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The Real Quantum Mechanics

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The Real Quantum Mechanics In A Brief History of Time, Stephen Hawking said: “Quan- tum mechanics underlies all of modern science and technology. It governs the behavior of transistors and integrated circuits, and is the basis of modern chemistry and biology”. However, many quantum mechanics textbooks are loaded with burden- some mathematics, such as complex numbers, Hilbert space, probability theory, as well as incomprehensible interpretations, causing quantum mechanics difficult to learn. In this entry-level quantum mechanics textbook, by elimi- nating those unnecessary digressions, the natural beauty of the central piece of modern physics emerges. Starting with the de Broglie wave, the Schr¨odinger equation arises intuitively and log- ically. Quantum mechanics appears as deterministic and as tan- gible as Maxwell’s theory of electromagnetism. There are no unintellegible paradoxes. The prerequisite for this course is el- ementary calculus at the high-school advanced-placement level. Partial differential equations are taught by examples in acous- tic waves and electromagnetism. Concepts in quantum physics are introduced by descriptions of experiments, including pho- toelectric effect, Millikan’s oil-drop experiment, diffraction and interference of electrons, and the Stern-Gerlach experiment. Ad- vanced topics including Hartree-Fock approximation with the Roothaan method, density functional theory (DFT) with local density approximation (LDA) and generalized gradient approx- imation (GGA), interaction of atomic systems with radiation, and elementary quantum electrodynamics are explained. While Dirac equation is only conceptually described, its non-relativistic version, Pauli equation, is presented mathematically. C. Julian Chen joined IBM T.J. Watson Research Center in 1985 as a Research Staff Member in the Department of Physical Sciences, doing experimental and theoretical research on scan- ning tunneling microscopy. He authored Introduction to Scan- ning Tunneling Microscopy (Oxford University Press 1993, 2007, 2020), a standard reference book in nanoscience. In 2007, he joined the Department of Applied Physics and Applied Math- ematics of Columbia University. For ten consecutive years, he teaches a graduate-level course Physics of Solar Energy, explain- ing the quantum physics of solar cells and photosynthesis. Contents Chapter 1: A Review of Classical Physics 1 1.1Newtonianmechanics...................... 1 1.1.1 Newton’s second law of mechanics ........... 1 1.1.2 Conservativesystemsandtheenergyintegral..... 3 1.1.3 Thependulum...................... 4 1.1.4 TheKeplerproblem................... 6 1.2Vibrationofstringsandmembranes.............. 9 1.2.1 Vibrationsofstrings................... 9 1.2.2 Vibrationofmembranes:thetimpani......... 15 1.3Soundwaveinair......................... 21 1.3.1 Derivationofthewaveequation............. 21 1.3.2 Thebugle......................... 23 1.3.3 Propagation of sound in open space .......... 24 1.3.4 Resonancefrequenciesofamusicroom......... 25 1.4Thesoundofbasketballs..................... 27 1.4.1 Waveequationinsphericalpolarcoordinates..... 27 1.4.2 Spherical harmonics ................... 29 1.4.3 Eigenfrequenciesofstandingwaves........... 31 1.4.4 Degeneracyandsymmetry................ 33 1.5Lightaselectromagneticwave.................. 34 1.5.1 Newton’scorpusculartheoryoflight.......... 34 1.5.2 Young’sdouble-slitexperiment............. 36 1.5.3 Maxwell’s theory of electromagnetic fields ....... 37 1.5.4 Electromagneticwaves.................. 42 1.5.5 Polarizationoflight................... 43 1.5.6 Doublerefraction..................... 47 1.6AtomicSpectra.......................... 49 Chapter 2: Wave and Quantum 53 2.1Einstein’senergyquantumoflight............... 53 2.1.1 Thephotoelectriceffect................. 55 2.1.2 Einstein’slawofphotoelectricaleffect......... 56 2.1.3 Millikan’s experimental verification ........... 57 2.1.4 Thethree-polarizerexperiment............. 59 2.1.5 Double-slit experiment with single photon detectors . 60 2.2Electronasaquantizedwave.................. 61 2.2.1 Experimentalstudiesofthecathoderay........ 61 ii Contents 2.2.2 Millikan’s oil-drop experiment .............. 63 2.2.3 ThedeBrogliewave................... 65 2.2.4 Low-energyelectrondiffraction............. 67 2.2.5 Double-slit experiment with single electron detectors . 69 2.3Electrostaticmassandradiusoftheelectron......... 70 2.4TheStern-Gerlachexperiment.................. 71 Chapter 3: Schr¨odinger’s Equation I 73 3.1 Time-independent Schr¨odingerequation............ 74 3.2 Wavefunctions in potential wells ................ 74 3.2.1 One-dimensionalpotentialwell............. 75 3.2.2 TheDiracnotation.................... 77 3.2.3 Two-dimensionalpotentialwell............. 78 3.2.4 Wavefunctions outside a spherical potential well . 80 3.3 The harmonic oscillator ..................... 81 3.3.1 Creationoperatorandannihilationoperator...... 82 3.3.2 Algebraic solution of the Schr¨odingerequation.... 83 3.3.3 Explicit expressions of the wavefunctions ....... 84 3.4Thehydrogenatom........................ 86 3.4.1 Thegroundstate..................... 87 3.4.2 EnergyeigenvaluesofexcitedStates.......... 89 3.4.3 Wavefunctions...................... 91 3.4.4 Nomenclatureofatomicstates............. 93 3.4.5 Degeneracy and wavefunction hybridization ...... 94 3.5 General properties of wavefunctions .............. 96 3.5.1 Normalization....................... 97 3.5.2 Orthogonality . ..................... 97 3.5.3 Completeness....................... 97 3.5.4 Chargedensitydistributions............... 98 Chapter 4: Many-Electron Systems 101 4.1 Many-electron Schr¨odingerequation..............101 4.2TheHartree-Fockmethod....................103 4.2.1 Theself-consistentfield.................103 4.2.2 Pauli exclusion principle and Slater determinants . 104 4.2.3 Theelectronspin.....................105 4.2.4 Exchangeinteraction...................106 4.3Theatoms.............................107 4.3.1 Atomic basis functions ..................107 4.3.2 TheRoothaan-Hartree-Fockmethod..........109 Contents iii 4.3.3 Lithium..........................110 4.3.4 Carbon..........................111 4.3.5 Accuracyofself-consistentcomputations........112 4.4 Density functional theory ....................112 4.4.1 TheHohenberg-Kohntheorem.............113 4.4.2 TheKohn-Shamequations...............114 4.4.3 Localdensityapproximation..............115 4.4.4 Generalizedgradientapproximation..........115 Chapter 5: The Chemical Bond 117 5.1Theconceptofchemicalbond..................117 5.1.1 Formationofmolecularorbitals.............118 5.1.2 Bondingenergyasasurfaceintegral..........119 5.2Thehydrogenmolecularion...................121 5.2.1 VanderWaalsforce...................123 5.2.2 Evaluationofthesurfaceintegral............125 5.2.3 Comparewiththeexactsolution............127 5.3Covalentbondsofmany-electronatoms............128 5.3.1 Theblack-ballmodelofatoms.............129 5.3.2 Wavefunctionsoutsidetheatomiccore.........131 5.3.3 Thederivativerule....................132 5.3.4 Typesofchemicalbonds.................134 5.3.5 Comparingwithexperimentaldata...........136 5.4 Imaging wavefunctions with AFM ...............140 Chapter 6: Schr¨odinger’s equation II 141 6.1 Time-dependent Schr¨odingerequations.............141 6.1.1 Properties of the real wavefunctions ..........143 6.1.2 ParallelismtoMaxwell’sequations...........144 6.1.3 Time-independent Schr¨odinger’sequation.......146 6.2Electronasamacroscopicparticle...............147 6.2.1 Born’sstatisticalinterpretation.............147 6.2.2 Wavepacketsasmacroscopicparticles.........148 6.2.3 Similarityofphotonsandelectrons...........150 6.3Ehrenfest’stheorem.......................151 Chapter 7: Perturbation Theories 153 7.1Stationaryperturbationtheory.................153 7.1.1 Polarizationofhydrogenatom.............155 + 7.1.2 The van der Waals force in H2 .............157 iv Contents 7.2Interactionwithradiation....................158 7.2.1 Time-dependentperturbationtheory..........158 7.2.2 Thegoldenrule......................159 7.3 Imaging wavefunctions with STM . ..............161 Chapter 8: Quantum Theory of Light 163 8.1Blackbodyradiation.......................163 8.1.1 Modesofelectromagneticwavesinacavity......164 8.1.2 Rayleigh–Jeanslaw....................166 8.1.3 PlanckformulaandStefan–Boltzmann’slaw......167 8.1.4 Einstein’sderivationofblackbodyformula.......168 8.2Therealquantumelectrodynamics...............170 8.2.1 Quantizationofelectromagneticwaves.........171 8.2.2 Indenticalparticles:bosonsandfermions.......173 8.2.3 Theanticommutationrelation..............174 8.2.4 Secondquantization...................174 8.2.5 Interactionofradiationwithatomicsystems.....174 Chapter 9: Spin and Pauli Equation 175 9.1ThePauliequation........................175 9.1.1 TheDiracequationandtheelectronspin.......175 9.1.2 TheRealPauliMatrices.................176 9.1.3 Pauliequationinamagneticfield...........176 9.2TheStern-Gerlachexperiment..................177 9.2.1 Similaritytopolarizationoflight............178 9.2.2 AnalysisbasedonEhrenfest’stheorem.........178 Appendix A: Units and Physical Constants 181 Appendix B: Vector Analysis 183 Appendix C: Bessel Functions 185 Appendix D: Statistics of Particles 189 D.1Maxwell–BoltzmannStatistics..................190 D.2Fermi–DiracStatistics......................192 D.3Bose-EinsteinStatistics.....................193 Contents v Appendix E: Real Matrix Formulation 195 E.1 Schr¨odingerequationinfreespace...............195 E.2Electroninafield.........................196 E.3AngularMomentum.......................197
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