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Modern Quantum Mechanics and Quantum Information Modern Quantum Mechanics and Quantum Information J S Faulkner Department of Physics, Florida Atlantic University, Boca Raton, Florida, FL, USA IOP Publishing, Bristol, UK ªIOPPublishingLtd2021 Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording orotherwise,withoutthepriorpermissionofthepublisher,orasexpresslypermittedbylawor undertermsagreedwiththeappropriaterightsorganization.Multiplecopyingispermittedin accordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency,theCopyright ClearanceCentreandotherreproductionrightsorganizations. PermissiontomakeuseofIOPPublishingcontentotherthanassetoutabovemaybesought [email protected]. JSFaulknerhasassertedhisrighttobeidentifiedastheauthorofthisworkinaccordancewith sections77and78oftheCopyright,DesignsandPatentsAct1988. ISBN 978-0-7503-2167-9(ebook) ISBN 978-0-7503-2165-5(print) ISBN 978-0-7503-2168-6(myPrint) ISBN 978-0-7503-2166-2(mobi) DOI 10.1088/978-0-7503-2167-9 Version:20211201 IOPebooks BritishLibraryCataloguing-in-PublicationData:Acataloguerecordforthisbookisavailable fromtheBritishLibrary. PublishedbyIOPPublishing,whollyownedbyTheInstituteofPhysics,London IOPPublishing,TempleCircus,TempleWay,Bristol,BS16HG,UK USOffice:IOPPublishing,Inc.,190NorthIndependenceMallWest,Suite601,Philadelphia, PA19106,USA I would like to express my gratitude to the ladies in my life. My wife Dora and my daughters Emilia and Lee Anne. Contents Preface xiii Acknowledgement xiv Author biography xv 1 Review of basics 1-1 1.1 About quantum mechanics 1-1 1.2 Hilbert space 1-2 1.3 Elementary quantum mechanics 1-4 1.4 Dirac and von Neumann 1-5 1.5 Rigged Hilbert space 1-6 1.6 Observables and Hermitean operators 1-10 1.7 The uncertainty relation 1-12 1.8 Commuting observables 1-14 1.9 Unitary operators 1-14 1.10 The Gaussian wave packet 1-15 1.11 Two-dimensional Hilbert space 1-16 1.12 Pairs of spins 1-22 1.13 Einstein, Podolsky, and Rosen 1-23 Problems 1-24 References 1-25 2 Non-relativistic quantum mechanics 2-1 2.1 Heisenberg’s matrix mechanics 2-1 2.2 The one-dimensional harmonic oscillator 2-2 2.3 Schrödinger’s wave mechanics 2-4 2.4 The one-dimensional harmonic oscillator (again) 2-6 2.5 Comparison of Heisenberg and Schrödinger theories 2-8 2.6 Wave mechanics in three dimensions 2-9 2.7 Angular momentum 2-10 2.8 Schrödinger equation for a spherically symmetric potential 2-14 2.9 Schrödinger equation for the hydrogen atom 2-15 2.10 Time-dependent wave equation 2-18 2.11 The time-evolution operator 2-20 2.12 The time dependence of Heisenberg’s operators 2-21 vii ModernQuantumMechanicsandQuantumInformation Problems 2-22 References 2-23 3 Relativistic quantum mechanics 3-1 3.1 The necessity for relativistic quantum mechanics 3-1 3.2 Klein–Gordon equation 3-1 3.3 Problems with the Klein–Gordon equation 3-4 3.4 Dirac theory 3-7 3.5 Proof of the Lorentz covariance of the Dirac equation 3-9 3.6 The fifth gamma matrix 3-15 3.7 Free particle solution of the Dirac equation 3-15 3.8 Angular momentum and spin 3-18 3.9 The magnetic moment of the electron 3-20 3.10 Scalar relativistic approximation 3-21 3.11 The Dirac theory of the hydrogen atom 3-23 3.12 Advantages and disadvantages 3-26 Problems 3-28 References 3-28 4 Symmetry 4-1 4.1 The importance of symmetry in physics 4-1 4.2 A simple example 4-2 4.3 Theory of finite groups 4-4 4.4 Representations of finite groups 4-6 4.5 Theory of infinite groups and Lie groups 4-9 4.6 Continuous groups in physics 4-10 4.7 Conservation laws from Noether’s theorem 4-16 4.8 Conservation laws from quantum mechanics 4-17 4.9 Continuous group representations 4-21 4.10 Groups of a Hamiltonian 4-25 4.11 Conclusions 4-26 Problems 4-27 References 4-27 5 Approximate methods 5-1 5.1 Rayleigh–Ritz variational method 5-1 5.2 Time-independent perturbation theory 5-6 viii ModernQuantumMechanicsandQuantumInformation 5.3 Time-dependent perturbation theory 5-8 5.4 The two-level Hamiltonian 5-10 5.5 Spin magnetic resonance 5-11 5.6 The maser 5-13 5.7 Fermi’s golden rule 5-13 5.8 An atom interacting with a plane electromagnetic wave 5-16 5.9 Approximate methods that use computers 5-19 Problems 5-21 References 5-21 6 Scattering and Green’s functions 6-1 6.1 Potential scattering 6-1 6.2 Position representation 6-2 6.3 The spherical scatterer 6-4 6.4 The optical theorem 6-6 6.5 The Born approximation 6-7 6.6 Green’s function and its adjoint 6-8 6.7 Green’s function with a scatterer 6-9 6.8 The non-spherical scattering potential with bounded domain 6-10 6.9 Spectral theory from scattering theory 6-13 6.10 Krein’s theorem 6-13 Problems 6-16 References 6-17 7 A practical tool 7-1 7.1 The exact equations 7-1 7.2 Pauli exclusion principle 7-4 7.3 Atomic structure 7-5 7.4 The hydrogen molecule 7-6 7.5 Covalent bonding 7-12 7.6 Ionic bonding 7-15 7.7 Bonding in metals 7-21 7.8 Conclusions 7-22 Problems 7-22 Reference 7-23 ix