 2.4 Commuting observables 66 Th. 3.1 Stone 122 Th. 5.1 Schmidt decomposition 185 Th. 6.1 Addition of angular momenta 229 Th. 7.1 Spin and statistics 249 Th. 8.1 Wigner 262