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 E.4HydrogenAtom...... 199 List of Figures

1.1 Newton’sPrincipia...... 2 1.2 The pendulum ...... 4 1.3 Energy conversion of a simple harmonic oscillator ...... 6 1.4 TheKeplerProblem...... 7 1.5 Derivationofthewaveequation...... 9 1.6 Running waves ...... 11 1.7 Overtonesonastring:ademo...... 13 1.8 Overtonesonastring:mechanism...... 14 1.9 Vibrationofacircularmembrane...... 15 1.10 Bessel functions ...... 16 1.11 Eigenfunctions of the vibration modes ...... 18 1.12 Nodes in eigenfunctions revealed by Chladni patterns ...... 19 1.13 Linear superposition of degenerate eigenfunctions ...... 20 1.14 Derivationofthewaveequation...... 22 1.15 Thebugle...... 23 1.16 Propagation of sound in open space ...... 24 1.17 Dimensionsofamusicroom...... 25 1.18 Resonancemodesinamusicroom...... 26 1.19 Laplaceoperatorinsphericalcoordinates...... 27 1.20 Modesofstandingwavesinabasketball...... 32 1.21 Newton’sOpticksandHuyges’Lumiere...... 34 1.22 DispersionofprismandNewton’sexplanation...... 35 1.23 Young’sdouble-slitexperiment...... 36 1.24 JamesClerkMaxwell...... 38 1.25 Observingandmappingmagneticfield...... 40 1.26 Observingandmappingelectricfield...... 41 1.27 Electromagneticwave...... 44 1.28 Thethreepolarizerexperiment:Step1...... 45 1.29 Thethreepolarizerexperiment:Step2...... 45 1.30 Thethreepolarizerexperiment:Step3...... 46 1.31 Thethreepolarizerexperiment:analysis...... 46 1.32 Doublerefractionofcalcite...... 47 1.33 Refractionofalaserbeambycalcite,caseA...... 48 1.34 Refractionofalaserbeambycalcite,caseB...... 48 1.35 Aschematicofdiffractiongrating...... 49 1.36 Absorptionatomicspectrainthevisiblerange...... 50 1.37 Emissionspectraofhydrogen...... 51

2.1 Einstein’spaperonenergyquantumoflight...... 54 2.2 Lenard’sapparatusforstudyingphotoelectriceffect...... 55 viii List of Figures

2.3 Result of Millikan’s experiment on photoelectric effect ..... 58 2.4 Albert Einstein and Robert Millikan ...... 59 2.5 ParadoxofDirac’sthree-polarizerexperiment...... 60 2.6 Paradoxofthedouble-slitexperiment...... 61 2.7 SchematicsofJ.J.Thomson’sexperiment...... 62 2.8 Schematics of Millikan’s oil-drop experiment ...... 64 2.9 LouisdeBroglie...... 65 2.10 SchematicsofLEED...... 67 2.11 WorkingprincipleandobservedpatternofLEED...... 68 2.12 Double-slit interference experiment with single-electron detection 70 2.13 Electrostaticmassofanelectron...... 71 2.14 SchematicsofStern-Gerlachexperiment...... 72

3.1 Austrian banknote with a portrait of Schr¨odinger...... 73 3.2 Wavefunctions in a one-dimensional potential well ...... 75 3.3 Energylevelsinaone-dimensionalpotentialwell...... 76 3.4 Wavefunctions in a two-dimensional potential well ...... 79 3.5 Wavefunctions outside a spherical potential well ...... 81 3.6 Energy levels and wavefunctions of a harmonic oscillator .... 85 3.7 Hydrogenatominsphericalpolarcoordinates...... 86 3.8 Wavefunction of ground-state hydrogen atom ...... 88 3.9 Wavefunctions of excited-states of hydrogen atom ...... 90 3.10 Hydrogen wavefunctions ...... 92 3.11 Hybrid sp1 wavefunctions ...... 94 3.12 Hybrid sp2 wavefunctions ...... 95 3.13 Hybrid sp3 wavefunctions ...... 96 3.14 Chargedensityofthe2pstates...... 98

4.1 Energydiagramofheliumatom...... 105 4.2 Observedspectrumofheliumatom...... 106 4.3 Electrondensitydistributionsofseveralbasisorbitals.....108 4.4 Data for the wavefunctions of lithium ...... 110 4.5 Electrondensitydistributionsoflithium...... 110 4.6 Data for the wavefunctions of carbon ...... 111 4.7 Electrondensitydistributionsofcarbon...... 111 4.8 Electrondensitydistributionofargon...... 112

5.1 Conceptofchemicalbond...... 118 5.2 Potentialcurveforthehydrogenmolecularion...... 122 5.3 Perturbationtreatmentofthehydrogenmolecularion.....123 5.4 Wavefunctions of the hydrogen molecular ion ...... 124 5.5 Evaluationofthecorrectionfactor...... 125 + 5.6 Accuracy of the perturbation treatment of H2 ...... 127 5.7 Theblack-ballapproximationforthechemicalbond...... 129 5.8 Wavefunctions outside the atomic core ...... 132 List of Figures ix

5.9 Molecular orbitals built from two s-typeatomicorbitals....135 5.10 The pσ and pσ∗ molecularorbitals...... 136 5.11 The pπa and pπ∗ molecularorbitals...... 136 5.12 Covalent bond energy and Morse function ...... 137 5.13 Comparingwithexperimentaldata...... 139 5.14 Schematicsofatomicforcemicroscopy...... 140

6.1 Spreadingofawavepacket...... 149

7.1 Ground-state and perturbative wavefunctions ...... 156 7.2 Conditionofenergyconservation...... 160 7.3 Schematics of scanning tunneling microscopy ...... 161

8.1 Blackbodyradiation...... 164 8.2 Radiationinacavity...... 165 8.3 Rayleigh-JeanslawandPlanck’slaw...... 167 8.4 Einstein’sderivationofblackbodyradiationformula...... 170

C.1 Spherical modified Bessel functions ...... 187 List of Tables

1.1Vibrationmodesonastring...... 13 1.2 Zeros of Bessel functions ...... 17 1.3 Spherical Harmonics ...... 30 1.4Modesofstandingwavesofabasketball...... 33 1.5Wavelengthsofhydrogenlines...... 52

2.1StoppingVoltageforPhotocurrent...... 56 2.2Wavelengthandappliedvoltage...... 69

3.1 Wavefunctions of the harmonic oscillator ...... 84 3.2 Wavefunctions of the hydrogen atom ...... 93 3.3Nomenclatureofatomicstates...... 94

4.1Atomicunits...... 103

5.1Potentialcurveofhydrogenmolecularion...... 128 5.2 Examples of atomic wavefunction data ...... 130 5.3Parametersofhomonucleardiatomicmolecules...... 137

A.1Unitsandphysicalconstants...... 181 Chapter 1 AReviewofClassicalPhysics

Classical physics dominated physics up to the year of 1900. Quantum me- chanics, developed during the first 30 years of the twentieth century, became the most successful and most useful theory in physics. However, in spite of the importance of quantum mechanics to the understanding of the atomic world, classical physics is still the base of our understanding of the macro- scopic world. Furthermore, in order to understand quantum mechanics, familiarity to the concepts in classical physics is absolutely essential.

1.1 Newtonian mechanics

The publication of Mathematical Principle of Natural Philosophy by Isaac Newton in 1686 marked the beginning of modern science. In this monu- mental monograph, among other items, Newton formulated the three laws of mechanics and the law of universal gravity, then explained the Kepler’s laws of the motions of planets and moons in the solar system, and the tan- gible world around us. For several hundreds of years, Newton’s mechanics was synonymous to physical science. According to Albert Einstein,

In accordance with Newton’s system, physical reality is char- acterized by concepts of space, time, the material points and force (interaction between material points). Physical events are to be thought of as movements according to the law of material points in space. The material point is the only representative of reality so far as it is subject to change. The concept of material points is obviously due to observable bodies; one conceived of the material point in the analogy of movable bodies by omitting characteristics of extension, form, spatial locality, and all their ‘inner’ qualities., retaining only inertia, translation, and addi- tional concept of force. ... . All happening was to be conceived of as purely mechanical, that is, merely as motions of material points according to the Newton’s laws of motion.

1.1.1 Newton’s second law of mechanics The core of classical mechanics is Newton’s second law. A material point is characterized by an intrinsic value of mass m, a measure of its inertia. At 2 A Review of Classical Physics

Fig. 1.1. Newton’s Principia. In 1686, Isaac Newton published his monumental monograph, Mathemati- cal Principles of Natural Philosophy. He defined three laws of mechanics, the law of gravitation, and explained the motion of planets and satellites of the planets among a large number of other subjects. Newton described the ma- terial world as composes of material points, each has a well-defined geomet- rical location and well-defined speed at any given time.

any well-defined time instant t, a material point has a well-defined position, represented a vector r =(x, y, z) in three-dimensional space, and a well- defined velocity v as the time derivative of r,

dr v = . (1.1) dt Newton’s second law of mechanics states that at any time, the accerelation of a material point, or the rate of change of its velocity, is proportional to the force F acting on it and inversely proportional to its mass,orinertia, an intrinsic property of the material point:

d2r dv m = m = F. (1.2) dt2 dt

According to Newton’s laws, the physical world is deterministic:Atany time, if the positions and the velocities of the material points and the laws of force are known, the system will evolve precisely according to the second law of mechanics, Eq. 1.2. By defining a momentum as the product of the mass and the velocity of the material point, p = mv, (1.3)

Newton’s second law can be written in a more compact form,

dp = F. (1.4) dt Because mass m is an intrinsic property of a material point, the validity of Eq. 1.4 is obvious. 1.1 Newtonian mechanics 3

1.1.2 Conservative systems and the energy integral In all cases we treat in this book, the force can be expressed as a gradient of a scalar function of the coordinates, the potential V (r),

F = −∇V (r). (1.5)

A necessary and sufficient condition for a conservative system is that the curl of force field is zero, ∇×F =0. (1.6) Newton’s second law is then dp + ∇V (r)=0. (1.7) dt

By multiplying both sides of Eq. 1.7 with dr, and integrate from r1 to r2, notice that dr 1 dr = dt = p dt, (1.8) dt m we have r r2 1 dp 1 2 p dt + dV (r) = p2 + V (r) . (1.9) r1 m dt 2m r1 The following identity is obtained: 1 1 p2 + V (r) = p2 + V (r) . (1.10) 2m r=r1 2m r=r2

The expression in the square bracket is independent of position and time. It is the total energy E of the system, which is a constant, only depends on the initial condition: p2 + V (r)=E. (1.11) 2m Thefirsttermiscalledthekinetic energy of the system,

p2 T = . (1.12) 2m Equation 1.11 can be written in a conceptually simpler form, such that the total energy is the sum of kinetic energy and potential energy,

T + V = E. (1.13)

In the following subsections, we will discuss two cases in detail, both are related to the understanding of quantum mechanics. 4 A Review of Classical Physics

1.1.3 The pendulum The pendulum, shown in Fig. 1.2, is a classical prototype of the harmonic oscillator in quantum mechanics. A material particle with mass m is hung with a flexible string of length L. A gravitational force mg is acting on the particle. A component of the gravitational force F drives the particle to its equilibrium position b. Based on elementary geometry, the component force F can be estimated as follows. The triangle marked light green is similar to the triangle marked yellow. The sides are proportional: F x = − . (1.14) mg L The negative sign means that the direction of F and x are opposite. If dis- placement x is much smaller than length L, the arc bc is practically identical to the horizontal line x. To resolve this problem according to Newton’s law, we apply the energy integral, Eq. 1.11 in the previous subsection. The potential energy function, by definition, is x x mg mg V (x)=− Fdx= xdx= x2. (1.15) 0 0 L 2L The momentum is a scalar, dx p = m . (1.16) dt Therefore, the energy integral is p2 mg + x2 = E. (1.17) 2m 2L

Fig. 1.2. The pendulum. In 1602, by watching the motion of a chandelier hanging in a cathedral, Galileo Galilee started to study the motions of pen- dulum. He discovered that the pen- dulum has a constant period, inde- pendent of the amplitude and the an- gle. The period only depends on the length L of the string, but indepen- dent of the weight m. In 1657, Christi- aan Huygens invented a mechanism to sustain the vibration of the pendulum. The mechanical clock was born. It re- mained to be the most accurate clock up to early twentieth century. 1.1 Newtonian mechanics 5

Using Eq. 1.16, it can be written in a convenient form dx 2 g 2 + x2 = E. (1.18) dt L m

When the particle moves to the extreme positions, x = a or x = −a,the kinetic energy is zero. The potential energy equals the total energy. Using the amplitude parameter a, Eq. 1.18 can be written as dx 2 g = a2 − x2 . (1.19) dt L

By taking a square root, the equation becomes directly integrable: dx g √ = dt. (1.20) a2 − x2 L Defining an angular frequency g ω ≡ , (1.21) L the integration gives x arcsin = ωt + φ, (1.22) a where the constant of integration φ is a phase angle. In other words,

x = a sin(ωt + φ). (1.23)

The pendulum makes a simple harmonic oscillation. Because the period of a sine function is 2π, the period of the pendulum is 2π L τ = =2π , (1.24) ω g and the frequency is ω 1 g f = = . (1.25) 2π 2π L Using angular frequency as a parameter, the energy integral, Eq. 1.17, can be written in a more general form, 1 m T + V = p2 + ω2x2 = E. (1.26) 2m 2 Figure 1.3 shows the simple harmonic oscillation of a pendulum, and the process of energy conversion. At t = 0, the material particle is at its right-most position, x = a, The potential energy is at its maximum, equals 6 A Review of Classical Physics

Fig. 1.3. Energy conversion of a simple harmonic oscillator. At time 0, the pendulum is at one of its extreme positions. the potential energy is at a maximum, and the kinetic energy is zero. At a quarter of a period τ, potential energy converts into kinetic energy. The process goes on. the entire total energy. The kinetic energy is zero. In a quarter of a period, t = τ/4, the particle moves to the equilibrium position x =0.Atthattime, the kinetic energy reaches its maximum, but the potential energy reaches its minimum. Because of inertia, the material particle continuous its motion to the left side, x<0. After another quarter period, the particle reaches its left most position and stops. The potential energy again reaches its max- imum, whereas the kinetic energy becomes zero. In the third quarter of a period, the particle moves in the positive direction, and regains speed. After passing the equilibrium position x = 0, the particle moves continuously to its original position, x = a, where the kinetic energy becomes zero and the potential energy reaches its maximum, that is, the total energy E.Thus the particle completes a full period τ.

1.1.4 The Kepler problem A central problem in Newtonian mechanics is the Kepler problem, where a planet is attracted by the Sun. The greatest achievement of Isaac New- ton was the interpretation of Kepler’s laws using his laws of mechanics and universal gravitation. It is also a central problem in quantum mechan- ics as a model of the hydrogen atom. The greatest achievement of Erwin Schr¨odinger is the explanation of the Rydberg formula using quantum me- chanics, see Sections 1.6 and 3.4. The Kepler problem is schematically shown in Fig. 1.4(A), in both rectangular coordinate system and polar co- ordinate system. The potential function is, K V (r)= , (1.27) r 1.1 Newtonian mechanics 7

Fig. 1.4. The Kepler Problem. (A), in Cartesian coordinate system. (B), in polar coordinate system. A planet of mass m is attracted by the Sun according to Newton’s inverse-square law of gravitation. The eccentricity vector e always points to the perihelion, where the planet is closest to the Sun. Thus e is a constant of motion. where K is a constant. According to Eq. 1.3, the force is

K r F = − . (1.28) r2 r The direction of the force is towards the Sun, marked as O, and the magni- tude is inversely proportional to the distance of the Sun and the planet r. Newton’s equation is dp K r = − . (1.29) dt r2 r Following Eq. 1.11, we can write down the energy integral,

1 K p2 + = E, (1.30) 2m r where m is he mass of the planet. An interesting solution is based on the eccentricity vector e, see Fig 1.4(B). As a consequence of the inverse square law of gravity, the eccentricity vec- tor e is invariant. Assume the planet moves in the xy-plane. The angular momentum, a vector in the z-direction, is defined as

dr L = r × p = mr × . (1.31) dt As a consequence of Eq. 1.29, the angular momentum is a constant, dL dr dr K r = m × + r × − =0. (1.32) dt dt dt r2 r 8 A Review of Classical Physics

Consider the time evolution of a vector p × L. Because L is a constant, only the time evolution of p has to be counted. Using Eq. 1.29, as well as the following identities, a × (b × c)=b(a · c) − c(a · b) (1.33) and dr dr r · = r , (1.34) dt dt we obtain d K r dr (p × L)=− × mr × dt r2 r dt mK dr dr = r r · − r2 r3 dt dt (1.35) 1 dr r dr = mK − r dt r2 dt d r = mK . dt r Therefore, the eccentricity vector p × L r e ≡ − (1.36) mK r is a constant. It points to the position of minimum distance r, called by astronomers as perihelion, the closest point to the Sun. Using the constant vector e, the trajectory of Kepler motion can be obtained. Denoting the magnitude of the eccentricity vector as e, and taking the aphelion point as the origin with θ =0,wehave r · e = −er cos θ. (1.37) On the other hand, from Eq. 1.36, r · (p × L) L · (r × p) L2 r · e = − r = − r = − r, (1.38) mK mK mK where L is the magnitude of angular momentum. The solution is L2 1 r = . (1.39) mK 1 − e cos θ Comparing with Fig 1.4, we have the semilatus rectum L2 p = , (1.40) mK the standard formula of the ellipse is obtained p r = , (1.41) 1 − e cos θ which explains the origin of the term eccentricity vector. 1.2 Vibration of strings and membranes 9

1.2 Vibration of strings and membranes

The images and properties of wavefunctions in quantum mechanics are very similar to the acoustic waves in macroscopic world. The mathematics is almost identical. Therefore, the vibrations of strings and membranes are instructive for an intuitive understanding of wavefunctions. In quantum mechanics, the concepts of superposition and orthogonality are essential. Those concepts are intuitive and obvious in terms of acous- tic waves. By familiarizing with the examples in acoustic waves, similar concepts in quantum mechanics can be easily understood.

1.2.1 Vibrations of strings The phenomena we are discussing here are related to any string instruments, for example, guitar, violin, cello, and piano. By doing experiments on those string instruments, one can make direct observations. Figure 1.5 shows a derivation of the wave equation. Consider a small section of a string, between x and x +Δx. The lateral displacement u(x, t) is a function of x and time t.AtensionT is applied on both sides. As shown in Fig. 1.5, the lateral force acting on the small section is ∂u ∂u ∂2u F = T (x +Δx) − T (x) ≈ T Δx. (1.42) ∂x ∂x ∂x2 The mass of the small section is ρΔx. According to Newton’s law, ∂2u F = ρΔx . (1.43) ∂t2 Combining Eqs. 1.42 and 1.43, we obtain ∂2u T ∂2u = . (1.44) ∂t2 ρ ∂x2 By denoting v = T/ρ, Eq. 1.44 is brought to a standard form, ∂2u ∂2u = v2 . (1.45) ∂t2 ∂x2

Fig. 1.5. Derivation of the wave equation. The lateral displacement u(x, t)isa function of x and time t. A tension T is applied on both sides of the small section Δx. The wave equation is a consequence of Newton’s law. 10 A Review of Classical Physics

Here we show that v is the velocity of sound. On a string of infinite length, the general solution of the wave equation Eq. 1.45 was obtained by French mathematician and physicist Jean le Rond d’Alembert in 1747,

u(x, t)=F (x − vt)+G(x + vt), (1.46) where F (x)andG(x) are two independent, arbitrary functions. The solu- tion can be proved by direct substitution. On one hand, ∂u(x, t) = vF(x − vt) − vG(x + vt), (1.47) ∂t thus ∂2u(x, t) = v2 F (x − vt)+v2 G(x + vt). (1.48) ∂t2 On the other hand, ∂2u(x, t) = F (x − vt)+G(x + vt). (1.49) ∂x2 Therefore, the d’Alembert solution satisfies the wave equation, Eq. 1.44. It is a combination of a wave F (x−vt) propagating in +x direction at velocity v,andawaveG(x + vt) propagating in −x direction at velocity −v. A special case of the d’Alembert solution is the sinusoidal wave. If the frequency is f, the variation of displacement with time is

u(x, t) ∼ sin(2πft + φ), (1.50) where φ is the phase. The factor 2π occurs frequently. It is convenient to introduce an angular frequency to eliminate it:

ω ≡ 2πf. (1.51)

Equation 1.50 is simplified to

u(x, t) ∼ sin(ωt + φ). (1.52)

Just as there are two d’Alembert solutions moving into two opposite directions, there could have two sinusoidal waves moving into opposite di- rections: A wave moving in +x direction is,

u(x, t)=a sin(kx − ωt + φ), (1.53) and a wave moving in −x direction is,

u(x, t)=a sin(kx + ωt + φ), (1.54) where k is the wave number, which has a dimension of L−1: ω k = . (1.55) v 1.2 Vibration of strings and membranes 11

Fig. 1.6. Running waves. (A) A running wave in positive x direction, Eq. 1.53. (B) A running wave in negative x direction, Eq. 1.54. Phase of wave is marked by colors; for example, red as positive, blue as negative. Intensity indicates amplitude. A complete period in length scale λ is a wavelength, which contains a positive half-wavelength marked red, and a negative half-wavelength marked blue.

Figure 1.6 shows the waves propagating in +x direction and in −x direction. The phases are marked by color. For example, red indicates positive, and blue indicates negative. The intensity of color indicates amplitude. The length of a complete period is a wavelength,withasymbolλ. By definition,

v 2πv 2π λ = = = . (1.56) f ω k

Principle of superposition and interference

The wave equation is linear. If a wave f1(x, t) is a solution of a wave equation, Eq. 1.45, and another wave f2(x, t) is also a solution, then any linear superposition of the two waves

f(x, t)=c1 f1(x, t)+c2 f2(x, t), (1.57) is also a solution of the same wave equation, where c1 and c2 are arbitrary constants. This statement can readily verified by inserting the expression of the new wave into Eq. 1.45. The superposition of waves gives rise to the interference of waves. Con- sider two sinusoidal waves of the same amplitude but difference phase,

u1(x, t)=a sin(kx − ωt + φ1), (1.58) and

u2(x, t)=a sin(kx − ωt + φ2), (1.59) 12 A Review of Classical Physics

The sum is

u(x, t)=u1(x, t)+u2(x, t),

= a sin(kx − ωt + φ1)+a sin(kx − ωt + φ2) (1.60) φ + φ φ − φ =2a sin kx − ωt + 1 2 cos 1 2 . 2 2

The result depends on the phase difference. If the phase difference is an integer multiple of 2π,thereisapositive interference. The amplitude is doubled, thus the power it quadrupled. If the phase difference is an odd integer multiple of π,thereisanegative interference. The amplitude is zero. The power vanishes. The interference phenomenon is unique to waves. By combining two beams of particles, the energy of the composite beam is the simple addition of the individual beams. For waves, depending on the relative phase, the energy of the composite beam could be much greater than the simple sum, or can be mutually cancelled, see Section 1.5.2. The principle of superposition is valid for all linear differential equations, including Schr¨odinger’s equation. Superposition in acoustic phenomena is easily visualized, which is helpful for the understanding of superposition in quantum mechanics.

Standing wave and the Helmholtz equation To describe the vibration of a string with both ends fixed at a fixed fre- quency, the standard way is to write the displacement u(x, t) as a product of a function of x and a sinusoidal function of time,

u(x, t)=u(x)sin(ωt + φ). (1.61)

Insert Eq. 1.61 into Eq. 1.45, we find a differential equation for u(x):

d2u(x) ω2 = − u(x). (1.62) dt2 v2 Using the wave vector k defined in Eq. 1.55, Eq. 1.62 becomes

d2u(x) = −k2u(x), (1.63) dx2 which is called a Helmholtz Equation.

Eigenvalues and eigenfunctions On string instruments, the strings are clamped at both ends, see Fig. 1.7. Let the ends be x = 0 and x = L. The values of displacement u(x)mustbe zero at both ends. The condition limits the values of k and ω in Eqs. 1.62 1.2 Vibration of strings and membranes 13

Fig. 1.7. Overtones on a string: a demo. An experiment on the string of note E1 of a grand piano. By lightly touching various points on the string and strike it, the overtones of the basic frequency of the string are excited. See Table 1.1 and Fig. 1.8. and 1.63, and also limits the waveform u(x).Theallowedvaluesfork and ω under the boundary conditions are called eigenvalues,andtheal- lowed waveforms are called eigenfunctions. The collection of eigenvalues and eigenfunctions are called vibration modes of the system. First, at x = 0, the string is fixed. A solution of the Helmholtz equation, Eq. 1.63, that is zero at x =0shouldbe

u(x)=a sin kx. (1.64)

At x = L, the string is also fixed. That boundary condition, u(x)=0at x = L, requires the eigenfunctions to be: nπ u(x)=a sin x ,n=1, 2, 3, ... . (1.65) L The wavevector eigenvalues are nπ k = ,n=1, 2, 3, ... . (1.66) n L Accordingly, the frequency eigenvalues are

ω vk nv f = n = n = ,n=1, 2, 3, ... . (1.67) n 2π 2π 2L

Table 1.1: Vibration modes on a string

Length (mm) 1370 685 456 343 274 Wave vector (m−1) 2.29 4.57 6.88 9.17 11.46 Frequency (Hz) 41.2 82.4 123.6 164.8 206.0 Note name E1 E2 B2 E3 G#3 14 A Review of Classical Physics

Fig. 1.8. Overtones on a string: mechanism. By lightly touching various points on a string and strike it with the hammer, the overtones are excited.

The n-th standing-wave solution is nπ nv u (x, t)=a sin x sin t + φ . (1.68) n n L 2L

Nodes and overtones Table 1.1 shows the vibration modes of an E1 string on a grand piano with overtones. The total length of the string is 1370 mm. By lightly touching the middle point of the string at 685 mm, the first overtone E2 with frequency 82.4 Hz is excited. By lightly touching the point of one fifth of the string at 274 mm, the fifth overtone G#3 with frequency 206 Hz is excited. The points with no displacement are called a node, and the collection of the nodal points are called a nodal pattern.

Orthogonality of eigenfunctions An important fact of the eigenfunctions is, for vibration modes of different eigenfrequencies, the eigenfunctions are orthogonal: L un(x) um(x) dx =0,n= m. (1.69) 0 From Eq. 1.68, it is obvious that unless m = n, the integral is zero. Another interesting fact worth noting is, for vibration modes of different eigenfrequencies, the number of nodes are different. In fact, the number of nodes for the five vibration modes are 0, 1, 2, 3, and 4, respectively. 1.2 Vibration of strings and membranes 15

1.2.2 Vibration of membranes: the timpani The vibration modes on a circular membrane show more resemblance to the quantum states in atoms. An example is the timpani, a key percussion instrument in a symphonic orchestra, see Fig. 1.9 (A). A membrane, called a head, is stretched across the opening of a bowl. The tension of the membrane can be adjusted by a number of screws, called tension rods. During playing, the tension, and consequently the frequency, can be temporarily adjusted by a pedal. The typical diameter is 50 to 80 cm. In the 19th century, there were a number of studies on its mechanism, and was described in detail in Lord Rayleigh’s classical treatise Theory of Sound. The wave equation of a membrane is similar to that for a string. In Cartesian coordinates, a similar argument would lead to ∂2u T ∂2u ∂2u ∂2u ∂2u = + = v2 + . (1.70) ∂t2 ρ ∂x2 ∂y2 ∂x2 ∂y2 Here the tension T is the force per unit length, and the density ρ is mass per unit area fixed frequency, following the same procedure as the one- dimensional case, using u = u(x, y)sin(ωt + φ), (1.71) we find the Helmholtz equation ∂2u(x, y) ∂2u(x, y) + = −k2u(x, y), (1.72) ∂x2 ∂y2 Following Eq. 1.55, ω k = . (1.73) v

Fig. 1.9. Vibration of a circular membrane. (A), timpani, a leading percussion instrument in an orchestra. For example, in the first movement of Beethoven’s violin concerto and the second movement of his ninth symphony, the timpani takes a prominent role. (B) polar coordinates for the analysis of the circular membrane. 16 A Review of Classical Physics

Fig. 1.10. Bessel functions. Values of first three Bessel functions, up to x = 12.

Apparently, polar coordinates r and θ suit better, see Fig. 1.9 (B). The Helmholtz equation in polar coordinates is

∂2u(r, θ) 1 ∂u(r, θ) 1 ∂2u(r, θ) + + = −k2u(r, θ). (1.74) ∂r2 r ∂r r2 ∂θ2 The standard method of solving Eq. 1.74 is through separation of vari- ables. By assuming a solution of u(r, θ) as a product of a function only of r and a function only of θ,

u(r, θ)=R(r)Θ(θ), (1.75) insert into Eq. 1.74, after a few simple algebraic moves, we obtain r2 d2R(r) 1 dR(r) 1 d2Θ(θ) + + k2R(r) = − . (1.76) R(r) dr2 r dr Θ(θ) dθ2

The left-hand side of the equation only depends on r. The right-hand side of the equation only depends on θ. Therefore, both sides must be a constant K. From the right-hand side of the equation,

d2Θ(θ) + KΘ(θ)=0. (1.77) dθ2 The solution can be a sine function, a cosine function, or exponential func- tion. The boundary condition that the function must be cyclic,

Θ(2π)=Θ(0), (1.78) requires that the function Θ(θ)mustbeeither

Θ(θ)=cos(nθ),n=0, 1, 2, 3, ... , (1.79) 1.2 Vibration of strings and membranes 17

Table 1.2: Zeros of Bessel functions

Index J0(x) J1(x) J2(x)

1 x01 = 2.4048 x11 = 3.8317 x21 = 5.1356 2 x02 = 5.5201 x12 = 7.0156 x22 = 8.4172 3 x03 = 8.6537 x13 = 10.174 x23 = 11.620

or Θ(θ)=sin(nθ),n=1, 2, 3, ... . (1.80) Therefore, the constant in Eq. 1.77 is K = n2. The differential equation for the function with r is then d2R(r) 1 dR(r) n2 + + k2 − R(r)=0. (1.81) dr2 r dr r2 This is the well-known Bessel equation, and the solutions are the Bessel functions, R(r)=Jn(kr). (1.82) Mathematical details of the Bessel functions are presented in Appendix A. Figure 1.10 shows the first three Bessel functions. Table 1.2 shows the first three zeros of the first three Bessel functions. The solution of Eq. 1.81 must satisfy the boundary condition that at therimofthetimpani,wherethemembraneisfixed.Atr = a, R(r) must be zero. This condition fixes the frequency eigenvalues. The vibration pattern of the membrane at a given frequency eigenvalue must conform to the corresponding eigenfunction. Denote the m-th zero of the n-th Bessel function be xnm,theallowedwavevectorsmustbe

knma = xnm. (1.83) The frequency eigenvalues are vk vx f = nm = nm , (1.84) nm 2π 2πa and the eigenfunctions of the vibration modes are either x r u(g) (r, θ)=J nm cos(nθ),n=0, 1, 2, 3, ... , (1.85) nm n a or x r u(u) (r, θ)=J nm sin(nθ),n=1, 2, 3, ... . (1.86) nm n a Here the notation (g) indicates that the eigenfunction is symmetric, and (u) indicates that the eigenfunction is antisymmetric. A notation we will use throughout the book. 18 A Review of Classical Physics

Fig. 1.11. Eigenfunctions of the vibration modes. Different color indicates polar- ity, and the density indicates the magnitude. The places with no or very little vibration are called nodes. The nodal pattern can be visualized by spreading powers on the mem- brane, which is called the Chladni pattern according to its discoverer.

Figure 1.11 shows graphical representations of the eigenfunctions. Dif- ferent colors indicate different phases, and the density indicates the magni- tude. The places with no vibration are indicated by white space, located between regions with different phases of vibration. Similar to the case of a string, those places are called nodes. The collection of nodes is called a nodal pattern. As shown, the geometry of the nodal patterns determines the nature of the vibration mode, or the nature of the eigenfunctions. In the parentheses in Fig. 1.11, the first digit n is the order of Bessel function, and the multiplier in sine and cosine functions. The second index is m, the index of zeros of the Bessel function. The letter g indicates a symmetric angular function, associated with cosine. The letter u indicates an antisymmetric angular function, associated with sine. With a timpani of about 660 mm in diameter, the fundamental frequency is 82 Hz. As shown, the frequencies of the higher vibration modes of a timpani are not integer multipliers of the frequency, as in the case of a string, see Table 1.1. Therefore, for a timpani with a bare membrane, the overtones are inharmonic. To make the overtones at least approximately harmonic, different types of perturbations are implemented. The vibration patterns, or the eigenfunctions, can be visualized by spread- ing powers on the membrane. While the membrane vibrates, only at places with no vibration, that is, the nodes, the powder stays. The nodal pattern of a vibrating membrane can be visualized. That method was invented by German physicist Ernst Chladni in late 18th century and called Chladni patterns. The patterns in Fig. 1.12 were acquired by a group at Northern Illinois University lead by Thomas D. Rossing in the 1980s. 1.2 Vibration of strings and membranes 19

Fig. 1.12. Nodes in eigenfunctions revealed by Chladni patterns. By spreading powders on the head of a timpani, during vibration, the powders concentrate at the nodes. The pattern was discovered by German physicist Ernst Chladni, and thus named Chladni pattern. The vibration eigenfunctions are than visualized.

Orthogonality of eigenfunctions An interesting fact is that the eigenfunctions of different frequency eigen- values are orthogonal. Two functions f1(r, θ)andf1(r, θ)onanarear

f1(a)=Jn(κ1a)=0, (1.90) and f2(a)=Jn(κ2a)=0. (1.91)

The eigenfunctions f1(r)andf2(r) should satisfy the Bessel equations, d df (r) n2 r 1 − f (r)+κ2rf (r)=0, (1.92) dr dr r2 1 1 1 20 A Review of Classical Physics and d df (r) n2 r 2 − f (r)+κ2rf (r)=0. (1.93) dr dr r2 2 2 2

Multiply Eq. 1.92 by f2(r), multiply Eq. 1.93 by f1(r), subtract one from another, then integrate it over (0,a), we obtain a df (r) df (r) a 2 − 2 2 − 1 (κ1 κ2) f1(r)f2(r)rdr = r f1(r) f2(r) . (1.94) 0 dr dr 0 Because of Eqs. 1.90 and 1.91, the right-hand side is zero. Consequently, if κ1 = κ2, the integral must be zero. The two eigenfunctions are orthogonal. In quantum mechanics, such scenario is quite frequent.

Linear superposition of degenerate eigenfunctions The symmetric eigenfunctions in Eq. 1.85 and 1.86 with the same order n, as displayed in the three right-hand rows in Fig. 1.11, have the same frequency eigenvalue. Those pairs of vibration modes are called degenerate. This situation is also very frequent in quantum mechanics. Because the two degenerate eigenfunctions satisfy the same Helmholtz equation, any linear combination of the two eigenfunctions is also an eigenfunction with the same frequency eigenvalue. Therefore, by making a linear superposition of the eigenfunctions with the same eigenvalue, a new eigenfunction with the same eigenvalue is constructed. Experimentally, by lightly touching some positions on the membrane and excite the timpani with a frequency source of 131 Hz, vibration modes with nodal lines passing the point of finger touching are generated. Figure 1.13 shows two such vibration patterns by touching the membrane lightly at different places. In quantum mechanics, with the presence of a weak per- turbation to a number of degenerate wavefunctions, new wavefunctions with the same energy eigenvalue but well-defined orientations are generated.

Fig. 1.13. Linear superposition of degenerate eigenfunctions. By lightly touch- ing some positions on the membrane and excite it, linear superposition of degenerate eigenfunctions with the same frequency eigenvalue can be formed. 1.3 Sound wave in air 21

1.3 Sound wave in air

Another example of waves in everyday life is the sound wave in air. As the waves on strings and membranes are transverse waves, where the displace- ment is perpendicular to the direction of propagation; the sound wave is longitudinal, the velocity of air particles is parallel to the direction of wave propagation. Especially, by focusing on perturbation pressure, or the devi- ation of local air pressure from its equilibrium value, the wave variable is a scalar, same as the electron wave. Although sound waves are not visible by naked eyes, its effects are easily perceptible.

1.3.1 Derivation of the wave equation Figure 1.14 shows the derivation of the wave equation for a one-dimensional sound wave.1 The focus is a small elementary volume of air, often called a particle, colored green. The most convenient wave variables are the velocity of the particle u(x, t) and the perturbation pressure p(x, t), which is the difference of the actual pressure and the pressure at rest, p0. During the process of derivation, two auxiliary variables are used: the displacement of a point in the particle ξ(x, t), and the perturbation density ρ(x, t), which is the difference of the local density and the average density ρ0. As shown in Fig. 1.14(a), it is clear that the perturbation density is associated to the differential of displacement by ∂ξ(x, t) ρ(x, t)=−ρ . (1.95) 0 ∂x Because the velocity is the time-derivative of the displacement, ∂ξ(x, t) u(x, t) ≡ , (1.96) ∂t by differentiating Eq. 1.95 with respect to t, using Eq. 1.96, we find ∂ρ(x, t) ∂u(x, t) = −ρ . (1.97) ∂t 0 ∂x Equation 1.97 is the equation of continuity. From the theory of thermodynamics, the relation between the perturba- tion pressure and the perturbation density is p(x, t) ρ(x, t) = γ . (1.98) p0 ρ0 The dimensionless constant c R γ = p ≡ 1+ (1.99) cv cv 1For more details of the derivation, see C. Julian Chen, Elements of Human Voice, Section 1.1, World Scientific Publishing Co. (2016). 22 A Review of Classical Physics

Fig. 1.14. Derivation of the wave equation. (a) The condition of continuity. (b) Application of Newton’s second law.

is a the ratio of the constant-pressure specific heat cp and the constant- volume specific heat cv,andR is the gas constant. For dry air, γ = 1.40. By combining Eqs. 1.97 and 1.98, we find ∂p(x, t) ∂u(x, t) = −γp . (1.100) ∂t 0 ∂x As shown in Fig. 1.14(b), the air particle colored green is experiencing a force as the pressure difference of the two sides. By applying Newton’s law on that air particle, one finds ∂u(x, t) ∂p(x, t) ρ = − . (1.101) 0 ∂t ∂x Equations 1.100 and 1.101 are a pair of first-order partial differential equations relating the particle velocity and the perturbation pressure. Dif- ferentiating both sides of Eq. 1.100 with regard to t, we obtain ∂2p(x, t) ∂2u(x, t) = −γp . (1.102) ∂t2 0 ∂t∂x On the other hand, differentiating Eq. 1.101 with regard to x yields ∂2u(x, t) ∂2p(x, t) ρ = − . (1.103) 0 ∂x∂t ∂x2 The differentiations should be independent of order. We then obtain a second-order differential equation for particle velocity u(x, t), ∂2p(x, t) ∂2p(x, t) ρ = γp . (1.104) 0 ∂t2 0 ∂x2 Defining the velocity of sound c by γp γRT c2 = 0 = , (1.105) ρ0 M where M is the molecular weight, we obtain the wave equation in air, ∂2p(x, t) 1 ∂2p(x, t) = . (1.106) ∂x2 c2 ∂t2 At 0◦C and one atmosphere pressure, the velocity of sound is c = 330 m/s. From Eq. 1.105, the velocity is proportional to the square root of the absolute temperature. At 20◦C, c = 342 m/s. 1.3 Sound wave in air 23

1.3.2 The bugle To achieve an intuitive understanding of the sound wave in air, the working principle of the bugle is analyzed, see Fig. 1.15(A). It is essentially a brass tube with both ends open. Nevertheless, a trained player can play many interesting melodies with four notes. For example, a bugle in C, a brass tube of length L = 1.3 m, can produce notes G4 (392 Hz), C5 (523 Hz), E5 (698 Hz), and G5 (784 Hz). See Fig. 1.15(B). The mechanism of producing those notes is shown in Fig. 1.15(C). At both ends, the values of perturbation pressure must be zero,

p(0,t)=p(L, t)=0. (1.107)

To obtain a differential equation of the standing sound wave inside the tube, the differential pressure is written as a product of a spatial factor and a temporal factor, p(x, t)=p(x)sinωt, (1.108) where ω =2πf is the circular frequency. Insert Eq. 1.108 into eq. 1.106, we obtain an ordinary differential equation,

d2p(x) + k2 p(x)=0. (1.109) dx2

Fig. 1.15. The bugle. (A) The bugle is very simple music instrument, essentially a brass tube with both ends open. (B) Notes produced by a bugle of C. (C) Patterns of perturbation pressure in a bugle for different notes. 24 A Review of Classical Physics

The parameter k,thewave vector, is defined as ω k = . (1.110) c Equation 1.109 is the Helmholtz equation in one-dimensional space. A direct inspection shows that the only possible solutions of Eq. 1.109 satisfying boundary conditions Eq. 1.107 are nπx p(x)=A sin ,n=1, 2, 3, .... (1.111) n L where An is the amplitude. Figure 1.15(C) shows the distribution of per- turbation pressure in a bugle for different notes. The sine function has both positive and negative values, the perturbation pressure are represented by two colors. Nevertheless, because of the temporal dependence, Eq. 1.108, the overall sign of the amplitude is not physically relevant.

1.3.3 Propagation of sound in open space Because perturbation pressure is a scalar quantity, Eq. 1.106 can be gener- alized to three-dimensional space. The wave equation is ∂2 ∂2 ∂2 1 ∂2p(r,t) + + p(r,t)= . (1.112) ∂x2 ∂y2 ∂z2 c2 ∂t2 Using the del operator ∂ ∂ ∂ ∇ = ix + iy + iz , (1.113) ∂x ∂j ∂z where ix, iy and iz are unit vectors in the three directions, Equation 1.112 canalsobewrittenas 1 ∂2p(r,t) ∇2p(r,t)= . (1.114) c2 ∂t2

Fig. 1.16. Propagation of sound in open space. (A) The wave vector in space, and its components, kx, ky, and kz. (B) A plain wave in open space. The wave vector is perpendicular to the wave front, where the perturbation pressure is a constant. 1.3 Sound wave in air 25

By direct substitution, a general solution of Eq. 1.112 is

p(r,t)=A sin(k · r − ωt + φ) (1.115) = A sin(kxx + kyy + kzz − ωt + φ), where the wave vector satisfies the relation ω k ≡|k| = k2 + k2 + k2 = , (1.116) x y z c and φ is an arbitrary phase constant. Figure 1.16(B) shows a plane wave is propagating in the direction of thewavevectork. It is a alternation of positive perturbation pressure and negative perturbation pressure with a wavelength λ, c 2πc 2π λ ≡ = = . (1.117) f ω k And the wave vector is perpendicular to the wavefront, a plain with a fixed value of the perturbation pressure.

1.3.4 Resonance frequencies of a music room Another example of the application of the wave equation, Eq. 1.112, is the resonance frequencies in a music room. Because the walls of a music room are made of hard materials, the sound wave reflects by the walls repeatedly. This resonance effect has advantages and disadvantages. First, the music sound is amplified by the resonance, which is welcome. Nevertheless, the geometry of the music room makes some of the frequencies stronger, and others weaker, which artificially colors the music. Such a harmful effect must be avoided. A well-studied problem in both music community and architecture community is to find the optimum dimensions of the room, such that the overall resonance is maximized while the resonance frequencies are evenly distributed in the relevant frequency range. The geometry is shown in Fig. 1.17. The lengths in the x, y,andz directions are denoted as Lx, Ly,andLz, respectively. Assuming the walls

Fig. 1.17. Dimensions of a music room. The lengths in x, y, and z directions of a music room are denoted as Lx, Ly, and Lz, respectively. 26 A Review of Classical Physics

Fig. 1.18. Resonance modes in a music room. The perturbative pressure distri- butions for several modes in a music room, near the floor where z ≈ 0. Only the mode indices of nx and ny are shown. See Eq. 1.122. are rigid, the normal velocity of the particles must be zero, and the normal derivatives of the pressure also must be zero. Therefore, the boundary conditions are ∂p(r,t) =0 x = 0 and x = L , ∂x x ∂p(r,t) =0 y = 0 and y = L , (1.118) ∂y y ∂p(r,t) =0 z = 0 and x = L . ∂z z Solutions of Eq. 1.114 with a fixed frequency having the form p(r,t)=p(r)sinωt. (1.119) Insert Eq. 1.119 into Eq. 1.114, we obtain ∇2p(r)+k2p(r)=0, (1.120) where the wave number is defined as ω k = . (1.121) c Equation 1.120 is the Helmholtz equation in three-dimensional space. The solutions of Eq. 1.120 satisfying boundary conditions Eq. 1.118 are n πx n πy n πz p(r)=A cos x cos y cos z , (1.122) Lx Ly Lz where nx, ny,andnz are natural numbers. Figure 1.18 shows some low- index standing waves. Again, the two colors indicate two polarities, positive and negative. It makes no difference by swapping the colors. 1.4 The sound of basketballs 27

Fig. 1.19. Laplace operator in spherical coordinates. (A), The standard bas- ketball is a sphere with inner radius of a = 11.6 cm. (B) The geometry of the spherical coordinate system and the elemental vectors, ir, iθ, and iφ.

1.4 The sound of basketballs

By holding a basketball in one hand and striking it with a baton, a metallic “ping” sound is excited, which could ring for a second or more. The ringing sound is due to the standing acoustic waves inside the basketball.2 It is similar to the quantum mechanics of atoms. Here is an analysis.

1.4.1 Wave equation in spherical polar coordinates To make an analysis of the problem, we first write the wave equation Eq. 1.114 in spherical coordinates, defined as

x = r sin θ cos φ, y = r sin θ sin φ, (1.123) z = r cos θ, see Fig. 1.19. Instead of the unit vectors in rectangular coordinates, those in spherical coordinates are used. The gradiant of pressure is

∂p 1 ∂p 1 ∂p A = ∇p = i + i + i . (1.124) ∂r r r ∂θ θ r sin θ ∂φ φ

The divergence of A is the total outgoing flow of A in three directions divided by the element volume. As shown in Fig 1.19, the three sides are dr, rdθ,andr cos θdφ. Therefore, the elementary volume is

dv = dr · rdθ · r sin θdφ = r2 sin θdrdθdφ. (1.125)

2See D. A. Russell, Basketballs as spherical acoustic cavities, American Journal of Physics, 78, 549-554 (2010). 28 A Review of Classical Physics

The flow in the r direction is ∂(r2 sin θA ) ∂ ∂p df = r dr dθ dφ =sinθ r2 dr dθ dφ, (1.126) r ∂r ∂r ∂r the flow in the θ direction is ∂(r sin θA ) ∂ ∂p df = θ dr dθ dφ = sin θ dr dθ dφ, (1.127) θ r∂θ ∂θ ∂θ and the flow in the φ direction is ∂(A ) 1 ∂ ∂p df = φ dr dθ dφ = dr dθ dφ. (1.128) φ r sin θ∂φ sin θ ∂φ ∂φ

Adding together, the Laplacian of pressure is

1 ∇2p = (df + df + df ) dv r θ φ (1.129) 1 ∂ ∂p 1 ∂ ∂p 1 ∂2p = r2 + sin θ + . r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2

The wave equation Eq. 1.114 is then 1 ∂ ∂p 1 ∂ ∂p 1 ∂2p 1 ∂2p r2 + sin θ + = . (1.130) r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2 c2 ∂t2

Using an angular momentum operator L2 defined as 1 ∂ ∂p ∂2p L2p ≡− sin θ sin θ + . (1.131) sin2 θ ∂θ ∂θ ∂φ2

Thewaveequationbecomes 1 ∂ ∂p 1 1 ∂2p r2 − L2p = . (1.132) r2 ∂r ∂r r2 c2 ∂t2

With a sinusoidal temporal variation, p = p(r)sinωt, the Helmholtz equation Eq. 1.120 becomes 1 ∂ ∂p(r) 1 r2 − L2p(r)+k2p(r)=0, (1.133) r2 ∂r ∂r r2 where k = ω/c isthewavevector. 1.4 The sound of basketballs 29

1.4.2 Spherical harmonics To find the solutions for the excited states, the standard method of resolving partial differential is used. The perturbative pressure p(r) is written as a product of a function of radius R(r) and a function of angle variables, Y (θ, φ), called spherical harmonics: p(r)=R(r)Y (θ, φ). (1.134) Insert Eq. 1.134 into Eq. 1.133, we find 1 d dR(r) 1 r2 + k2 r2 = L2Y (θ, φ). (1.135) R(r) dr dr Y (θ, φ) The left-hand side of the equation depends only on r, and the the right-hand side of the equation of the spherical harmonics depends only on angular variables θ and φ. Therefore, both sides must be a constant. In the following, using the differential equation for the function Y (θ, φ), Eq. 1.131, we look for the explicit value of constant λ: 1 ∂2Y (θ, φ) 1 ∂ ∂Y (θ, φ) − − sin θ = λY (θ, φ). (1.136) sin2 θ ∂φ2 sin θ ∂θ ∂θ The angular variables can be separated by writing Y (θ, φ) as a product, Y (θ, φ)=Θ(θ)Φ(φ). (1.137) Insert Eq. 1.137 into Eq. 1.136, we obtain sin2 θ 1 d dΘ 1 d2Φ sin θ + λΘ = − . (1.138) Θ sin θ dθ dθ Φ dφ2 The left-hand side only depends on θ, and the right-hand side only depends on φ. Both sides must be a constant. The function Φ(φ) should be a sinusoidal function of an integer multiple of φ,either Φ(φ)=cosmφ (1.139) or Φ(φ)=sinmφ, (1.140) to satisfy the ordinary differential equation d2Φ + m2 Φ=0. (1.141) dφ2 For historical reasons, the index m is called the magnetic quantum number. The right-hand side of Eq. 1.138 is then 1 d dΘ m2 sin θ − Θ+λΘ=0. (1.142) sin θ dθ dθ sin2 θ 30 A Review of Classical Physics

To solve Eq. 1.142, we use a trail function Θ(θ)=sinm θ. (1.143) Insert Eq. 1.143 into Eq. 1.142, after a short algebra, we obtain −m(m +1)Θ+λΘ=0. (1.144) Therefore, it is a good solution with λ = m(m +1). (1.145) Because λ is a parameter for Θ, a function of polar angle θ, we define an azimuthal quantum number l by λ = l(l +1). (1.146) For the case of l = m, the expression of Θ(θ) in Eq. 1.143 is correct. In general, m could be smaller than l. The expressions for a general case is

Table 1.3: Spherical Harmonics

Mathematical Chemist’s Formula in angular In Cartesian notation Name variables coordinates 1 1 Y00(θ, φ) s √ √ 4π 4π 3 3 z Y10(θ, φ) pz cos θ 4π 4π r g 3 3 x Y11(θ, φ) px sin θ cos φ 4π 4π r 3 3 y Y u (θ, φ) p sin θ sin φ 11 y 4π 4π r 2 5 3 2 1 5 3 z 1 Y (θ, φ) d 2 cos θ − − 20 z 4π 2 2 4π 2 r2 2 g 15 15 zx Y21(θ, φ) dxz cos θ sin θ cos φ 2 4π 4π r u 15 15 zy Y21(θ, φ) dyz cos θ sin θ sin φ 2 4π 4π r 2 − 2 g 15 2 15 x y 2 2 Y22(θ, φ) dx −y sin θ cos 2φ 2 16π 16π r 15 15 xy Y u (θ, φ) d sin2 θ sin 2φ 22 xy 16π 4π r2 1.4 The sound of basketballs 31 complicated. However, we need only a few more cases, and the expressions are fairly simple. Here is a complete list:

Θ10(θ)=cosθ, l =1,m= 0; (1.147)

2 Θ20(θ)=3cos θ − 1,l=2,m= 0; (1.148) and Θ21(θ)=cosθ sin θ, l =2,m=1. (1.149) The correctness of those solutions can be verified by Eq. 1.142, 2 1 d dΘlm(θ) m sin θ − Θlm(θ)+l(l +1)Θlm(θ)=0, (1.150) sin θ dθ dθ sin2 θ which is left as an exercise. For applications in chemistry and molecular biology, expressions of spher- ical harmonics in Cartesian coordinates are useful. And it looks simpler. Table 1.3 lists the first nine spherical harmonics in real variables, which represents a complete list of spherical harmonics often used in chemistry and molecular biology. In the Table, superscript g means symmetric ver- sus x, and superscript u means antisymmetric versus x. In chemistry and molecular biology, the real spherical harmonics are preferred. Chemists have special names for those angular dependences, as shown in the√ second column, the chemist’s names. The numerical constant such as 1/ 4π is to normalize the spherical harmonics over a spherical surface of radius 1.

1.4.3 Eigenfrequencies of standing waves The radial function R(r) is a solution of Eq. 1.135 d dR(r) r2 − l(l +1)− k2r2 R(r)=0, (1.151) dr dr where the parameter l defined by Eq. 1.146 is the azimuthal quantum num- ber specifying the spherical harmonics, an integer. By direct insertion, one can verify the solutions of Eq. 1.151 are

R(r)=jl(kr), (1.152) where sin ρ j (ρ)= , (1.153) 0 ρ sin ρ cos ρ j (ρ)= − , (1.154) 1 ρ2 ρ and 3 1 3cosρ j (ρ)= − sin ρ − , (1.155) 2 ρ3 ρ ρ2 32 A Review of Classical Physics for l = 0, 1, and 2, respectively. The general expression is 1 d l sin ρ j (ρ)=(−ρ)l . (1.156) l ρ dρ ρ

The parameter k, the wavevector, is determined by the boundary condi- tion: at the inner surface of the basketball r = a, the velocity of air particle is zero, and the pressure reaches a stable value, d jl(kr) =0. (1.157) dr r=a

For each l, there are a series of values satisfying Eq. 1.157. Let ρnl be the n-throotofthederivativeofjl(ρ). The eigenfrequency of the n-th standing-wave mode with azimuthal quantum number l is cρ f = nl . (1.158) nl 2πa

Fig. 1.20. Modes of standing waves in a basketball. Patterns of perturbation pressures in a basketball. Different color means different polarity, or phase. There are only two different phases, positive and negative. They are equivalent. By swapping the coler, the physics is unchanged. The mode index are: n is the index of roots of the radial function with azimuthal quantum number l. For modes with different magnetic quantum number m, the eigenfrequencies are the same: they are degenerate. 1.4 The sound of basketballs 33

Table 1.4: Modes of standing waves of a basketball

Mode Root ρnl Eigenfrequency (Hz) (1, 1,m) 2.082 979 (1, 2,m) 3.342 1572 (2, 0, 0) 4.493 2113 (2, 1,m) 5.940 2795 (2, 2,m) 7.290 3431 (3, 0, 0) 7.725 3638

The term eigenfrequency originated from German literature in quantum mechanics, which means proper frequency. The velocity of sound is c = 343 m/sec, and the inner radius of a standard basketball is a = 0.116 m. The numerical value of the eigenfrequency in Eq. 1.158 is

fnl = 470.6 ρnl Hz. (1.159)

Table 1.4 shows several prominent modes of standing waves of a basketball. The first column is the mode index, in (n, l, m). The first index n,often called the principle quantum number, is the index of the root of a radial function with azimuthal quantum number l. The magnetic quantum number m is defined by Eqs. 1.139 and 1.140. Modes with different m have the same eigenfrequency, thus become degenerate.

1.4.4 Degeneracy and symmetry In Fig. 1.20, the patterns of several standing wave modes are shown. For modes with l>0, there are several different magnetic quantum numbers m with identical eigenfrequencies. It is obvious from the graphs that those degenerate modes are related by a coordinate rotation. From an intuitive point of view, the eigenfrequency should be the same. It is called degeneracy. It is a result of the spherical symmetry of the basketball. The origin of degeneracy from the symmetry of the system is very impor- tant in quantum mechanics. The symmetry of a system can be mathemati- cally described as a group. And the behavior of eigenfrequency degeneracy is described by representations of a symmetry group. In the case of a basketball, eigenfrequencies with different azimuthal quantum number l are different. Nevertheless, in the case of hydrogen atom, there is an additional symmetry due to the property of the Coulomb force, as we described in Section 1.1.4. As a result, the energy levels of hydrogen atom only depend on the principle quantum number n. 34 A Review of Classical Physics

1.5 Light as electromagnetic wave

The corpuscular and wave theories of light, as two completely incompatible theories, coexisted for more than two centuries. In 1690, Christiaan Huy- gens published Traite de la Lumiere (A Treatise on Light), systematically expounded his wave theory. In 1704, Isaac Newton published a monograph Opticks: A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light, criticized Huygens’ wave theory of light, and expounded his cor- puscular theory of light. The centuries-old debate is still instructive for the understanding of the modern concept of wave-particle duality.

1.5.1 Newton’s corpuscular theory of light Besides research in mechanics represented by the epoch monograph Prin- cipia, Isaac Newton made numerous discoveries and inventions in optics. Besides a mathematical genius, he was also a super handy experimental- ist. He invented and built the first working reflecting telescope by his own hands in 1668. It remains today the prototype of most of the professional astronomical telescopes as well as amateur telescopes. A highly influential experimental discovery of Newton is that white light

Fig. 1.21. Newton’s Opticks and Huyges’ Lumiere. Left, Newton’s Opticks, pub- lished in 1704, systematically presented the corpuscular theory of light. Right, Huygens’ Treatise on Light, published in 1690, systematically presented the wave theory of light. 1.5 Light as electromagnetic wave 35

Fig. 1.22. Dispersion of prism and Newton’s explanation. (A), a schematics of the experiments of Isaac Newton, by decomposing the sunlight into colors using a prism. (B), Newton’s explanation of dispersion based on his corpuscular theory of light, that the mass of red-light particles is greater than the mass of the violet-light particles, thus the violet light bends more than the red light. is a mixture of a number of colored light rays, each of the colored rays is original and unchanged during propagation. In 1666, using a glass prism, he decomposed white sunlight into a spectrum, see Fig. 1.22(A). In Opticks, Query 29, on pages 533 – 538, he made a detailed explanation of this dis- covery by the corpuscular theory of light, see Fig. 1.22(B). Newton’s corpuscular theory of light is completely in line with his gen- eral mechanical view of the physical world, as summarized by Einstein, quoted in Section 1.1. In Newton’s own words, rays of light consist of ”very small Bodies emitted from shining Substances”. Those material points are structureless, dimensionless, perfectly elastic, characterized only by its ge- ometrical location (x, y, z) and mass. Its motion is determined only by the forces from other material points in the medium, governed by the law of mechanics. Accordingly, in vacuum, the light particles move in a straight line. In a uniform medium such as glass, because the force from the con- stituent particles of glass are balanced, light also moves in a straight line. Only at the interface between two different media, the speed changes. As shown in Fig. 1.22(B), as the particles of light enters from vacuum to glass, the material particles constituting glass attracts the particles of light, and the vertical speed of the light particles is increased. Newton assumed that the particles constituting red light are heavier than the particles constitut- ing violet light. At the interface, red light particles gain less vertical speed than violet light particles, thus red light bends less than violet light. He derived Snell’s sine law of refraction from that idea mathematically. He further presented evidence that the particles of red light are heavier than the rest, because when red light is absorbed, it generates more heat than for example violet light particles, characterized by Newton as smaller. Experiments in the 19th century and on overturned Newton’s corpuscu- lar theory of light. First, the speed of light in transparent media such as 36 A Review of Classical Physics glass was found slower than the speed of light in vacuum. The speed of light in vacuum was found to be the upper limit of any speed. Second, although Einstein’s theory of photon assigns a mass to light particles, the photons of the red light have a mass smaller than those of violet-light. Especially, Robert Young’s double-slit experiment made a definitive proof of the wave nature of light. According to Young, color is associated to wavelength.

1.5.2 Young’s double-slit experiment Figure 1.23 shows a schematic of the Young double-slit experiment. From the left side, a plane wave of light with wavelength λ falls on a screen with to slits S1 and S2. The distance between the two slits is a. In the Figure, the phase of light wave is marked by colors; for example, red as positive, and blue as negative. The intensity of colors represents amplitude. Consider a point x on the detection screen D. The distance from point x to the first slit is a 2 r = L2 + x − . (1.160) 1 2 and to the second slit is a 2 r = L2 + x + . (1.161) 2 2

The distance from the slits to the detection screen L is often much greater

Fig. 1.23. Young’s double-slit experiment. A plane wave of light with wavelength λ falls on a screen with to slits S1 and S2. The distance between the two slits is a.The two light beams propagate and produce an interference pattern on the detection screen D. Here the phase of the wave is marked by colors: red is positive, blue is negative. The intensity of color represents amplitude. 1.5 Light as electromagnetic wave 37 that x and a. Using Newton’s binomial formula, √ u 1+u =∼ 1+ , (1.162) 2

A fairly accurate formula of the difference of r2 and r1 is obtained 1 a 2 1 a 2 xa r − r = L 1+ x + − 1+ x − =∼ . (1.163) 2 1 L2 2 L2 2 L

At the point x, the wave amplitude is the sum of rays from two slits:

u(x)=a sin(kr1 − ωt)+a sin(kr2 − ωt) r + r r − r (1.164) =2a sin k 1 2 − ωt cos k 1 2 . 2 2

According to Eq. 1.56, λk =2π. If r2 − r1 equals an integer multiple of wavelength λ, the cosine factor is ±1. The light intensity reaches a maxi- mum. If r2 − r1 equals an odd integer multiple of one half of wavelength λ, the cosine factor is zero. The two rays interfere destructively. The distance between two adjacent intensity minima Δx (see Fig. 1.23) is λL Δx = . (1.165) a Young’s two-slit experiment is essential to the understanding of the wave nature of the electron and other particles. We will repeatedly refer to that experiment in the discussions of wave-particle duality.

1.5.3 Maxwell’s theory of electromagnetic fields Up to the middle of the nineteenth century, electromagnetic phenomena and light have been considered as totally independent entities. In 1865, in a monumental paper A Dynamic Theory of the Electromagnetic Field,James Clerk Maxwell (Fig 1.24) presented a complete set of partial differential equations explaining electromagnetic phenomena, and inferred that light is an electromagnetic wave. In an article written for the centenary of Maxwell’s birth entitled Maxwell’s Influence on the Development of the Conception of Physical Reality, Einstein said thusly: Before Maxwell, people thought of physical reality – in so far as it represented events in nature – as material points, whose changes consist only in motions which are subject to total differ- ential equations. After Maxwell, they thought of physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations. This change in the conception of reality is the most profound and the most fruitful that physics has experienced since Newton. 38 A Review of Classical Physics

Fig. 1.24. James Clerk Maxwell. Scot- tish physicist (1831–1879), one of the most in- fluential physicists along with Isaac Newton and Albert Einstein. He developed a set of equations describing electromagnetism, known as the Maxwell’s equations. In 1865, based on those equations, he predicted the existence of electromagnetic waves and proposed that light is an electromagnetic wave. He also pioneered the kinetic theory of gases, and created a sci- ence fiction character Maxwell’s demon.Por- trait courtesy of Smithsonian Museum.

Quantum mechanics represents a further extension of the conceptual breakthrough started by Maxwell. Wavefunctions, the physical reality in quantum mechanics, also subject to partial differential equations.

Maxwell’s equations In Gaussian unit system,3 Maxwell’s equations are

∇·E =4πρ, (1.166)

∇·B =0, (1.167)

1 ∂B ∇×E = − , (1.168) c ∂t

1 ∂E 4π ∇×B = + J, (1.169) c ∂t c where E is the electric field intensity, B is the magnetic field intensity, J is the electrical current density, ρ is the electric charge density, and c is the speed of light. Maxwell’s equations were mathematical descriptions of experimental observations of Charles-Augustin de Coulomb, Andr´e-Marie Amp`ere, Alessandro Volta, especially those of Michael Faraday. Only the first term of the right-hand side of Eq. 1.169, the displacement current, were added by Maxwell for logical consistency.

3Although in engineeing literature, the SI unit system is the standard, in this book, the Gaussian unit system is used. All formulas are simplest in Gaussian units. And the conversion to the atomic units in Chapter 4 becomes more natural. 1.5 Light as electromagnetic wave 39

Vector Potential and Scalar Potential To treat the electromagnetic field in space, a convenient method is to use the vector potential. From Eq. 1.167, it is possible to construct a vector field A which satisfies B = ∇×A. (1.170) Then, Eq. 1.167 is automatically satisfied. Substituting Eq. 1.170 into Eq. 1.168, one obtains 1 ∂A ∇× E + =0. (1.171) c ∂t

For any function φ(r), ∇×[∇φ(r)] = 0. Therefore, it is always possible to set up the vector potential A such that 1 ∂A E = − −∇φ, (1.172) c ∂t where φ is the electrostatic potential arising from the charges. The choice of the vector potential is not unique. By adding a gradient of an arbitrary function to it, values of the electric field and magnetic field do not change. This is called the gauge invariance of the vector potential. It is possible to define a vector potential which satisfies the condition

∇·A =0. (1.173)

Equation 1.173 is called the Coulomb gauge, which is the most convenient gauge to treat nonrelativistic problems of an atomic system and an indepen- dent electromagnetic wave. In fact, using Eq. 1.173 and the first Maxwell equation Eq. 1.166, one obtains

∇2φ = −4πρ, (1.174) which means that the scalar potential is generated by the static charges only. It is thus convenient for treating the problems of interactions between the radiation field and atomic systems. From Eq. 1.174, we can derive an expression of the potential energy for a generalized Coulomb’s law. Assuming the charge density is spherically symmetric, that it only depends on r. The symmetry implies that the scalar potential is a function of r only. Equation 1.174 becomes 1 d dφ(r) r2 = −4πρ(r), (1.175) r2 dr dr

Because the total charge Q(r) in a sphere of radius r is r Q(r)= 4πr2ρ(r)dr, (1.176) 0 40 A Review of Classical Physics taking a similar integration on the left-hand side of Eq. 1.175 yields r 2 1 d 2 dφ(r) 2 dφ(r) 4πr 2 r dr =4πr . (1.177) 0 r dr dr dr Combining Eqs. 1.175 and 1.177, we obtain dφ(r) Q(r) = . (1.178) dr r2 Taking an integration again, we have r Q(r) Q(r) φ(r)= 2 dr = . (1.179) ∞ r r Therefore, for a spherically symmetric charge density, the scalar potential at a point r is the same as an electrical charge Q(r) at the origin. For the hydrogen atom, with a proton of positive charge e at the origin andanelectronwithanegativecharge−e at r,thepotentialenergyis e2 V (r)=− . (1.180) r

Electromagnetic fields as a physical reality In Newtonian physics, the physical reality is material particles. The po- sition and momentum of each particle can be observed without significant disturbance. If the position and momentum of a particle is known at a given time, the subsequent values are determined by Newton’s laws.

Fig. 1.25. Observing and mapping magnetic field. By placing a white paper above a magnet and sprinkle iron filings on it, the iron filings follows the lines of magnetic field and shows a map of the magnetic field. The magnetic field is an observable physical reality, which exists regardless of being measured or not, and only marginally disturbed by the mapping process. 1.5 Light as electromagnetic wave 41

Fig. 1.26. Observing and mapping electric field. Using a piece of conducting paper or a tray of salt water, the electric field can be probed and mapped by a probe and a galvanometer. A potentiometer defines or selects a value of potential voltage. When the galvanometer shows no current, the spot of the same potential voltage is identified. The entire electric field can be imaged. Similar to the magnetic field, the electric field is an observable physical reality, which exists regardless of being measured or not, and only marginally disturbed by the measurement process.

In Maxwellian physics, the physical reality is the electrical fields and the magnetic fields. The observations are not the position and momentum of individual material particles, but rather the continuous fields. The meth- ods of observing those fields are different from those in Newtonian physics. However, from a philosophical point of view, there is no difference. Although not visible by naked eyes, magnetic field can be observed, measured, and mapped by simple experiments. Figure 1.25 shows an ex- periment easily done in a high-school classroom. By placing a piece of white paper on top of a magnet and sprinkle iron filing over it, the lines of magnetic field are displayed. The experiment was described in Faraday’s laboratory notebooks, and motivated Maxwell to represent the fields with continuous functions governed by partial differential equations. More ac- curate measurements of magnetic fields can be performed using Hall-effect probes, frequently used by scientists and engineers. Note that the magnetic fields around a permanent magnet is an objective physical reality, which ex- ists regardless of being measured or not. The iron filings and the Hall-effect probe does not perturb the magnetic field significantly. Similarly, the electric field can be observed, measured, and mapped by a simple device. Although not visible by naked eyes, the experiments can also be performed in a high-school classroom, as shown in Fig. 1.26. A battery and a pair of electrodes create a pattern of electrical field on a piece of conducting paper. To find the equal potential lines at a given voltage provided by a potentiometer, a probe with a galvanometer is used. By connecting the points with zero current together, the electrical field is mapped. Again, the electrical field pattern is an objective reality. It exists 42 A Review of Classical Physics regardless whether a measurement is conducted or not. The probe, to find the points with no current literally makes no disturbance to the objective physical reality of the electrical field. The electromagnetic fields are governed by Maxwell’s equations, which are first-order in time. Once the electromagnetic fields are known at a given time, the subsequent evolution is completely determined by Maxwell’s equations. Therefore, similar to Newtonian physics, Maxwell’s physics of electromagnetic fields is completely causal.

1.5.4 Electromagnetic waves In this section, we study the electromagnetic waves in free space, that is, where the electric charge ρ and current J are zero. Take the curl of Eq. 1.168, then using Eq. 1.169, we have

1 ∂ 1 ∂2E ∇×∇×E = −− ∇×B = − . (1.181) c ∂t c2 ∂t2 On the other hand, using the identity

∇×∇×E ≡∇(∇·E) −∇2E (1.182)

Equation 1.181 becomes 1 ∂2E ∇2E = . (1.183) c2 ∂t2 which is a wave equation with velocity c. Using a similar procedure, we can show that the magnetic field intensity satisfy the same wave equation,

1 ∂2B ∇2B = . (1.184) c2 ∂t 2 Radiation power and Poynting vector Let us study the energy balance in an electromagnetic field by considering a unit volume with relatively uniform fields. If the current density is J and the electric field intensity is E, the ohmic energy loss per unit time per unit volume is J · E. Using Eq. 1.169, the expression of energy loss becomes c 1 ∂E J · E = E · (∇×B) − E · . (1.185) 4π 4π ∂t Using the mathematical identity

E · (∇×B)=−∇ · (E × B)+B · (∇×E), (1.186)

Eq. 1.185 becomes c 1 ∂E J · E = −∇ · E × B + B · (∇×E) − E · . (1.187) 4π 4π ∂t 1.5 Light as electromagnetic wave 43

Using Eq. 1.168, Eq. 1.187 becomes c ∂ 1 J · E = −∇ · E × B − E2 + B2 . (1.188) 4π ∂t 8π The right-hand side of Eq. 1.188 has a straightforward explanation. The energy density of the electromagnetic fields is 1 W = E2 + B2 , (1.189) 8π and the power density of the electromagnetic field per unit area is c S = E × B. (1.190) 4π The vector S represents the power flow of electromagnetic waves, and called the Poynting vector after its discoverer. According to Eqs. 1.189 and 1.190, the electromagnetic field is not only a medium to transfer forces among material points. It is a physical reality by itself, possessing energy density and energy flow. According to the Einstein relation, E = mc2, the electromagnetic field also has mass density and mass flow. As emphasized by Einstein, continuous field as a physical reality, the Maxwellian point of view, is distinctive from the Newtonian point of view, that only the material points represent physical reality.

1.5.5 Polarization of light Polarization of light is a phenomenon well known in classical physics. But it is closely related to the basic concepts of quantum physics. A good understanding from the classical-physics point of view will be helpful to understand quantum mechanics. To make a simple and clear picture, consider a sinusoidal electromagnetic wave with angle frequency ω =2πf propagating in the z-direction. See Fig. 1.27. In general, the electrical field intensity is

E = E0 sin(kz − ωt), (1.191) where ω k = (1.192) c is the wavevector. Because the field intensities only depends on z,infree space, the first Maxwell equation Eq. 1.166 becomes ∂E z =0. (1.193) ∂z In other words, the z-component of the electrical field intensity is a constant, not a wave. The wave is transverse, which has x and y components only. The x-polarized wave is

Ex(z,t)=Ex0 sin(kz − ωt). (1.194) 44 A Review of Classical Physics

Using Eq. 1.169, we have

By(z,t)=−Ex0 sin(kz − ωt)=−Ex(z,t). (1.195)

The power density of the wave, according to Eq. 1.190, is c S = E2 sin2(kz − ωt). (1.196) 4π x0 Because the average of the square of a sine function is 1/2, the average power density is c S¯ = E2 . (1.197) 8π x0 The y-polarized wave is

Ey(z,t)=Ey0 sin(kz − ωt) (1.198) and using Eq. 1.168, we have

Bx(z,t)=Ey0 sin(kz − ωt)=Ey(z,t). (1.199)

The average power density is c S¯ = E2 . (1.200) 8π y0 Typically, for natural light, such as sunlight or from an electrical lamp, the two polarizations are balanced. The magnitude of Ex0 and the magni- tude of Ey0 are roughly equal. By using a lineal polarizer, for example, in the x direction, the y-component is blocked. The light comes out from the polarizer is lineally polarized in the x-direction.

Fig. 1.27. Electromagnetic wave. The electromagnetic wave is transverse,where the intensity vectors E and B are perpendicular to the direction of propagation. The electric field intensity E is perpendicular to the magnetic field intensity B. The energy flux vector S =(c/4π)E × B is formed from E and B by a right-hand rule. 1.5 Light as electromagnetic wave 45

Fig. 1.28. The three polarizer experiment: Step 1. If both polarizers (C) and (D) are in x-direction, all light goes through polarizer (C) passes (D).

Figures 1.28 to 1.30 show an experimental setup for the demonstration of superposition of polarized light, as described in Chapter 1, Section 2 of Dirac’s The Principles of Quantum Mechanics, sometimes referred to as Dirac’s three-polarizer experiment. In all three Figures, (A) is a light source. (B) is a lens. (C) and (D) are linear polarizers, the blue sidebar indicates the direction of polarization. (E) is a screen. All elements are mounted by magnetic and elastic clampers to facilitate adjustment. In Fig. 1.28, both polarizers (C) and (D) are in x-direction, all light goes through (C) passes (D). In Fig. 1.29, polarizer (C) is in x-direction, but polarizer (D) is in y-direction. All light goes through (C) is blocked by polarized (D). By inserting a third polarizer (F) between (C) and (D) with a tilting angle, as shown in Fig. 1.30, some light can pass all three polarizers.

Fig. 1.29. The three polarizer experiment: Step 2. If polarizer (C) is in x- direction, but polarizer (D) is in y-direction, all light goes through (C) is blocked by polarized (D). 46 A Review of Classical Physics

Fig. 1.30. The three polarizer experiment: Step 3. By inserting a third polarizer (F) between (C) and (D) with a tilting angle, some light can pass all three polarizers.

Figure 1.31 shows an analysis of the role of the tilted polarizer. Assum- ing that the angle is α. The insertion of polarizer (F) is equivalent to a coordinate transformation in the (x, y)-plane. As shown in Fig. 1.31(A), the component of original electrical field intensity, in x0-direction, to the new x1-direction is E1 = E0 cos α. (1.201) The electrical field intensity on the subsequent polarizer (D) is equiva- lent to another coordinate transformation in the (x, y)-plane. As shown in Fig. 1.31(B), the component of the electrical field intensity in the x2- direction is 1 E = E sin α = E sin α cos α = sin 2α. (1.202) 2 1 0 2 Because the power is proportional to the square of the electrical field inten- sity, the outgoing radiation power intensity S¯2 is related to the incoming

Fig. 1.31. The three polarizer experiment: analysis. (A) By inserting a third polarizer with an angle α shifted from the first polarizer, it is equivalent to do a coordinate transformation from (x0,y0)to(x1,y1). The projection of the original electrical field intensity E0 on the transformed x1 axis is E1 = E0 cos α. (B) The projection of E1 to the x2 axis of the final polarizer is E2 = E1 sin α. 1.5 Light as electromagnetic wave 47

power intensity S¯0 by S¯ S¯ = 0 sin2 2α. (1.203) 2 4 Therefore, if the polarization direction of the third polarizer (F) is either in the x-direction or in the y-direction, the output radiation power is zero. At α =(2n +1)π/4=(2n +1)× 45◦,wheren is an integer, the outgoing radiation power density is at maximum. The predicted results can be easily verified by direct experiments.

1.5.6 Double refraction An interesting phenomenon with polarized light is double refraction,as shown in Fig. 1.32. Certain types of crystals, such as calcite (CaCO3), a very common mineral, have anisotropic optical properties. The diffractive coefficients in different crystallographic orientations are different. Looking through a piece of calcite, as shown in Fig. 1.32, a single image splits into two. By using a polarizer as used in Section 1.5.5, it is shown easily that the two images are made of light with different polarizations. The red light from a laser pointer is polarized. By shining such a laser beam through a piece of calcite, interesting phenomenon are observed, see Figs. 1.33 and 1.34. By rotating the axis of the laser pointer, the direction of polarization is changed, and the outgoing beam is changed. In general, the laser beam splits into two, as shown in Fig. 1.33, case A. However, if the polarization of the laser beam is aligned with one of the axes of the crystal, there is only one outgoing beam, see Fig. 1.34, case B.

Fig. 1.32. Double refraction of calcite. The optical property of calcite (Iceland spar) is anisotropic. It responses differently to light with different polarization. Through a piece of calcite, a single image splits into two. Using a polarizer, it shows that the two images are made of light polarized in different directions. 48 A Review of Classical Physics

Fig. 1.33. Refraction of a laser beam by calcite, case A. By submerging the crystal in water mixed with a drop of milk, the light beams become visible. In general, the laser beam splits into two beams with different polarizations.

By assigning the direction of light as the z-axis, using Maxwell’s equa- tions, Eqs. 1.168 and 1.169, the differential equations for those laser beams can be established. For the pair of Ex and By,wehave

1 ∂E ∂B x = − y , (1.204) c ∂t ∂z 1 ∂B ∂E y = − x . (1.205) c ∂t ∂z The electromagnetic waves with a circular frequency ω are

Ex = Ex0 sin(kz − ωt), (1.206)

By = Ex0 sin(kz − ωt)=Ex, (1.207)

Fig. 1.34. Refraction of a laser beam by calcite, case B. Same as Fig. 1.33, the light beams are made visible in water with a drop of milk. If the polarization of the laser beam is aligned with one of the optical axes, a single beam comes out. 1.6 Atomic Spectra 49

where k = ω/c. And for the pair of Ey and Bx,wehave

1 ∂E ∂B y = x , (1.208) c ∂t ∂z 1 ∂B ∂E x = y . (1.209) c ∂t ∂z And the electromagnetic waves are

Ey = Ey0 sin(kz − ωt), (1.210)

Bx = −Ey0 sin(kz − ωt)=−Ey. (1.211) To make a complete description of the two polarized light beam, four components of electromagnetic fields are needed. Those four components are grouped into two pairs, independent from each other. An important experimental fact is, while turning the laser beam along its axis, the positions of the two outgoing beams never change. Only the relative intensity of the two beams changes with the angle. For every 90◦ rotation, the intensities of the two beams alternate.

1.6 Atomic Spectra

As we have presented in Section 1.5.1, Isaac Newton discovered that by using a prism, sunlight can be dispersed into a variety of colors. That was the starting point of optical spectroscopy. In the 19th century, as a result of

Fig. 1.35. A schematic of diffraction grating. A beam of light from the source is going through a thin slit, then spread on a collimating mirror to become parallel light, falling on a diffraction grating. After diffraction, the light is focused by a camera mirror onto a detector. Light beams of different wavelength, for example, red light and green light, will hit on different places of the detector. 50 A Review of Classical Physics

Fig. 1.36. Absorption atomic spectra in the visible range. Absorption lines in the spectrum of solar radiation, originally discovered by Joseph von Fraunhofer in 1814, thus named as the Fraunhofer lines. By comparing with the emission spectra of various elements, the origin of those absorption lines were gradually known. Of special interest are the four strong absorption lines at 656.2 nm, 486.1 nm, 434.0 nm, and 410.0 nm, identified as from the hydrogen atom, and renamed as Hα,Hβ,Hγ, and Hδ. Hydrogen is by far the most abundant element in the atmosphere of the Sun. the advances in precision machining, a more powerful and accurate device for dispersion of light, diffraction gratings, was developed. Figure 1.35 is a schematic of diffraction grating. A grating is made by engraving a large number of groves on a flat piece of metal. The spacing of two adjacent grooves d should be greater than the wavelength of the light of interest, but not excessively greater. A beam of light is incident through a slit on a collimating mirror to become parallel light, falls on the grating. After diffracted by the grating, the parallel light beam is focused by a camera mirror on a detector, typically a CCD camera. Due to interference, light of different wavelength have different condition for maximum intensity. For example, the condition of maximum intensity for red light is that the path difference equals an integer multiple of the wavelength,

d(sin θi − sin θr)=mλr, (1.212) where θi is the incident angle, θr is the angle of maximum intensity for red light with wavelength λr,andm is an integer of the number wavelengths of the interference path. Likewise, for green light, the condition for maximum intensity is d(sin θi − sin θg)=mλg, (1.213) where θr is the maximum angle for green light with wavelength λr. The resolution power with a grating with N grooves of order m is λ = mN, (1.214) Δλ where λ is the wavelength of the light, and Δλ is the smallest resolvable wavelength difference. Simply speaking, the more lines on the gratings, the finer its resolution. In late 19th century, a typical gratings has more than 10,000 lines, capable of a resolution power of 10,000 or more. 1.6 Atomic Spectra 51

In 19th century, using the diffraction gratings, a vast field of science and technology, atomic spectroscopy, emerged. Each atom has its characteristic group of spectral lines. Those lines can be either observed as absorption lines or emission lines. In 1814, German physicist Joseph von Fraunhofer discovered a number of absorption lines in the observed spectrum of sun- light. After years of continuing research, thousands of so-called Fraunhofer lines were identified. Later, by using flames with known elements as the light source, thousands of emission lines of atoms were discovered. The ab- sorption lines and the emissions lines are correlated one by one. Over the 19th century, optical spectroscopy gradually became one of the most impor- tant methods for chemical analysis. A number of elements were discovered first by the observation of spectral lines of unknown origin, including Cs, Rb, and Tl (1860-61); In (1863); Ga (1875); Tm (1870); Nd and Pr (1885); Sm and Ho (1886). An interesting case is helium. Several strong absorption lines were first discovered in the absorption spectrum of the Sun. In 1868, by emission spectroscopy, helium was found as a rare element in the air. The name, helium, was referred to its origin as from the Sun. Of particular interest are four Fraunhofer lines, initially observed in the

Fig. 1.37. Emission spectra of hydrogen. The observed spectra of hydrogen in the infrared, visible, and ultraviolet regions can be explained by the Ritz combination rule: There is a series of terms, marked by a natural number, 1, 2, 3, ... etc. The frequencies of each spectral line is a difference of the frequencies of spectral terms, defined by the Rydberg formula, Eq, 1.215. A systematic explanation of the spectral lines of hydrogen is the first historical triumph of quantum mechanics, see Chapter 3. 52 A Review of Classical Physics

Table 1.5: Wavelengths of hydrogen lines

n1 =1 n1 =2 n1 =3 n1 =4 Lyman Balmer Paschen Brackett

n2 = 2 121.6 nm n2 = 3 102.6 nm 656.3 nm n2 = 4 97.2 nm 486.1 nm 1875 nm n2 = 5 94.9 nm 434.0 nm 1282 nm 4.05 μm n2 = 6 93.8 nm 410.2 nm 1094 nm 2.62 μm n2 = 7 93.0 nm 397.0 nm 1005 nm 2.16 μm n2 = 8 92.6 nm 388.9 nm 955 nm 1.94 μm n2 = ∞ 91.2 nm 364.6 nm 820 nm 1.46 μm spectrum of sunlight, named C, F, G and h. Later identified as from the hydrogen atom, and renamed Hα,Hβ,Hγ,andHδ lines. See Fig. 1.36. Because of the high resolution of diffraction gratings, the accuracy of wave- lengths of those lines reached six to seven significant digits. In 1885, Swiss mathematician Johann Jakob Balmer (1825 – 1898) found an accurate em- pirical formula for the four hydrogen lines. In 1889, Swedish physicist Jo- hannes Robert Rydberg (1854 – 1919) extended the Balmer formula to include hydrogen spectral lines in the infrared and ultraviolet regions. The key of their success was to use wavenumber, the inverse of wavelength, to represent the data. Rydberg’s formula is 1 1 − 1 =Ry 2 2 , (1.215) λ n1 n2 where Ry is the Rydberg constant, n1 and n2 are integers. For n1 =1,the ultraviolet lines in the Lyman series are represented. For n1 = 2, the visible lines of the Balmer series are represented. See Table 1.5 and Fig. 1.37. Towards the end of the 19th century, the Rydberg formula was well val- idated. The Rydberg constant was the most accurately measured physical constant. However, despite many trials, no interpretation of the Rydberg formula was discovered until the rise of the quantum theory. Chapter 2 Wave and Quantum

On November 9, 1922, the Royal Swedish Academy of Sciences an- nounced that the Nobel Prize in Physics 1921 was awarded to Albert Ein- stein “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.” The announcement was a surprise to the public and news media, as Einstein was already a celebrity because of his theory of relativity. In fact, the Nobel Committee was correct: The paper about the photoelectric effect is the most revolutionary, the most original, the most consequential, and also the most controversial of Einstein’s papers. It started the era of quantum physics, the scientific theory having the most profound impact on human society in the 20th century. The concept of wave-quantum duality, defined in that paper for light, was later generalized to all particles by Louis de Broglie, then became the conceptual basis of quantum mechanics.

2.1 Einstein’s energy quantum of light

Figure 2.1 shows the title and a key paragraph of Einstein’s paper, which ex- plained the concept of energy quantum of the electromagnetic wave. Expect- ing a strong opposition, the title was scrupulously worded: On a Heuristic Viewpoint Concerning the Production and Transformation of Light. No- tice that the word propagation is avoided. The adjective “heuristic”, rarely used in scientific literature, is an attempt to disperse objections. The para- graph showed on Fig. 2.1 said emphatically that the wave theory of light, “has worked well in the representation of purely optical phenomena and will probably never be replaced by another theory”, such as diffraction, reflec- tion, refraction, dispersion, etc. However, the continuous wave theory may “lead to contradictions with experience when it is applied to the phenomena of emission and transformation of light”. Einstein argued that in order to explain experimental observations dur- ing production and transformation of light, energy is always transferred in an integer multiple of an indivisible unit he called energy quantum ,

 = hf, (2.1) here h is the Planck constant, f is the frequency. Max Planck proposed this postulation in 1900 as a mathematical trick to explain his blackbody radia- 54 Wave and Quantum

Fig. 2.1. Einstein’s paper on energy quantum of light. The title and a key paragraph of the paper cited for the 1921 Nobel Prize in Physics published on Annalin der Physik in 1905. First, Einstein emphasized the correctness of the wave theory of light. Then, he discussed four experimental observations during the generation and conversion of light that directly contradicted the wave theory of light, and proposed the concept of light quantum. The word “heuristic” was attempted to disperse objections. tion formula. Einstein took it as a basic law of physics. In that monumental paper, Einstein discussed four experiments: blackbody radiation, photoflu- orescence, photoelectric effect, and ionization of gases by ultraviolet light. The first six sections are about blackbody radiation, where the validity of the Planck’s postulation was discussed in detail. Einstein pointed out that by comparing the Planck formula of blackbody radiation with experimental data, the Avogadro constant is found to be 6.17×1023/mole, consistent with the value from other sources. In Section 7, he cited the Stokes’s rule of pho- toluminescence of light that the frequencies of outgoing radiation is always lower than the frequencies of incoming radiation. By assuming that the energy of higher frequency light quantum is always higher than the energy of lower frequency light quantum, Stokes’s rule can be explained naturally. Section 8 discussed the photoelectric effect. He made an order of magnitude estimate that the light quanta of ultraviolet light should be greater than the energy to eject an electron from typical metals, or the work function of the metals. Section 9 is about the ionization of gases with ultraviolet light. Experimentally, the threshold voltage to ionize a gas is about 10 V, and the largest wavelength of light to ionize a gas is 1.9×10−5 cm. Using Eq. 1.8, the two thresholds are roughly equivalent. Attention from the academic community was concentrated to Section 8, where Einstein proposed an experimentally verifiable formula relating the frequency of incoming light and the kinetic energy of outgoing electrons. 2.1 Einstein’s energy quantum of light 55

Fig. 2.2. Lenard’s apparatus for studying photoelectric effect. A quartz window allows the UV light from an electric arc lamp to shine on a target. The voltage between the target and the counter electrode is controlled by an adjustable power sup- ply. An ammeter is used to measure the electric current generated by the UV light, the photocurrent. By gradually increasing the voltage (with the polarity as shown), the pho- tocurrent is reduced. The voltage with which the photocurrent becomes zero is recorded as the stopping voltage.

The formula, referred to the “law of photoelectric effect” by the Nobel Committee, was tested by Robert Millikan with a series of meticulous ex- periments for 10 years, and found to be surprisingly accurate.

2.1.1 The photoelectric effect The photoelectric effect was discovered accidentally by Heinrich Hertz in 1887 during experiments to generate electromagnetic waves. Since then, a number of studies have been conducted in an attempt to understand the phenomena. Around 1900, Phillip Lenard did a series of critical studies on the relation of the kinetic energy of ejected electrons with the intensity and wavelength of the impinging light. His results were in direct conflict with thewavetheoryoflight.1 Figure 2.2 shows schematically the experimental apparatus of Phillip Lenard. The entire setup was enclosed in a vacuum chamber. An electric arclamp,usingcarbonrodsorzincrodsastheelectrodes,generatesstrong UV light. A quartz window allows such UV light to shine on a target made of different metals. The target and a counter electrode are connected to an adjustable power supply. An ammeter is used to measure the electric

1His results were published on P. Lenard, Annalen der Physik, 8, 167 (1902). 56 Wave and Quantum

Table 2.1: Stopping Voltage for Photocurrent

Rod Driving Distance Photocurrent Stopping material current (A) to target (cm) (pA) voltage (V)

Carbon 28 33.6 276 -1.07 Carbon 20 33.6 174 -1.12 Carbon 28 68 31.7 -1.10 Carbon 8 33.6 4.1 -1.06 Zinc 27 33.6 2180 -0.85 Zinc 27 87.9 319 -0.86 current generated by the UV light, the photocurrent, especially when the voltages between the two electrodes are very small. By gradually increasing the voltage, which tends to reflect the electrons back to the target, the photocurrent is reduced. The voltage with which the photocurrent becomes zero is recorded as the stopping voltage. The stopping voltage is apparently related to the kinetic energy of the electrons ejected from the target:

1 eV = mv2. (2.2) 2 Understandably, the photocurrent varies with the intensity of light. By changing the magnitude of the current that drives the arc or the distance from the arc lamp to the target, the photocurrent could change by two orders of magnitude: for example, from 4.1 to 276 pA. An unexpected and dramatic effect Lenard observed was that no matter how strong or how weak the light is, and no matter how large or how small the photocurrent is, the stopping voltage does not change; see Table 2.1. The stopping voltage changes only when the material for the electric arc lamp changes. However, for a given type of arc, the stopping voltage stays unchanged. The effect Lenard observed has no explanation in the framework of the wave theory of light. According to the wave theory of light, the more intense the light is, the more kinetic energy the electrons acquire.

2.1.2 Einstein’s law of photoelectrical effect Einstein proposed an explanation that the energy of light can only be trans- ferred as an integer multiple of energy quantum , solely depending on its frequency f,

 = hf, (2.3) 2.1 Einstein’s energy quantum of light 57 where h =6.63×10−34 J·s is the Planck constant. Einstein also proposed a simple formula that can be tested experimentally. When a photon interacts with an electron in the metal, it transfers the radiation energy  to the electron. The electron could escape from the metal by overcoming the work function of the metal, for example tungsten, approximately 4.3 electron volts. If the energy of the photon is smaller than the work function of the metal, the electron would stay in the body of the metal. If the energy of the photon is greater than the work function of the metal, then the electron can escape from the metal surface with an excess kinetic energy, 1 T = mv2 = hf − W. (2.4) 2 Einstein took an example of near ultraviolet light, λ =0.29 μm, and the frequency is 1.03×1015 s−1. The energy of the photon is 6.9×10−19J, or 4.3 eV, in line with the work function of the metal tungsten used by Lenard. The kinetic energy of an escaping electron can be measured by an external voltage, or electric field, to turn it back onto the target. Voltage just enough to cancel the kinetic energy is called the stopping voltage, 1 eV = mv2 = hf − W, (2.5) stop 2 where e is the electron charge, 1.60 × 10−19 C. According to Einstein’s theory of light quantum, the stopping voltage is linearly dependent on the frequency of the radiation and independent of the intensity of light.The slope should be a universal constant, which provides a direct experimental method to determine the Planck constant h, ΔV h stop = . (2.6) Δf e The quantities in Eq. 2.6 are well-defined. It can be verified by direct experiments and to obtain the value of the Planck constant.

2.1.3 Millikan’s experimental verification Einstein’s theory of photons was rejected by a number of prominent physi- cists for many years, including Max Planck, Niels Bohr, and notably Robert Millikan. Starting in 1905, for 10 years Millikan tried very hard to exper- imentally disprove Einstein’s theory. Comparing with the primitive exper- iments of Philip Lenard, Millikan had much more resources to do preci- sion measurements to verify the Planck-Einstein relation  = hf. Using six spectral lines from a powerful mercury lamp with wavelengths carefully mea- sured, showing in the blue boxes, Millikan’s determination of the Planck’s constant was definitive and straightforward. In 1916, Millikan published a long paper on Physical Review, entitled A Direct Photoelectric Determina- tion of Planck’s h. The conclusion reads as follows: 58 Wave and Quantum

Fig. 2.3. Result of Millikan’s experiment on photoelectric effect. Figure 3 of Millikan’s 1916 Physical Review paper on experiment on photoelectric effect. Six spectral lines of a mercury lamp are used, with wavelengths in nm marked with the blue boxes. Planck’s constant was directly obtained from the slope of the curve with frequencies of the radiation and the stopping voltages.

1. Einstein’s photoelectric equation has been subject to very searching tests and it appears in every case to predict exactly the observed results. 2. Planck’s h has been photoelectrically determined with a pre- cision of about .5 percent.

An interesting fact is that in the same paper Millikan emphatically re- jected Einstein’s theory of photons. He said that Einstein’s photon hypothe- sis “may well be called reckless first because an electromagnetic disturbance which remains localized in space seems a violation of the very conception of an electromagnetic disturbance, and second because it flies in the face of the thoroughly established facts of interference.” Millikan wrote that Einstein’s photoelectric equation, although accurately representing the experimental data, “cannot in my judgment be looked upon at present as resting upon any sort of a satisfactory theoretical foundation.” Millikan’s objection was not all unreasonable. Einstein was clear in his 1905 paper that the quantization of radiation energy only occurs at its emission and conversion; in other words, only occurs upon interacting with atomic systems. Nevertheless, there is a widespread misunderstanding that light behaves like Newtonian material points during propagation in the free space. According to such a point of view, a photon might be conceived as a geometrical point with a well-defined trajectory in space. The experiments of Millikan only proved that when light interacts with atomic systems, the amount of radiation energy transfer is always an integer 2.1 Einstein’s energy quantum of light 59

Fig. 2.4. Albert Einstein and Robert Millikan. Both Einstein and Millikan won a Nobel Prize for their contributions to the photoelectric effect. Photograph taken in 1930 when Robert Millikan invited Albert Einstein to a conference in California. Original photograph courtesy of Smithsonian Museum, slightly cleaned up by the author. multiple of the light quantum , determined by its frequency f. It is not a proof that light is a spray of particles during propagation. In the following subsection, the nature of light quantum is clarified through the analysis of two experiments, the three-polarizer experiment and the double-slit interference experiment of Thomas Young.

2.1.4 The three-polarizer experiment As emphasized by Einstein in his Nobel-Prize winning 1905 paper, energy quantization occurs only during the emission and transformation of radia- tion. During propagation, as a wave, light follows the Maxwell theory of electromagnetism, never behaves like a spray of Newtonian material points, as such a view would lead to absurdity. Consider first the three-polarizer experiment discussed in Section 1.5.5. By passing a beam of natural light through a polarizer, the outgoing light in compartment B is fully polarized, for example, in x-direction. By placing a second polarizer in y-direction, the light is blocked. However, by placing a third polarizer with a slanted polarization angle, light can go through. Maxwell’s theory of light provides a perfect quantitative explanation to the results, see Section 1.5.5. Note that Maxwell’s theory is strictly linear, thus the result is independent of the intensity of light. If light is a stream of material points, when the intensity of light is weak, a light beam is composed of widely separated individual photons. After the first polarizer, each photon should be polarized in x-direction. As shown in 60 Wave and Quantum

Fig. 2.5. Paradox of Dirac’s three-polarizer experiment. The concept that light is composed of individual material points during propagation in free space leads to ab- surdity, as shown by the three-polarizer experiment. (1) In compartment A, the photons of the natural light are a mixture of different polarizations. After passing polarized P1, all photons are x-polarized. No such photons can pass the y-polarizer P2, as expected. (2) By inserting a polarizer P3 in compartment B with a slanted direction, some of the photons can go through. However, there is no mechanism to switch those photons’ po- larization to y-direction. Therefore, there is no light comes out in compartment C. To solve this paradox, one should stand by Maxwell’s theory that during propagation, light is an electromagnetic wave, not as a spray of particles.

Fig. 2.5(1), no such photons can go through the y-polarizer. From a particle point of view, this is still understandable. By inserting a third polarizer P3 with a slanted angle between P1 and P2, experimentally, some light goes through into compartment C. The par- ticle view encounters serious difficulties. If each photon in compartment B is polarized in x-direction, with a slanted polarization angle, some of the photons can go through. However, there is no mechanism to switch the po- larization of an x-polarized particle to y-polarized. Therefore, the particle view is in conflict with experimental observations. The solution to this paradox is to stick to Einstein’s original statement, that radiation energy is quantized only during emission and transformation. During the process of propagating, for example, in compartment (B), light is a wave governed by Maxwell’s theory, not a spray of particles. In order to understand the experimental observations, one must insist that during propagation, there is no such thing as photons described as material points. During propagation, the electromagnetic radiation is a wave, strictly follows Maxwell’s equations. Energy is quantized only when the radiation is generated or interacting with atomic systems.

2.1.5 Double-slit experiment with single photon detectors Second, consider the double-slit experiment, as shown in Fig. 2.6. It is known for many decades that even at extremely low intensity, that individ- ual photons are observed on a sensitive light sensor such as a CCD chip, interference still persists. If light is composed of individual particles, each such particle can only go through one of the two slits. Interference can never happen. To resolve this paradox, one should stand by Einstein’s original 2.2 Electron as a quantized wave 61

Fig. 2.6. Paradox of the double-slit experiment. The concept that light is composed of individual particles during propagation in free space leads to absurdity, as shown by the double-slit experiment. According to the particle point of view, each photon can only pass through one of the two slits. When the light intensity is very low, the photon passes slit (A) and the photon passes slit (B) are unrelated. No interference pattern can be observed. To resolve this paradox, one should stand by Einstein’s statement in his original publication that during propagation, light as an electromagnetic wave, follows Maxwell’s theory of light, not as a spray of particles. statement that during propagation, light follows Maxwell’s theory of light as an electromagnetic wave, not as a spray of particles. In the advanced quantum theory of light, quantum electrodynamics, outlined in Chapter 8, Einstein’s original view is represented precisely by rigorous mathematics. For example, in compartment B of Fig. 2.5, light is represented by standing waves. Each mode of the standing waves fills the entire compartment. And even the energy of each mode of standing wave spreading in the space is quantized, such electromagnetic wave still conforms to the rules of superposition as a consequence of Maxwell’s equations. Only when interacting with atomic systems, radiation energy can be transferred by an integer multiple of the light quantum hf.

2.2 Electron as a quantized wave

2.2.1 Experimental studies of the cathode ray In 1897, J. J. Thomson studied the beam emitted from the cathode (negative electrode) in a vacuum tube, see Fig. 2.7. Naturally, it is called cathode ray. Typically, the cathode is heated with a filament to red hot. The cathode ray is then accelerated by a positive anode, then flows through a hole. Although cathode ray is not perceptible by naked eye, when it hits a fluorescent film inside the vacuum tube, visible light is generated. To study the properties of the cathode ray, Thomson placed a pair of electrodes on the way the cathode ray passes. By applying a voltage on the pair of electrodes, the cathode ray is deflected, indicating as a stream of negatively charged material particles. Assuming each such particle has 62 Wave and Quantum

Fig. 2.7. Schematics of J. J. Thomson’s experiment. In a vacuum tube, the cathode (negative electrode), especially when heated, emits a ray of negatively charged material towards the anode (positive electrode). By placing a pair of deflection plates in its path, the cathode ray bends toward the positive plate. The charge-to-mass ratio of the particles can be determined by direct measurements.

an electrical charge e and mass me,thecharge-to-mass ratio e/me of the cathode ray can be determined experimentally, see Fig. 2.7. To make an intuitive picture, the electron beam can be imagined as particles of a mass m. Before entering the electrical fields of the deflection plates, the particles are accelerated to a velocity in the x-direction, vx. Denote the distance of the two deflecting electrodes as D, with a deflecting voltage V , the force acting on each particle is eV F = . (2.7) D Denote the length of the deflection electrodes as L, the time the particle passes through is L t = . (2.8) vx According to Newton’s law, Eq. 1.2, after passing the deflection plates, the particle gains a velocity in the y-direction, Ft eV L vy = = . (2.9) me mevxD By traveling through a distance x to the fluorescent screen, the y-displacement of the particle is determined by

y vy eV L = = 2 . (2.10) x vx mevxD The charge-to-mass ratio is then e v2Dy = x . (2.11) me VLx Through experiments, Thomson found the ratio is e ≈ 1.759 × 1011 C/Kg. (2.12) me 2.2 Electron as a quantized wave 63

In the 19th century, there was a controversy of whether the cathode ray is a wave or a stream of particles. The interpretation of Thomson’s experiment is based on Newtonian mechanics. However, soon it was found that the electron beam is a wave, and the Young’s double-slit experiment of interference was observed, especially in the electron microscopes. In the 20th century, the cathode ray found a widespread application in TV sets, computer displays, and oscilloscopes, called CRT (cathode-ray tubes). Although CRT was largely replaced by the solid-state displays, one application of the cathode ray is still alive and strong, that is the elec- tron microscope. The resolution of the optical microscopes is limited by the wavelength of the visible light, which is a fraction of a micron. The wavelength of electron beams can be much smaller than a nanometer. Nev- ertheless, in the interpretation of high-resolution electron microscopy, the wave nature of electrons is conceptually indispensible. The double-slit interference experiments for electrons were conducted even with very weak electron beams, where from a na¨ıve point of view the electrons are well separated. The observed interference pattern indicates that in free space, the electrons must be described as waves. In an advanced formalism of quantum mechanics, the path-integral approach of Richard Feynman, the interference effect in the double-slit experiment is natural: Even for a single electron to travel from point A to point B, all possible paths in the entire space must be taken into account.

2.2.2 Millikan’s oil-drop experiment

In 1923, the Nobel Committee announced that Robert Millikan won the Nobel Prize in physics “for his work on the elementary charge of electricity and on the photoelectric effect”. Before working on verifying Einstein’s law of photoelectric effect, he spent five years (1908-1913) to study the quantization of electric charge. Figure 2.8 shows a schematics of the experimental setup. A spray gun creates tiny oil drops from the upper chamber, then fall down to the lower chamber through a small hole. The X-ray ionizes the air and generates free electrons. Some of the electrons are attached to some oil drops. A power supply provides a controllable voltage on the electrodes. Each oil drop is subject to gravity force mg, pulling downwards, see Fig.2.8(A); and the electrostatic force from the high-voltage electrodes on the electrical charges in the oil drop. If electrical charge is quantized with a unit e, then the electrical force should depend on the number of electrons in the oil drop, see Fig.2.8(B) and (C). By adjusting the voltage V ,anoil drop could stay immobile when the net force is zero

neV = mg, (2.13) D 64 Wave and Quantum where n is the number of electrons in the oil drop, V is the electrical field intensity and D is the distance between the two electrodes. The mass of the oil drop can be determined by turning off the voltage, then the oil drop is subject to gravity and the viscosity force of air. Because the viscosity force is proportional to the radius r and the mass is propor- tional to r3, the radius of the oil drop can be determined by the steady speed of free fall, and its mass can be calculated. In 1913, Millikan published a paper on Physical Review, entitled On the Elementary Electrical Charge and the Avogadro Constant. He reported that the observed charge on the oil drop “showed a very exact multiple relation- ship under all circumstances – a fact which demonstrated very directly the atomic structure of the electric charge”. The value he reported is within 1% to the currently recognized value,

e =1.603 × 10−19 C. (2.14)

Combing with the value of charge-to-mass ratio, Eq. 2.12, the mass of the electron is −31 me =9.109 × 10 Kg. (2.15) Because the value of the elementary charge is the Faraday constant divided

Fig. 2.8. Schematics of Millikan’s oil-drop experiment. Using an atomizer, drops of oil is formed in the upper chamber of the setup. Some of the oil drops fall through a hole to the lower chamber. The air in the lower chamber can be ionized by X-ray to generate free electrons. The electrons can attached to the oil drops to make it charged. (A), without attached electrons, an oil drop is pulling downwards by gravity force mg. (B), with one attached electron, the oil drop is pulling upwards by an electric force eV/D. (C), with two attached electrons, the oil drop is pulling upwards by an electric force 2eV/D. The electric charge can be determined by adjusting the voltage to make the oil drop stationary. The schematics is not on scale, as the radius of the oil drops is only about a micron. 2.2 Electron as a quantized wave 65 by the Avogadro constant, Millikan’s experiment also gave the most accurate value of Avogadro constant at that time. Millikan’s oil-drop experiment proved that electrical charge is quantized: the total electrical charge on any isolated subject must be an integer multiple of the elementary charge e. However, as shown in Section 2.3, an electron can never be described as a geometrical point. On the contrary, to reduce the total energy of the system, the charge of a single electron is basically spread over the entire oil drop.

2.2.3 The de Broglie wave Several years before the wave nature of electrons were observed experimen- tally, French physicist Louis de Broglie extended Einstein’s postulation that light can be both wave and quantum to all particles, including the electron, in his 1924 Ph.D. thesis entitled Recherches sur la th´eorie des quanta.Here is a heuristic argument leads to the wave property of electrons. As shown in Eq. 1.53, a general form of a wave is

u(x, t)=a sin(kx − ωt + φ), (2.16) here ω =2πf is the angular velocity, and k = ω/v is the wavevector. According to Einstein, from Eq. 2.1, for light, the energy quantum  is

 = hf = ω, (2.17) where  is a reduced Planck’s constant often called the Dirac constant h  ≡ =1.054 × 10−34 J · s. (2.18) 2π

Fig. 2.9. Louis de Broglie. French physicist (1892–1987), proposed a the- ory of general wave-quantum duality in his 1924 Ph.D. thesis, then won the 1929 Nobel Prize in physics. He was born to a noble family in France and became the 7th duc de Broglie in 1960. In 1942, he was elected as the Perpetual Secretary of the French Academy of Sciences. Af- ter the second world war, he proposed the establishment of multi-national re- search laboratories, leading to the estab- lishment of European Organization for Nuclear Research (CERN). 66 Wave and Quantum

To describe a wave, a wave vector k is also required. Intuitively, de Broglie guessed that k is related to the momentum p of the particle, and the relation can be obtained through a heuristic argument as follows. In general, the the momentum p is defined as in Eq. 1.3,

p = mv. (2.19)

For a photon with energy , according to Einstein’s relation between mass and energy, the mass is  m = . (2.20) c2 Because the velocity of the photon is c, its momentum is  p = mc = . (2.21) c Using Eq. 2.17, Eq. 2.21 becomes

ω p = . (2.22) c According to Eq. 1.55, the relation between angular frequency and wave vector is through the velocity of the wave v, ω k = . (2.23) v Because the speed of light is c, Eq. 2.22 becomes

p = k. (2.24)

Equation 2.24 is the second equation correlating a particle with a wave as de Broglie proposed for all particles, including electrons. Therefore, as proposed by de Broglie, both light and the cathode ray – electrons – are waves. For light, as an electromagnetic wave, the energy is quantized. The elementary unit, the energy quantum, is proportional to the frequency. For electrons, as a wave, the electrical charge is quantized. The unit is a universal constant, the elementary charge e. The electron has a well-defined charge-mass ratio, see Section 2.2.1. Consequently, the mass of the electron is also quantized. According to the historical records, in 1925, Einstein received a preprint of de Broglie’s thesis. Einstein highly appreciated the idea, then immedi- ately sent a letter to Schr¨odinger for his attention. In just a few months, based on the idea of de Broglie, Schr¨odinger formulated his wave equa- tion of electrons and derived the Rydberg formula for the hydrogen atom. Quantum mechanics in its most productive form was born. 2.2 Electron as a quantized wave 67

2.2.4 Low-energy electron diffraction The de Broglie concept that electron is a wave was verified experimentally by C. Davisson and G. P. Thomson, both received a physics Nobel prize in 1937 for the discovery of electron waves. An interesting fact is, G. P. Thomson is the son of J. J. Thomson, who measured the charge-mass ratio of electron, see section 2.2.1. In some popular literature, there was a misconception that J. J. Thomson discovered the electron as a particle, and his son G. P. Thomson discovered the electron as a wave. To speak correctly, J. J. Thomson never showed that electron is a material particle. In fact, the electron wave has a well-defined charge-mass ratio, measured by the father decades before the son measured its wavelengths. The electron wave of de Broglie is the basis of a frequently used tech- nology in surface science, low energy electron diffraction,widelyknownas LEED. A schematics of LEED is shown in Fig. 2.10. The entire setup is under high vacuum. An electron gun, consists of a filament and an acceler- ation voltage U, directs an electron beam to the sample surface. According to de Broglie relation, the electron beam is a wave with wavelength 2π h λ = = . (2.25) k p here we used the relation of Planck’s h and Dirac’s  in Eq. 2.18. The mo- mentum of the electron is related to its kinetic energy Ek, thus determined by the applied voltage U, p = 2meEk = 2meeU. (2.26)

Fig. 2.10. Schematics of LEED. Using an electron gun, an electron beam with wavelength λ is directed to a sample surface. The wavelength of the electron wave is determined by the voltage U, typically 20 V to 200 V. The sample, typically a crystal, serves as a diffraction gratings. The diffracted electrons contain crystallographic infor- mation of the sample. To filter out the randomly scattered electron, a retarding voltage between Grid 1 and Grid 2 is applied. Then the electrons are accelerated by a high positive voltage on the fluorescent screen to make a bright pattern. 68 Wave and Quantum

Fig. 2.11. Working principle and observed pattern of LEED. (A) An electron beam of wavelength λ impinges on a sample with a lattice constant a. A diffracted beam appears at an angle θ where λ = a sin θ. (B) A typical LEED pattern.

Using the known values of the mass me and the charge e of the electron, an approximate equation between the wavelength of the electron λ in unit of nanometer (nm) and the applied voltage U in unit of volt is 1.504 λ = . (2.27) U Table 2.2 shows some typical values of the applied voltage U and the wave- length λ of the electron. The working principle and a sample diffraction pattern obtained by a LEED instrument are shown in Fig. 2.11.2 The impinging electron beam is a wave of wavelength λ determined by the de Broglie relation, expressed by Eq. 2.27. The sample is a crystal with lattice constant a. Each atomic site is a diffraction center. As shown in Fig. 2.11(A), if the diffracted wave and the incoming wave has an angle θ such that λ = a sin θ, (2.28) the waves scattered from each atomic site are in phase, thus the waves reinforce each other. From the above condition, the lattice constant can be determined as λ a = . (2.29) sin θ And in general, as shown in Section 1.6, if for any non-zero integer n,the diffraction peaks occur at nλ = a sin θ. (2.30)

2For details on LEED, see for example M. A. Van Hove, W. H. Weinberg and C.- M. Chan, Low energy Electron Diffraction, Experiment, Theory, and Surface Structure Determination, Springer Series in Surface Science 6. Springer Verlag 1986. 2.2 Electron as a quantized wave 69

Table 2.2: Wavelength and applied voltage Voltage (V) 25 50 100 200 Wavelength (nm) 0.245 0.173 0.123 0.087

The real surface structure is usually more complicated than a simple lattice. The diffraction pattern from LEED experiments can determine the details of the surface crystallographic structure. Figure 2.11(B) shows an example. Because the typical lattice constant in in the range of 0.3 nm to 1 nm, the wavelengths of the electrons from the range of applying voltage, 20 to 200 V, as shown in Table 2.2, are suitable. The low-energy electron diffraction is one of the best demonstration of de Broglie’s concept of the wave nature of electrons.

2.2.5 Double-slit experiment with single electron detectors Because the wavelengths of a typical electron beam is much smaller than those of visible light, the double-slit interference experiment is difficult. Es- pecially, the more interesting experiments with single electron detection only became practical in the 1970s. Because its high value in physics education, many reports were published in American Journal of Physics, the official journal of American Association of Physics Teachers.3 Figure 2.12 shows the general observations. Typically those interference experiments were preformed at a very low electron beam intensity. Single- electron detectors such as sensitive CCD camera chips are used. The arrival of each individual electron can be displayed on a TV screen. If the elec- trons are considered as individual particles, the large distances between the electrons could be interpreted that those electrons are independent to each other. At the beginning of the experiment, single electrons arrive at the CCD chip randomly as shown in Fig. 2.12(a). As time goes on, more elec- trons are recorded at places expected to have interference peaks, as shown in Fig. 2.12(b). By waiting long enough, well-defined interference pattern emerges, see Fig. 2.12(c). The observed phenomenon conflicts with the view that electrons are individual point-like particles. The only possible explanation of those ex- periments is, no matter how dilute it is, the electron beam is as wave. Only when the electrons collide with the detectors, the electrical charge and mass are quantized: only integer multiples of the elementary charge and el- ementary mass are possible. The situation is similar to the electromagnetic waves. In free space, while propagating, the only correct description of light

3For example, American Journal of Physics, 41, 639 (1973); 42, 4 (1974); 44, 306 (1976); 57, 117 (1989); 75, 1057 (2007). 70 Wave and Quantum

Fig. 2.12. Double-slit interference experiment with single-electron detection. The interference experiment was performed with an electron beam of very low intensity and using single electron detectors. (a) at the beginning of the experiment, electrons occurred randomly. (b) as time goes on, a faint interference pattern emerges. (c) by waiting long enough, well defined interference pattern appears no matter how dilute the electron beam is. is an electromagnetic wave. Remember the title of Einstein’s Nobel-prize winning paper is A heuristic viewpoint of the production and conversion of light. Only during production and conversion, the energy of electromag- netic wave is quantized according to the Planck-Einstein formula  = ω. Electron is no difference. Only during the production and absorption, the electrical charge and mass of the electron wave are quantized. It is worth noting that the sharp resolution on the CCD chip for the naked eye does not mean that the location of an electron is observed as a geometric point. Because each detecting cell in a CCD chip is made of thousands of atoms, the signaling of the detection of an electron by one of the CCD cells only reflects the charge density of the electron beam over that cell. We will come back to this point in Chapter 6.

2.3 Electrostatic mass and radius of the electron

In some popular science literature, an electron is described as a geometrical point of charge. This is impossible simply because of the mass associated with the electrostatic field outside the electron. Outside an electron at a distance r from its center, according to Eq. 1.178, the magnitude of electrical field intensity is

dφ(r) e |E| = = . (2.31) dr r2 2.4 The Stern-Gerlach experiment 71

Fig. 2.13. Electrostatic mass of an electron. According to Einstein’s relation between mass and energy, the electrical field outside an electron is as- sociated with a distribution of mass. The smaller the radius, the greater the mass. If the radius is zero, the electro- static mass becomes infinity.

According to Eq. 1.189, the energy density of the electrical field is

1 e2 W = |E|2 = . (2.32) 8π r4

If the electron has a radius r0, the total energy of the static electrical field outside the electron is ∞ e2  = 4πr2Wdr = . (2.33) r0 2r0 The mass of the static electric field of the electron is roughly

2 e2 m = 2 = 2 . (2.34) c c r0 The factor of 2 is a convention by including the field inside the electron. When the radius is smaller than a fixed value re, the mass of the electrostatic field becomes greater than the rest mass of the electron,

e2 ≈ × −15 re = 2 2.818 10 m. (2.35) c me It is called the classic radius of the electron. The radius of the electron must be greater than re, otherwise it becomes absurd. Therefore, it is impossible to represent the electron as a geometrical point.

2.4 The Stern-Gerlach experiment

As shown in Sections 1.5.6, electromagnetic waves are polarized. It has two polarizations, which can be revealed by a crystal with anisotropic refractive indices. In 1922, Stern and Gerlach showed experimentally that electron waves are similarly polarized. Instead of an anisotropic optical crystal, it responds to an anisotropic magnetic field, as shown in Fig. 2.14. Using an oven, silver atoms are evaporated and streamed through a slot, then letting through an anisotropic magnetic field between two magnetic poles of different shapes. A single beam of silver atoms was split into two beams under an anisotropic magnetic field. 72 Wave and Quantum

Fig. 2.14. Schematics of Stern-Gerlach experiment. Abeamofsilveratomsis generated by an oven then goes through an anisotropic magnetic field. the beam of silver atoms split into two, representing different polarizations of the electron wave.

Although the experiment was performed with silver atoms, the subject is the electron. A silver atom is a closed core with a single outer electron. Later, the same experiment was performed with hydrogen atoms, which has only one electron, and produced the same results. Around the time that experiment was conducted, the electron was con- ceived as a particle with internal structure as a spinning top. Because the electron has electrical charge, the circular electrical current generated by the spin creates a magnetic moment, which responds to the magnetic field. The phenomenon was thus termed electron spin. For electromagnetic waves propagating in the z-direction, the two polarizations are often identified as x-polarized and y-polarized, respectively. For electrons, the two polariza- tions are often called spin up and spin down, respectively. By comparing the Stern-Gerlach experiment with the polarization of electromagnetic waves described in Section 1.5.6, the similarity of the ob- served phenomena is apparent. The electron wave has two polarizations, similar to electromagnetic waves. While an anisotropic crystal separates the two polarizations of light, an anisotropic magnetic field separates the two polarizations of the electron wave. Descriptions of the electron spin in terms of Pauli equation will be discussed in Chapter 9. Chapter 3 Schr¨odinger’s Equation I

The year 1905 was Albert Einstein’s annum mirabilis when he published four papers on Annalen der Physik that literally started the modern physics. Similarly, the year 1926 was Erwin Schr¨odinger’s annum mirabilis with six papers published on the same journal Annalen der Physik that defined non- relativistic quantum mechanics. Thus commented Paul Dirac in 1929: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known”. Those papers, belonging to the defining publications of modern science, are still worth reading. Here is a brief summary of the important ones: The first paper, Quantization as an Eigenvalue Problem, Part I,re- ceived by Annalen der Physik on January 27, 1926, defined a wavefunction ψ, which is “everywhere real, single-valued, finite, and continuously differ- entiable up to the second order”. A differential equation of ψ is presented as a variation of the Hamilton-Jacobi equation. By applying that equation to the hydrogen atom, the Rydberg formula was explained. The second paper, Quantization as an Eigenvalue Problem, Part II,pre- sented a parallelism of the relation of quantum mechanics and classical mechanics with the relation of wave optics and geometrical optics. He em- phasized that the wave nature of electrons is fundamental, and the particle view is a macroscopic approximation. He also presented an even simpler way to introduce the differential equation of the wavefunction based on de

Fig. 3.1. Austrian banknote with a portrait of Schr¨odinger. It is a rare honor for a scientist to have a portrait printed on a banknote. Note the large value. 74 Schr¨odinger’s Equation I

Broglie wave. Three further problems were treated: the harmonic oscillator, rigid rotor and non-rigid rotor, all related to molecular physics. The time-dependent Schr¨odinger’s equation was introduced in the sixth paper, received by Annalen der Physik on June 23, 1926. The problem of interaction of radiation with atomic systems was resolved. We will introduce Schr¨odinger’s equations in three stages. In this Chap- ter, the time-independent version for a single electron is introduced. In Chapter 4, the many-electron version is introduced. In Chapter 6, the time-dependent Schr¨odinger’s equation is introduced.

3.1 Time-independent Schr¨odinger equation

Time-independent Schr¨odinger equation can be derived by applying the de Broglie postulate to the classical energy integral, Eq. 1.11, p2 + V (x)=E. (3.1) 2me According to de Broglie, the electron is a wave. A bound electron is a standing wave ψ(r), satisfying the Helmhotz equation, Eq. 1.120, ∇2ψ(r)+k2ψ(r)=0. (3.2) The wave vector is then ∇2ψ(r) k2 = − . (3.3) ψ(r) According to the de Broglie’s relation, Eq. 2.24, p = k. (3.4) Combining Eq. 3.3 and Eq. 3.4, one finds 2∇2ψ(r) p2 = − . (3.5) ψ(r) Insert Eq. 3.5 into Eq. 3.1, multiply both sides by ψ(r), a differential equa- tion for wavefunction ψ(r) is obtained, 2 − ∇2ψ(r)+V (r)ψ(r)=Eψ(r). (3.6) 2me This is the time-independent Schr¨odinger equation.

3.2 Wavefunctions in potential wells

To illustrate the meanings of the Schr¨odinger equation and wavefunctions, the problems of electrons in potential wells are studied. The similarity to sound waves is explained. The Dirac notations are introduced. 3.2 Wavefunctions in potential wells 75

3.2.1 One-dimensional potential well Consider first a one-dimensional potential well of length L. Within the well, the potential is zero. Schr¨odinger’s equation is 2 2 − d ψ(x) 2 = Eψ(x). (3.7) 2me dx Assuming the wall is infinitely high. The boundary conditions are: at the boundaries, x = 0 and x = L, the wavefunction must be zero. Introduce a wave vector k defined as √ 2m E k = e , (3.8)  the Schr¨odinger’s equation Eq. 3.7 becomes

d2ψ(x) = −k2ψ(x). (3.9) dx2 The general solution of Eq. 3.9 is

ψ(x)=C sin(kx + φ), (3.10) where C is a normalization constant, and φ is a phase angle. Because at x = 0, the wavefunction must be zero, one should have φ =0:

ψ(x)=C sin(kx). (3.11)

The boundary condition at the other end, ψ(L) = 0, requires that nπ k = , (3.12) L where n =1, 2, 3, ... is an integer. The wavefunctions are nπx ψ (x)=C sin . (3.13) n L The constant C will be determined shortly. Figure 3.2 shows the wavefunc- tions. It is similar to the sound waves in a bugle, see Fig. 1.15.

Fig. 3.2. Wavefunctions in a one-dimensional potential well. The wavefunctions are labeled by quantum number n, similar to the sound waves in a bugle. See Fig. 1.15. 76 Schr¨odinger’s Equation I

Fig. 3.3. Energy levels in a one-dimensional potential well. The wavefunctions are labeled by quantum number n. The wavefunctions are similar to the vibration of string, see Fig. 1.8. The energy eigenvalue is proportional to n2.

In classical mechanics, as in Eq. 3.1, energy can take any value. In quantum mechanics, because of condition Eq. 3.12, energy is quantized:it can only take discrete values determined by Eqs. 3.8 and 3.12: 2 22 2 n π En = n E1 = 2 , (3.14) 2meL Those allowed values of energy are called the energy eigenvalues, adapted from German, the proper values of energy. Because the Schr¨odinger equation is linear to the wavefunction, the con- stant C does not affect the determination of energy eigenvalues. According to Schr¨odinger, the square of the wavefunction is proportional to the charge density distribution of the electron as a field in space:

ρ(x)=−eψ2(x). (3.15)

Because the total charge of an electron over the space equals to one elemen- tary charge −e, the integral over the entire space must equal to 1. In the current situation, the electron is confined in a well of width L, L ψ2(x)dx =1. (3.16) 0 The average value of the square of sine function over any number of half periods is 1/2. Therefore, for all quantum numbers, the constant is 2 C = . (3.17) L 3.2 Wavefunctions in potential wells 77

The wavefunctions are 2 nπx ψ (x)= sin . (3.18) n L L

It is straightforward to show that eigenstates with different quantum numbers n are orthogonal. In fact, using Eq. 3.18, L dx ψn(x) ψm(x)=δnm, (3.19) 0 which is zero when n = m, and is 1 when n = m. Furthermore, the set of wavefunctions is complete. Any function f(x)in the interval [0,L] can be expanded as a sum of those wavefunctions, ∞ f(x)= bnψn(x), (3.20) n=1 with coefficnents L bn = f(x) ψn(x) dx. (3.21) 0 This is a special case of the Fourier theorem, the proof can be found in any mathematics textbook with Fourier series. Nonetheless, if Eq. 3.20 is true, it is easy to prove that the expression of the coefficients, Eq. 3.21, is correct. In fact, because of the orthonormnal relation Eq. 3.19, L ∞ ∞ ψn(x)dx bmψm(x)= δnmbm = bn. (3.22) 0 m=1 m=1

3.2.2 The Dirac notation The notations of wavefunctions, Eq. 3.18, and the integrals, Eqs. 3.16 and 3.19, occurs very often in quantum mechanics. In the third edition of Prin- ciples of Quantum Mechanics, Dirac introduced the bra and ket notations, that greatly simplifies mathematical notations in quantum mechanics. In the real formulation of quantum mechanics, bra and ket are equivalent. A wavefunction can be denoted either as a bra or as a ket. For the case of electrons in a one-dimensional potential well, it is either 2 nπx n| = sin , (3.23) L L or 2 nπx |n = sin . (3.24) L L 78 Schr¨odinger’s Equation I

A complete bracket represents an integral of two wavefunctions, L n|m = dx ψn(x) ψm(x). (3.25) 0 Obviously, the real Dirac notation is symmetric, or commutative,

n|m = m|n . (3.26)

The orthogonization and normalization condition in Dirac notation is

n|m = δnm, (3.27) and the completeness of the wavefunctions means that any function |f in the same interval [0,L] can be expanded as a sum of the wavefunctions, ∞ |f = n|f |n . (3.28) n=1 The Schr¨odinger equation, Eq. 3.6, can be written as

Hˆ |ψ = E|ψ (3.29) by defining a Hamiltonian operator or simply Hamiltonian

2 Hˆ ≡− ∇2 + V (r). (3.30) 2me A hat is marked to denote that it is not an ordinary number, but an operator that makes sense only when acting upon a wavefunction.

3.2.3 Two-dimensional potential well The treatment of the one-dimensional potential well can be generalized to a two-dimensional potential well. The potential function is

V (x, y)=0 0>x>L and 0 >y>L, (3.31) V (x, y)=∞ elsewhere.

Within the square, 0

Similar to the arguments in Section 3.2.1, the general mathematical expression of the wavefunctions is n πx n πy |n n = ψ (x, y)=C sin x sin y , (3.33) x y nx,ny L L where nx and ny are integers. Because the average value of the square of sine function over any number of half periods is 1/2, for all quantum numbers, the constant is 2 2 2 C = × = . (3.34) L L L The normalized wavefunctions are 2 n πx n πy |n n = sin x sin y , (3.35) x y L L L as shown in Fig. 3.4. Obviously, eigenstates with different quantum numbers nx,ny are orthogonal. In fact, using Eq. 3.35, L L | dx dy nxny mxmy = δnx,mx δny ,my (3.36) 0 0 which is none zero only when nx = mx and ny = my.

Fig. 3.4. Wavefunctions in a two-dimensional potential well. The wavefunctions are labeled by the two quantum numbers, nx and ny, in Dirac notation, see Section 3.2.2. The wavefunctions are largely determined by the nodal structures. Wavefunctions with different nx and/or ny are orthogonal, see Eq. 3.27. 80 Schr¨odinger’s Equation I

3.2.4 Wavefunctions outside a spherical potential well In Section 1.4, the solutions of the wave equation inside a basketball are discussed. Similar solutions of the Schr¨odinger equation inside a spheri- cal potential well can be obtained. We leave this as a problem. In this section, a similar but more useful problem is discussed: the solutions of the Schr¨odinger equation outside a spherical potential well. Such solutions have applications in solid-state physics to describe the wavefunctions of elec- tronic bands in the space surrounding the atomic cores, and the formation of covalent bonds in chemical physics. Outside the sphere where r>r0, the potential is zero. The Schr¨odinger equation Eq. 3.6 becomes

2 − ∇2ψ(r)=Eψ(r). (3.37) 2me The energy E is negative, see Fig 3.5. Introducing a decay constant κ, √ −2m E κ = e , (3.38) 

Equation 3.37 becomes

∇2ψ(r) − κ2ψ(r)=0. (3.39)

Following the mathematics in Section 1.4, by writing the wavefunction as a product of a radial function R(r) and a spherical harmonics,

ψ(r)=R(r)Y (θ, φ), (3.40) the differential equation for the radial function R(r)is d dR(r) r2 − l(l +1)+κ2r2 R(r)=0, (3.41) dr dr where the parameter l is an index of the spherical harmonics, see Section 1.4.2. Equation 3.41 is almost identical to Eq. 1.135 except that −k2 is replaced by κ2. It is expected that the solutions are similar. By direct insertion, one finds the solutions of Eq. 3.41 are

R(r)=kl(κr), (3.42) where 1 k (ρ)= e−ρ, (3.43) 0 ρ 1 1 k (ρ)= + e−ρ, (3.44) 1 ρ ρ2 3.3 The harmonic oscillator 81

Fig. 3.5. Wavefunctions outside a spherical potential well. (A) The energy diagram. (B) through (E), the wavefunctions are specified by the azimuthal quantum number l and the magnetic quantum number m, determined by the wavefunctions inside the sphere, which can be understood as the vacuum tails of the wavefunctions inside the sphere. Wavefunctions with the same l but different m are degenerate. and 1 3 3 k (ρ)= + + e−ρ, (3.45) 2 ρ ρ2 ρ3 for l = 0, 1, and 2, respectively. The general expression is d l e−ρ k (ρ)=(−ρ)l . (3.46) l ρdρ ρ Although those functions are elementary, due to their origin, they are called spherical modified Bessel functions, see Appendix D. Figure 3.5(A) shows the energy diagram and wavefunctions. A wave- function inside the sphere ψ(r) is a solution of a Schr¨odinger equation inside the sphere with potential function U(r). Simialr to the acoustic waves in Section 1.4, it is a product of a radial function R(r) and a spherical harmon- ics Y (θ, φ). The spherical harmonics is labeled by an azimuthal quantum number l and a magnetic quantum number m. Outside the sphere, the potential is zero. The wavefunctions are shown in Fig. 3.5 (B) through (E). At the boundary r = r0, their values match those of the wavefunctions inside the sphere. As a vacuum tail of a wave- function inside the sphere, it decays with radius r. Nevertheless, the angular dependence characterized by the spherical harmonics is perserved.

3.3 The harmonic oscillator

In quantum mechanics, the harmonic oscillator is of fundamental impor- tance. It describes the oscillation of molecules and solids near its equilib- rium point. The electromagnetic wave can be decomposed into a number 82 Schr¨odinger’s Equation I of simple harmonic oscillators. Using the quantization procedure presented here, the electromagnetic waves can be quantized. It is the basis of quantum electrodynamics, the complete theory of radiation and matter. From Eqs. 1.26 and 3.6, the Schr¨odinger equation for a one-dimensional harmonic oscillator is 2 d2 m − + ω2x2 ψ(x)=Eψ(x). (3.47) 2m dx2 2 By introducing a dimensionless coordinate q defined as mω q ≡ x, (3.48)  the Schr¨odinger equation Eq. 3.47 is standardized to 1 d2 − + q2 ωψ(q)=Eψ(q). (3.49) 2 dq2

3.3.1 Creation operator and annihilation operator As we have presented in previous sections, the basic concept in quantum mechanics is the wavefunction, and it is governed by a partial differential equation, the Schr¨odinger equation. In some cases, the process of obtaining solutions of partial differential equations can be greatly simplified with dif- ferential operators using algabraic methods. This is especially true for the solution of the harmonic oscillator. In order to find an algebraic solution of the harmonic oscillator, a pair of operators are introduced: an annihilation operator, 1 d aˆ = √ q + , (3.50) 2 dq and a creation operator 1 d aˆ† = √ q − . (3.51) 2 dq The meanings of these terms well be clarified soon. Those operators not only greatly simplify the solution of the quantum-mechanical harmonic oscillator problem, but also serve the basis of quantum field theory. By acting on any function of q, a simple algebra shows that the two operators satisfy the following commutation relation, [ˆa, aˆ†] ≡ aˆaˆ† − aˆ†aˆ =1. (3.52) Also by a simple algebra, the Schr¨odinger equation Eq. 3.49 becomes 1 aˆ†aˆ + ωψ(q)=Eψ(q). (3.53) 2 3.3 The harmonic oscillator 83

3.3.2 Algebraic solution of the Schr¨odinger equation In this Section, we show how to utilize Eq. 3.52 to solve the Schr¨odinger equation Eq. 3.53. Denoting the wavefunction with Dirac notation |n (see Section 3.2.2), it is sufficient to solve the following algebraic equation,

† aˆ aˆ|n = un|n . (3.54)

Here the eigenstates are labeled by a number n with eigenvalue un.By comparing Eq. 3.53 with Eq. 3.54, the energy eigenvalues are 1 E = u + ω. (3.55) n n 2

As a consequence of Eq. 3.52, if |n is an eigenstate with eigenvalue un, thena ˆ|n is also an eigenstate,

† † aˆ aˆ aˆ|n =(ˆaaˆ − 1)ˆa|n =(un − 1)a ˆ|n (3.56)

† with eigenvalue un − 1. Because n|aˆ aˆ|n must not be negative, there must be an eigenstate with minimum value, un = 0. For such a state,

aˆ†aˆ|0 =0. (3.57)

On the other hand, also as a consequence of Eq. 3.52, if |n is an eigenstate † with eigenvalue un,thenˆa |n is also an eigenstate

† † † † † aˆ aˆaˆ |n =ˆa (ˆa aˆ +1)|n =(un +1)ˆa |n (3.58) with eigenvalue un + 1. Starting with the lowest eigenstate |0 , by applying aˆ† many times, we have

n n aˆ†aˆ aˆ† |0 = n aˆ† |0 . (3.59)

Because the eigenvalue of the zeroth eigenstate |0 of the operatora ˆ†aˆ is zero, and each time a creation operatora ˆ† is applied, the eigenvalue is added by 1, the eigenvalues of the operatora ˆ†aˆ equals to an integer, the number of times the a creation operator applied. Therefore, except for a numerical constant, it is the n-th eigenstate of the operatora ˆ†aˆ: † n |n = Cn aˆ |0 . (3.60)

The operatora ˆ†aˆ is deservedly called the particle number operator,

Nˆ ≡ aˆ†a,ˆ (3.61) because its eigenvalue is the number of energy quanta,

Nˆ|n = n|n . (3.62) 84 Schr¨odinger’s Equation I

The normalization constant Cn can be determined as follows. The zeroth-order state is by definition normalized, 0|0 =1. (3.63) By applyinga ˆ† many times, we have n 0| (ˆa)n aˆ† |0 = n!. (3.64)

−1/2 Therefore, Cn =(n!) ,and

1 n |n = √ aˆ† |0 . (3.65) n! Following Eq. 3.55, the energy eigenvalues of the harmonic oscillator is 1 E = n + ω. (3.66) n 2 The energy level of the harmonic oscillator is thus quantized, with energy quanta ω. The operatora ˆ† adds an energy quanta to the oscillator, thus named a creation operator; the operatora ˆ removes an energy quanta from the oscillator, thus named an annihilation operator.

3.3.3 Explicit expressions of the wavefunctions The above algebraic solution provides the simplest approach to find explicit expressions of the wavefunctions. First, from Eqs. 3.57 and 3.50, the zeroth-

Table 3.1: Wavefunctions of the harmonic oscillator

State Energy wavefunction

1 1 2 |0 ω √ e−q /2 2 π 3 1 2 |1 ω √ 2qe−q /2 2 2 π

5 1 2 |2 ω √ 2q2 − 1 e−q /2 2 2 π

7 1 2 |3 ω √ 2q3 − 3q e−q /2 2 3 π

9 1 2 |4 ω √ 4q4 − 12q2 +3 e−q /2 2 24 π

11 1 2 |5 ω √ 4q5 − 20q3 +15q e−q /2 2 60 π 3.3 The harmonic oscillator 85

order wavefunction |0 = ψ0(q) should satisfy d q + ψ (q)=0. (3.67) dq 0

The solution is q2 ψ (q)=C exp − . (3.68) 0 0 2

The normalization constant C0 can be determined directly by ∞ ∞ √ 2 2 − 2 2 ψ0(q)dq = C0 exp( q )dq = C0 π =1, (3.69) −∞ −∞ which gives −1/4 C0 = π . (3.70)

All wavefunctions of the harmonic oscillator |n = ψn(q) can be obtained from Eqs. 3.51 and 3.65 using Eqs. 3.68 and 3.70: 1 d n q2 ψn(q)= √ q − exp − . (3.71) 2nn! π dq 2

The first few wavefunctions are listed in Table 3.1, and graphically dis- played in Fig. 3.6. The solid curve, a parabola, represents the potential

Fig. 3.6. Energy levels and wavefunctions of a harmonic oscillator. The solid curve is the potential energy. The wavefunctions of the first five eigenstates are shown. The red shade indicates positive phase, and blue shade indicates negative phase. The y-position of the energy eigenvalue is the baseline for the wavefunction. 86 Schr¨odinger’s Equation I energy as a function of the normalized coordinate q. The horizontal lines represent the energy levels of the eigenstates. For the region inside the potential curve, energy level E is greater than the potential energy. The wavefunction resembles a sinusoidal wave. Outside the potential curve, the energy level is lower than the potential curve. The wavefunction resembles an exponential function decaying into the barrier. The lowest eigenstate 1  has an energy value of 2 ω. The wavefunction has no node. The energy eigenvalue increases by ω each step, while a new node is added. Those nodal structures make the wavefunctions orthogonal to each other.

3.4 The hydrogen atom

Hydrogen atom is a central subject of quantum mechanics. The accurate interpretation of the Rydberg formula marked a groundbreaking triumph. Many predictions of relativistic quantum electrodynamics are verified by measurements on the hydrogen atom. The hydrogen wavefunctions are the foundation for the understanding of complex atoms and atomic systems. There are only two real-world systems that the Schr¨odinger equation has + analytic solutions: the hydrogen atom, and the hydrogen molecular ion, H2 , which is the basis to understand the chemical bond, and how condensed matter is formed. The ground-state wavefunction of hydrogen atom is also + the starting point of the treatment of H2 . In Gaussian unit system, the potential energy function V (r) is the at- tractive force from the positively charged proton, see Eq. 1.180,

e2 V (r)=− . (3.72) r

Because the electron mass me is much smaller than the proton mass, to a good approximation, the Schr¨odinger equation Eq. 3.6 is

2 e2 − ∇2ψ − ψ = Eψ. (3.73) 2me r

Fig. 3.7. Hydrogen atom in spherical polar coordinates. The center of the coordinate system is the positively charged proton. The force and potential energy V only depends on radius r. Therefore, it is natu- ral to use spherical polar coordinates, with radius r, polar angle θ, and az- imuth φ. 3.4 The hydrogen atom 87

The potential only depends on r. It is natural to solve the equation in polar coordinates, similar to the problem of basketball in Section 1.4. In spherical polar coordinates, Eq. 3.73 becomes 2 1 ∂ ∂ψ 1 e2 − 2 − 2 − 2 r 2 L ψ ψ = Eψ. (3.74) 2me r ∂r ∂r r r where the angular momentum operator L2 is defined as 1 ∂2ψ ∂ ∂ψ L2ψ ≡− +sinθ sin θ . (3.75) sin2 θ ∂φ2 ∂θ ∂θ

The solutions of Eq. 3.75 are spherical harmonics, see Sections 1.4.2.

3.4.1 The ground state First, we study the ground state, where the wavefunction ψ only depends on r. The Schr¨odinger equation Eq. 3.74 becomes 2 1 d dψ e2 − 2 − 2 r ψ = Eψ, (3.76) 2me r dr dr r

Intuitively, since the electron is being attracted by the positively changed proton, the wavefunction should concentrate near the proton, and decays with distance r. Therefore, to resolve Schr¨odinger’s equation Eq. 3.76, we use the following trial function

ψ = Ce−r/a, (3.77) where a is a parameter to be determined, and C is a normalization constant. Insert Eq. 3.77 into Eq. 3.76, eliminate the common factor ψ, we obtain

2 2 2 − − e 2 = E. (3.78) mear 2mea r The solution should be valid for all values of r. The two terms with common factor 1/r must cancel each other. It implies

2 a = 2 . (3.79) mee Therefore, the trial function Eq. 3.77 is a valid solution of Eq. 3.76, and the decay length is a, given by Eq. 3.79. The rest of Eq. 3.78 provides an expression of the energy eigenvalue

2 4 − −mee E = 2 = 2 . (3.80) 2mea 2 88 Schr¨odinger’s Equation I

Fig. 3.8. Wavefunction of ground-state hydrogen atom. (A) The density plot of the wavefunction. (B) the amplitude profile of the wavefunction.

The parameter a in Eq. 3.79 is the Bohr radius, 2 ≡ ≈ a0 2 52.9pm. (3.81) mee The absolute value of the energy eigenvalue in Eq. 3.80 is the Rydberg con- stant, which agrees well with experimental findings, 4 2 ≡| | mee e ≈ Ry E0 = 2 = 13.6eV. (3.82) 2 2a0 In order to determine the normalization constant C,wenotethatac- cordingtoSchr¨odinger, the square of the wavefunction is proportional to the charge density distribution of an electron as a field in space, ρ = −eψ2, (3.83) requiring that the integration of ψ2 over the entire space is 1, ∞ ∞ 4πψ2r2dr = 4πC2e−2r/a0 r2dr =1, (3.84) 0 0 which yields 1 C = . (3.85) 3 πa0 The ground-state wavefunction of hydrogen atom is then 1 |1s ≡ψ = e−r/a0 . (3.86) 1s 3 πa0 The subscript 1s indicates that it is the lowest state and is spherically symmetric. A density plot and an amplitude contour of the ground-state wavefunction of hydrogen atom are shown in Fig. 3.8. Wavefunction is the form of existence of the electron. The ground state represents an electron at rest. There is no point charge, and no motion at all. The electrical charge of the electron is spread in space around the proton, with the highest charge density in the immediate vicinity of the proton. 3.4 The hydrogen atom 89

3.4.2 Energy eigenvalues of excited States Following the mathematics in Section 1.4, by writing the wavefunction as a product of a radial function R(r) and a spherical harmonics,

ψ(r)=R(r)Y (θ, φ), (3.87) the differential equation for the radial function R(r)is 2 1 d dR(r) l(l +1)R(r) e2 − 2 − 2 r 2 + R(r)=ER(r), (3.88) 2me r dr dr r r where l is the azimuthal quantum number, a parameter of the spherical harmonics, see Section 1.4.2. To resolve Eq. 3.88, by intuition, we use the following trial function R(r)=Crbe−r/a, (3.89) where the parameters a and b are to be determined by Eq. 3.88, and C is a normalization constant. Insert Eq. 3.89 into Eq. 3.88, eliminate the common factor R(r), we obtain an algebraic equation 2 2 − b(b +1)− l(l +1) 2(b +1)− 1 e 2 2 + 2 + = E. (3.90) 2me r r ar a r To cancel the two terms with 1/r2, a sufficient condition is

b = l. (3.91)

To cancel the two terms with 1/r, one must have 2 a =(b +1) 2 =(b +1)a0. (3.92) mee The remaining terms in Eq. 3.90 determine the energy eigenvalue 2 − − Ry E = 2 = 2 . (3.93) 2mea (b +1) In all those formulas, (b + 1) occurs. Define an integer index n,

n = b +1, (3.94) which is the principle quantum number. The wavefunction is

n−1 −r/na0 ψ = Cr e Ylm(θ, φ). (3.95) The energy eigenvalue only depends on the principle quantum number n, perfectly explains the experimentally discovered Rydberg formula, Ry E = − . (3.96) n n2 90 Schr¨odinger’s Equation I

The length scale of the wavefunction, a = na0, is a useful parameter for further study of the hydrogen wavefunction. It is  a = na0 = √ . (3.97) −2meE Similar to Eq. 3.84, the normalization constant is found to be 22n+1 C = 2n+3 . (3.98) π(2n +2)!(na0) In Fig. 3.9, we show some interesting cases. Following Section 1.4.2, for m = l, the spherical harmonics up to a normalization constant is g ∝ l Yll (θ, φ) sin θ cos lφ, (3.99) and u ∝ l Yll (θ, φ) sin θ sin lφ. (3.100) The wavefunctions shown in Fig. 3.9 are on the plane z =0,wheresinθ =1. Using Eqs. 3.91 and 3.94, l = n − 1. The explicit expressions are

n−1 −r/na0 ψg ∝ r e cos(n − 1)φ, (3.101) and n−1 −r/na0 ψu ∝ r e sin(n − 1)φ, (3.102)

Fig. 3.9. Wavefunctions of excited-states of hydrogen atom. Wavefunctions in Eqs. 3.101 and 3.102, standing waves on the horizontal plane z = 0. For each principle quantum number, there is an even wavefunction symmetric regarding the x-axis, and an wavefunction antisymmetric regarding the x-axis. The energy level, Eq. 3.96, depends only on the principle quantum number n, explains the Rydberg formula. 3.4 The hydrogen atom 91 where the subscript g and u indicates even and odd with respect to x-axis. Those wavefunctions represent standing waves on the z =0plane.As remarked by Schr¨odinger, the energy-level quantization in hydrogen atom resembles the nodes of a vibrating string in Section 1.2.1, and especially the eigenfrequencies in a basketball in Section 1.4. The electron forms standing waves around the proton. Similar to the vibrating string and the basketball, different patterns of standing waves with different number of nodes are formed, resulting in different energy levels.

3.4.3 Wavefunctions In the previous Subsection, following an elementary mathematical proce- dure, special solutions of Eq. 3.88 are derived. Due to a high degree of degeneracy, all energy eigenvalues are obtained. Especially, for the cases of l = n − 1andm = l, wavefunctions are shown in Figs 3.8 and 3.9. Nevertheless, Eq. 3.88 does have other useful solutions, here we find them. Inspired by the special solutions, a dimensionless variable ρ is introduced by scaling the radius r using the length a = na0 in Eq. 3.97, r ρ = . (3.103) a Equation 3.88 becomes 1 d dR(ρ) l(l +1) 2n ρ2 + −1 − + R(ρ)=0. (3.104) ρ2 dρ dρ ρ2 ρ

In the process of the algebra, Eq. 3.81 is applied to obtain

2 2meae a 2 = = n. (3.105)  a0 Analogous to Eqs. 3.89 and 3.91, we make a substitution

R(ρ)=ρl e−ρ F (ρ). (3.106)

Insert Eq. 3.106 to Eq. 3.104, the differential equation for F (ρ) is obtained:

d2F (ρ) dF (ρ) ρ +2(l +1− ρ) +2(n − l − 1)F (ρ)=0. (3.107) dρ2 dρ It is the differential equation for associate Laguerre polynomials, well known for mathematicians for two centuries. The general formula is fairly cumber- some. In condensed-matter physics, chemistry and molecular biology, only a few more cases are needed. Here is a complete list. 1 r F (r)=1− ,n=2,l= 0; (3.108) 2 a0 92 Schr¨odinger’s Equation I

2 r 2 r 2 F (r)=1− + ,n=3,l= 0; (3.109) 3 a0 27 a0 1 r F (r)=1− ,n=3,l=1. (3.110) 6 a0 Those expressions can be verified by directly inserting those polynomials F (r) into Eq. 3.107. It is left as an exercise. By combining the radial functions in Eq. 3.106, using Eqs. 3.108 through 3.110 and the spherical harmonics in Table 1.3, we found the first nine wavefunctions of the hydrogen atom, listed in Table 3.2, and in Fig. 3.10. In Table 1.3, the first column shows the chemist’s name of the wavefunction. The second column is in Dirac notation. The expression of the wavefunction, column 3, is in Cartesian coordinates with Bohr radius a0 as unit. The last column is the average size of the wavefunction, also in unit of a0.

Fig. 3.10. Hydrogen wavefunctions. The phase, either positive or negative, is shown in different color. The value is indicated by intensity, but scaled slightly to facilitate visualization. As shown in the last column of Table 3.2, the size of wavefunctions with different principle quantum number n varies dramatically. All the figures showing here are accurately sized according to the Scale at the bottom of the Figure. 3.4 The hydrogen atom 93

Table 3.2: Wavefunctions of the hydrogen atom

Name State Wavefunctionr ¯(a0) 1 1s |1s0 √ e−r 1.5 π 1 1 2s |2s0 √ (1 − r) e−r/2 6 2 2π 2

1 −r/2 2px |2p1u √ xe 5 4 2π

1 −r/2 2py |2p1g √ ye 5 4 2π

1 −r/2 2pz |2p0 √ ze 5 4 2π 1 2 2 3s |3s0 √ (1 − r + r2) e−r/3 13.5 3 3π 3 27

2 1 −r/3 3px |3p1u √ (1 − r)xe 12.5 27 6π 6

2 1 −r/3 3py |3p1g √ (1 − r)ye 12.5 27 6π 6

2 1 −r/3 3pz |3p0 √ (1 − r)ze 12.5 27 6π 6

2 2 1 2 −r/3 3dz2 |3d0 √ (z − r ) e 10.5 27 6π 3

2 −r/3 3dxz |3d1u √ xz e 10.5 27 6π

2 −r/3 3dyz |3d1g √ yz e 10.5 27 6π

2 −r/3 3dxy |3d2u √ xy e 10.5 27 6π

4 2 2 −r/3 3dx2−y2 |3d2g √ (x − y ) e 10.5 27 6π

3.4.4 Nomenclature of atomic states The wavefunctions of the hydrogen atom are the foundation of the nomen- clature of electron states in many-electron atoms, see Table 3.3. The principle quantum number n identifies the shells, labled by K, L, M, etc. The azimuthal quantum number l identifies the subshells. The labels, s, p, d etc., have its origin in atomic spectrum. The spectral lines starting from l = 0 states are often sharp, thus named s. The spectral lines starting 94 Schr¨odinger’s Equation I

Table 3.3: Nomenclature of atomic states n Shell Maximum l Subshell Maximum electrons electrons 1 K 2 0 1s 2 2 L 8 0 2s 2 12p 6 3M 18 03s 2 13p 6 23d 10

from l = 1 states are often intensive, thus named p, means principle. The spectral lines starting from l = 2 are often diffuse, thus named d. Because of spin, see Section 2.4, each wavefunction has two electrons.

3.4.5 Degeneracy and wavefunction hybridization The energy eigenvalues of the hydrogen atom only depends on the principle quantum number n. For each principle quantum number, there are n2 differ- ent wavefunctions: it is n2-fold degenerate. For n = 2, there are four states, |2s , |2px , |2py ,and|2pz .BecausetheSchr¨odinger equation is linear, any linear superposition of wavefunctions with the same energy eigenvalue is also a good wavefunction with the same energy eigenvalue. Especially, an s-wavefunction can make linear superposition with p-wavefunctions, to form hybrid wavefunctions. This concept is fundamental in chemistry, such as for carbon and silicon, especially in organic chemistry. Here we show the concept of hybridization using the hydrogen wavefunctions.

Fig. 3.11. Hybrid sp1 wavefunctions. The phase of the wavefunction is shown by color, positive in red, negative in blue. The equal-value contours are also shown. The main lobe spans a wide angle. It is difficult to show both on the same figure. 3.4 The hydrogen atom 95

Figure 3.11 shows two sp1 hybrid wavefunctions. (A) is with a positive 2px wavefunction, 1 1 |2sp1+ = √ |2s + √ |2px , (3.111) 2 2 resulting in a wavefunction preferentially concentrated in the +x direction; and (B) is with a negative 2px wavefunction, 1 1 |2sp1− = √ |2s −√ |2px , (3.112) 2 2 resulting in a wavefunction preferentially concentrated in the −x direction. This happens for example in acetylene C2H2, for both the σ-bond between the two carbon atoms, and the two C-H bonds. Figure 3.12 shows three sp2 hybrid wavefunctions. The formulas are 1 2 |2sp2u = √ |2s + |2py , (3.113) 3 3 1 1 1 |2sp2l = √ |2s −√ |2px −√ |2py , (3.114) 3 2 6 and 1 1 1 |2sp2r = √ |2s + √ |2px −√ |2py , (3.115) 3 2 6 respectively. It is the basic structure of grapheme and carbon nanotubes. In ethylene, C2H4,thesp1 hybrid wavefunctions are the basis of both the σ-bond between the two carbon atoms, and the four C-H bonds. A prevailing hybridization is the sp3 mode, where one s-wavefunction is linearly superposed with three p-wavefunctions to form four hybrid wave- functions pointing to the four vertices of a tetrahedron, see Fig. 3.13. A set

Fig. 3.12. Hybrid sp2 wavefunctions. The three sp2 hybrid wavefunctions are pointing to the three vertices of a regular triangle, 120◦ apart in the same plane. The phase, positive or negative, is shown by color. The equal-value contours are also shown. 96 Schr¨odinger’s Equation I

Fig. 3.13. Hybrid sp3 wavefunctions. (A), one of the four hybrid wavefunctions. The other three have the same shape but different orientation, pointing to the four vertices of a regular tetrahedron. (b), a regular tetrahedron. of sample formula is 1 |t111 = (|2s + |2px + |2py + |2pz ) , 2 1 |t11¯ 1¯ = (|2s −|2px + |2py −|2pz ) , 2 1 (3.116) |t1¯11¯ = (|2s −|2px −|2py + |2pz ) , 2 1 |t11¯1¯ = (|2s + |2px −|2py −|2pz ) . 2

It is the backbone of all alkanes, including methane, ethane, propane, bu- tane, etc., and the crystalline structures of diamond and silicon. In some quantum chemistry textbooks, those hybrid wavefunctions are depicted as sharply oriented long and narrow bulbs, to emphasize the direc- tional effect. Often, three or four such long and narrow bulbs are shown in a single drawing. As shown here, the accurate amplitude contours of wave- functions of sp-hybridizations span a broad angle. It is difficult to depict three or four such hybrid wavefunctions on a single graph. By comparing Fig. 3.6 with the vibrational modes of a string, Fig. 1.28, a striking resemblance is found. As an objective reality, wavefunction is of the same nature as the standing waves in classical physics.

3.5 General properties of wavefunctions

In this Chapter, we presented wavefunctions as the solution of the Schr¨odinger equation through three example. Here are the general properties of the wavefunctions. Those properties are similar to those in acoustics, see Sec- tion 1.2. For simplicity, we use Dirac notations. 3.5 General properties of wavefunctions 97

3.5.1 Normalization According to Schr¨odinger, the square of the wavefunction is proportional to the charge density of the electron as a field in space

ρ(r)=−eψ2(r). (3.117)

The total charge of an electron is −e. Therefore, the wavefunction is nor- malized over the entire space, ψ|ψ ≡ ψ2(r) d3r =1. (3.118)

3.5.2 Orthogonality Wavefunctions with different energy eigenvalues are orthogonal. For two wavefunctions with different energy eigenvalues E1 = E2, the Schr¨odinger equations are Hˆ |ψ1 = E1|ψ1 (3.119) and Hˆ |ψ2 = E2|ψ2 . (3.120)

Multiply Eq. 3.119 by ψ2| and multiply Eq. 3.120 by ψ1|, the difference is

ψ1|Hˆ |ψ2 − ψ2|Hˆ |ψ1 =(E2 − E1) ψ1|ψ2 . (3.121)

By definition, the right-hand side of Eq. 3.121 is zero. Therefore, if E2 − E1 = 0, one must have ψ1|ψ2 =0. (3.122)

3.5.3 Completeness If the eigenfunctions of a Hamiltonian are

Hˆ |ψn = En|ψn ,n=1, 2, ...∞, (3.123)

For an arbitrary function |χ , define a coefficient

cn = χ|ψn ,n=1, 2, ...∞, (3.124) then the set of eigenfunctions is complete means ∞ |χ = cn|ψn . (3.125) n=0 For the three examples in this Chapter, the series of eigenfunctions are complete. If this is true, then an arbitrary function can be expended into a series of eigenfunctions from that Hamiltonian. 98 Schr¨odinger’s Equation I

3.5.4 Charge density distributions For a single electron wavefunction, according to Schr¨odinger, the charge density is proportional to the square of the wavefunction,

ρ(r)=−eψ2(r). (3.126)

If the electronic states are degenerate, a linear superposition of wavefunc- tions of the same energy eigenvalues is also a good wavefunction for that energy eigenvalue. The square of the new wavefunction also represents elec- tron charge density. Here is an example. The 2px-state and 2py-state are degenerate. The wavefunctions from Table 3.2 are

1 −r/2 ψ2px = √ e r sin θ cos φ (3.127) 4 2π and 1 −r/2 ψ2py = √ e r sin θ sin φ. (3.128) 4 2π The charge density distributions are e ρ = − e−r r2 sin2 θ cos2 φ (3.129) 2px 32π and e ρ = − e−r r2 sin2 θ sin2 φ, (3.130) 2py 32π as shown in Fig. 3.14(1) and (2). Because any linear combination of the wavefunctions Eq. 3.127 and 3.128 is also a good wavefunction, by rotating the coordinate system 45◦,the following wavefunctions are legitimate: 1 −r/2 π ψ2p1 = √ e r sin θ cos φ − (3.131) 4 2π 4 and 1 −r/2 π ψ2p2 = √ e r sin θ sin φ − . (3.132) 4 2π 4

Fig. 3.14. Charge density of the 2p states. (1) and (2), charge density distributions of the 2px- and 2py-states. (3) and (4), the 2p-states rotated around the z-axis by 45◦. (5) The sum of (1) and (2), or the sum of (3) and (4), represents the physical reality. 3.5 General properties of wavefunctions 99

The charge density distributions are e π ρ = − e−r r2 sin2 θ cos2 φ − (3.133) 2p1 32π 4 and e π ρ = − e−r r2 sin2 θ sin2 φ − , (3.134) 2p2 32π 4 as shown in Fig. 3.14(3) and (4). Therefore, the electrical charge density of the individual states depends on the choice of coordinate system. The sum of Eqs. 3.129 and 3.130 is coordinate-independent. In a plane z =0,itis e ρ = − e−r r2, (3.135) 2p 32π and identical to the sum of Eqs. 3.133 and 3.134. The charge density of an individual electronic state may not represent physical reality, but the sum of electrical charges of all states ρ(r)=−e ψ2(r) (3.136) is invariant under a coordinate rotation. Experimental observation is on the total density distribution of electrons. For example, in the Hartree- Fock approximation of the the sodium atom, the outer valence electron is moving in the field of the six 2p states. Only the collective effect of the six 2p states, forming a spherical symmetric electrical charge, is effective. This observation hints to a very general and powerful theorem in quan- tum mechanics: the Hohenberg-Kohn theorem, that at least for the ground state, the total charge density distribution function of the entire system contains complete information about the system, as we shell present in Chapter 4. Nevertheless, the individual wavefunctions, also called orbitals, are necessary and indispensible in the computation and understanding of the atomic system. Take an analogy, although the mechanical phenomena is independent of the choice of coordinate system, and invariant under a rota- tion of the coordinate system, to study and calculate a mechanical process, a specific coordinate system is a necessity. 100 Schr¨odinger’s Equation I Chapter 4 Many-Electron Systems

Motivated by the initial success of the Schr¨odinger equation, in 1929, Paul Dirac made the following statement in a paper entitled Quantum Mechanics of Many-Electron Systems:

The general theory of quantum mechanics is now almost com- plete, the imperfections that still remain being in connection with the exact fitting in of the theory with relativity ideas. These give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reac- tions, in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei.

Only for two real chemical systems, the Schr¨odinger equation have an- + alytic solutions: the hydrogen atom and the hydrogen molecular ion, H2 . Both are single electron system. For atoms and molecules with two or more electrons, approximations are required. Over the last century, many effec- tive approximate methods have been developed, culminated with two Nobel Prizes in Chemistry to John Pople and Walter Kohn in 1998. It is still an active research field. New methods and improvements of existing meth- ods are intensively pursued. In this Chapter, the principles and some most important approximate methods are presented.

4.1 Many-electron Schr¨odinger equation

Following the argument leading to the single-electron Schr¨odinger’s equation in Chapter 3, a many-electron wave equation can be introduced from the classical energy integral and the de Broglie relation. Because the mass of an nucleus is thousands of times greater than the mass of an electron, in almost all computations, the coordinates of the nuclei are treated as fixed input parameters. The accuracy of such approximation was analyzed mathematically by Born and Oppenheimer in 1927. Now, one can take 102 Many-Electron Systems that Born-Oppenheimer approximation as granted. For a system with M electrons with coordinates rj in a potential field formed by N nuclei, the classical energy integral, similar to that for a single electron, Eq. 3.3, is j=M p2 i=M e2 j + v(r ) + = E. (4.1) 2m j |r − r | j=1 e i>j>0 i j where the external potential v(rj) is the total attractive potential of all nuclei on the j-th electron at position rj,

l=N 2 Zl e v(rj)=− , (4.2) |rj − Rl| l=1 where Rl is the position of the l-th nucleus with atomic number Zl.The second sum in Eq. 4.1 is the repulsive potential between pairs of electrons, and the condition i>jis to avoid double counting. The wavefunction as a function of the positions of the electrons is

ψ = ψ(r1, r2, ...,rM ). (4.3) The wave vector associated to the j-th electron is 1 k2 = − ∇2ψ, (4.4) j ψ j where the Laplacian for the j-th electron is ∂2 ∂2 ∂2 ∇2 j = 2 + 2 + 2 . (4.5) ∂xj ∂yj ∂zj Using the de Broglie relation, 2 2 2 pj = kj , (4.6) we obtain the Schr¨odinger equation for the wavefunction ψ, j=M 2∇2 i=M e2 − j + v(r ) ψ + ψ = Eψ. (4.7) 2m j |r − r | j=1 e i>j>0 i j By using atomic units, see Table 4.1,

 = e = me =1, (4.8) Equation 4.7 is simplified to

j=M 1 i=M 1 − ∇2 + v(r ) ψ + ψ = Eψ, (4.9) 2 j j r j=1 i>j>0 ij 4.2 The Hartree-Fock method 103

Table 4.1: Atomic units Quantity Name Notation Value in SI −31 Mass electron mass me = 1 9.109 ×10 kg Charge electron charge |e| = 1 1.602 ×10−19 C Action Dirac constant  = 1 1.0546 ×10−34 J·s 2 2 Length Bohr radius a0  /mee 0.05291 nm 4 2 Energy hartree mee / 27.211 eV

where rij = |ri − rj|, and Eq. 4.2 becomes

l=N Zl v(rj)=− . (4.10) |rj − Rl| l=1

4.2 The Hartree-Fock method

Equation 4.9 is complicated. In 1928, Douglas Hartree invented a method to resolve Eq. 4.9 by writing the wavefunction of M electrons as a product of M single-electron wavefunctions,

ψ(r1, r2, ...,rM )=ψ1(r1) ψ2(r2) ... ψM (rM ). (4.11)

Equation 4.9 is then decomposed into M differential equations. The indi- vidual wavefunctions, or orbitals, are orthogonal and normalized, 3 ψi(r)ψj(r) d r = δij. (4.12)

4.2.1 The self-consistent field Take an example of helium atom, where M =2,N =1,andZ = 2, and set the nucleus at the origin of the coordinate system, R =0, 1 2 1 2 1 − ∇2 − − ∇2 − − 1 2 + E ψ1(r1) ψ2(r2)=0, (4.13) 2 r1 2 r2 r12 where E is the energy eigenvalue to be determined. Multiply Eq. 4.13 by ψ2(r2) and integrate over the space, using Eq. 4.12, one finds 1 2 − ∇2 − − 1 + vee(r1) E1 ψ1(r1)=0, (4.14) 2 r1 104 Many-Electron Systems where 1 2 3 ∇2 E1 = E + d r2 ψ2(r2) 2 + ψ2(r2) (4.15) 2 r2 is a new energy eigenvalue for electron 1 to be determined. The integration in Eq. 4.15 is obviously independent of the variable r1. The electron-electron repulsion potential in Eq. 4.14 is | |2 3 ψ2(r2) vee(r1)= d r2 . (4.16) r12

Because the electric charge is defined as |e| = 1, the square of the wavefunc- tion of electron 2 is its charge density,

2 ρ2(r2)=|ψ2(r2)| , (4.17) the electron repulsion potential in Eq. 4.14 is then 3 ρ2(r2) vee(r1)= d r2 . (4.18) r12 Equation 4.14 has an intuitive explanation. Electron 1 is moving in the potential field of the nucleus and the charge density of electron 2. Similarly, a differential equation for electron 2 can be derived, 1 2 − ∇2 − − 2 + vee(r2) E2 ψ2(r2)=0, (4.19) 2 r2 where vee(r2) is the electron-electron repulsion potential on electron 2 gen- 2 erated by the charge density of electron 1, ρ1(r1)=|ψ1(r1)| . Equations 4.14 and 4.19 are the one-electron equations for the two-electron system. It can be extended to systems of more than two electrons. The two equations are correlated to each other. To solve the equation for electron 1, the orbital of electron 2 is required. Those equations are always resolved using iteration: By first guess a set of initial orbitals, do the computation, then used the results of the first round of computation as the basis for the second round, and so on, until the results converge. This method, invented by Hartree, is termed self-consistent field.

4.2.2 Pauli exclusion principle and Slater determinants The Hartree self-consistent field method presented in the previous section is in many cases inaccurate because the basic statistical property of the electron is not implemented. Even for simply explaining the stability of the atomic system, the Pauli exclusion principle is necessary, that no two electrons can occupy the same quantum state. The standard representation 4.2 The Hartree-Fock method 105 of the exclusion principle, as proposed by John Slater in 1930, is to write the wavefunction as a determinant. For a two-electron system, 1 ψ1(r1) ψ1(r2) 1 ψ = √ = √ [ψ1(r1) ψ2(r2) − ψ1(r2) ψ2(r1)] . (4.20) 2 ψ2(r1) ψ2(r2) 2

By exchanging any two rows or any two columns, the wavefunction changes sign. Therefore, if any two rows are identical, the Slater determinant be- comes zero. The square root of 2 is added for normalization.

4.2.3 The electron spin

As we presented in Section 2.4, each electron wavefunction can have two spin states. Helium is an interesting example to show the effect of spin, see Fig. 4.1. The two electrons can have opposite spin states, one spin up and one spin down, to form a state called parahelium. The two electrons can also have the same spin state, both spin up or both spin down, to form a state called orthohelium. Those types of states cannot make a transition to the other one by optical means. For parahelium, the spin states of the two electrons are different. The spatial wavefunction can be identical. The loweststateis1s, similar to the ground state of hydrogen. The energy level is -24.47 eV, almost twice as that of hydrogen, as expected.

Fig. 4.1. Energy diagram of helium atom. For parahelium, the spins of the two electrons are antiparallel, and the spatial wavefunction can be identical. The orbitals of the lowest are 1s, with no nodes. For orthohelium, the spin states are parallel. One of the spatial orbitals must have a node. The energy level is higher. 106 Many-Electron Systems

4.2.4 Exchange interaction For orthohelium, the wavefunction of the entire atom is a Slater determi- nant. The Schr¨odinger equation is 1 2 1 2 1 − ∇2 − − ∇2 − − ψ1(r1) ψ1(r2) 1 2 + E =0. (4.21) 2 r1 2 r2 r12 ψ2(r1) ψ2(r2) Following the same procedure leading to Eq. 4.14, by multiplying the equa- tion with ψ2(r2) then integrate over r2, one finds 1 2 − ∇2 − − 1 + vee(r1) E1 ψ1(r1)=vxc(r1) ψ2(r1), (4.22) 2 r1 where 3 ψ2(r2)ψ1(r1) vxc(r1)= d r2 (4.23) r12 is the exchange potential that acting to the second orbital. Similarly, 1 2 − ∇2 − − 2 + vee(r2) E2 ψ2(r2)=vxc(r2) ψ2(r2), (4.24) 2 r2 where the exchange interaction potential is 3 ψ1(r1)ψ2(r2) vxc(r2)= d r1 . (4.25) r12 With the exchange potentials, Eqs. 4.23 and 4.25, the accuracy is greatly improved, while the computation is much more demanding. The above equa- tions, often called the Hartree-Fock method, is widely used in the quantum- mechanical computations of atoms and small molecules.

Fig. 4.2. Observed spectrum of helium atom. The most intensive spectral lines in the visible range are shown. Wavelengths are in nanometers. 4.3 The atoms 107

The two orbitals in Eq. 4.21, ψ1 and ψ2, must be orthogonal. If one of them is an 1s-state, the other one must have a node. Such an orbital with lowest energy is 2s, see Fig. 3.10. With a node, the energy level is much higher, see Fig. 4.1. Although the lowest state of orthohelium is about 20 eV higher than the lowest state of parahelium, no optical process can make a transition to a state on parahelium. The ground state of orthohelium can stay for a long time. It is called a metastable state. The observed spectrum of helium is shown in Fig. 4.2. Seven strongest spectral lines in the visible region are shown. For parahelium, the spec- tral lines to or from the ground state are in deep ultraviolet region. For orthohelium, the lines are in visible and infrared region.

4.3 The atoms

The Hartree-Fock method has been successfully applied to compute the wavefunctions of atoms. The results are listed in the open literature, in the form of expansions in terms of atomic basis functions.

4.3.1 Atomic basis functions The hydrogen radial wavefunctions are products of a polynomial of r and an exponential function, see Table 3.2. For practical computations, those functional forms are still too complicated. In 1930, John Slater defined a set of functions as simplified versions of the hydrogen wavefunction, and then widely used in numerical computations involving atoms and molecules. The full wavefunction is a radial function times a spherical harmonics, see Table 1.3. The radial function is frequently expressed as a sum of Slater-type orbitals, abbreviated as STO, defined as

n−1 −ζr Sn(r)=Nn r e , (4.26) where n is the principle quantum number. The normalization constant is

1 (2ζ)(n+ 2 ) Nn = . (4.27) (2n)! The explicit formulas for n =1throughn = 3 are (2ζ)3 S (r)= e−ζr, (4.28) 1 2! (2ζ)5 S (r)= re−ζr, (4.29) 2 4! and (2ζ)7 S (r)= r2e−ζr, (4.30) 3 6! 108 Many-Electron Systems

The Slater-type orbitals are quite similar to the authentic hydrogen wavefunctions, and are widely used to represent atomic orbitals. Neverthe- less, the integrals such as in Eqs. 4.16 and 4.23 must be evaluated numer- ically. Another type of atomic base functions, the Gaussian-type orbitals, abbreviated as GTO, are more frequently used in calculation of molecules. The radial functions are defined as 2(4α)n 2α 2 G (r)= rn−1e−αr . (4.31) n (2n − 1)!! π

The explicit formulas for n =1throughn = 3 are

1 3 4 8α 2 G (r)=2 e−αr , (4.32) 1 π

1 5 4 8α 2 G (r)=4 re−αr , (4.33) 2 9π and 1 7 4 8α 2 G (r)=8 r2 e−αr . (4.34) 3 225π A significant advantage of the GTO is, all integrals involved have analytic expressions. Numerical computations are greatly simplified. Charge density plots of those atomic basis orbitals are shown in Fig. 4.3. To make a fair comparison, the radius dependence of the electron density

Fig. 4.3. Electron density distributions of several basis orbitals. For STOs, the parameter ζ is 2.0. For GTOs, the parameter α is 1.0. All electron density distributions are normalized to r, such that the area below each curve is 1. 4.3 The atoms 109

ρ(r) from a basis function f(r) is expressed as

ρ(r)dr = r2|f(r)|2dr, (4.35) such that the density distribution is normalized in radius r, ∞ ρ(r)dr =1. (4.36) 0 The STOs are very close to the hydrogen wavefunctions, and the tails extend more to the space. the GTOs are cut off sharply for large r.There- fore, to accurately represent an atomic wavefunction, a larger number of GTOs are required. A complete orbital is a product of a radial factor and a spherical har- monics. Examples of GTOs in terms of Cartesian coordinates are,

3 2α 4 2 g (α, r)= e−αr , (4.37) s π 1 5 4 128α 2 g (α, r)= ze−αr , (4.38) z π3 and 1 7 4 2048α 2 g (α, r)= xy e−αr . (4.39) xy π3 For more details on GTOs, see the monograph AB Initio Molecular Orbital Theory coauthored by John Pople, and many more recent books.

4.3.2 The Roothaan-Hartree-Fock method In 1951, Clemens Roothaan invented a method to compute atomic orbitals by expanding each one as a sum of STOs, see Eq. 4.26, Rnl(r)= CjlnSjl (4.40) j An atomic orbital with principle quantum number n and azimuthal quantum number l is a sum of several STOs with expansion coefficients Cjln. Each STO is described by an integer njl and an exponent ζjl, which determine the normalization constant Njl, see Eqs. 4.26, and 4.27,

njl−1 −ζjlr Sjl = Njl r e . (4.41)

The coefficients Cjln and the exponents ζjl are then computed using the Hartree-Fock equations with a self-consistent procedure, similar to Eqs. 4.22 and 4.24. A widely used table of the Roothaan-Hartree-Fock atomic wave- functions was published in 1993, and available online.1

1See www.ccl.net/cca/data/atomic-RHF-wavefunctions/tables.html. 110 Many-Electron Systems

Fig. 4.4. Data for the wavefunctions of lithium. The data set is adapted from open literature, see Section 4.3.2. Each item is annotated for easy understanding.

The published data for atomic wavefunctions is the starting point of un- derstanding their chemical property, and the ab-initio quantum-mechanical computations. Here we show two examples of lithium and carbon.

4.3.3 Lithium Figure 4.4 shows part of the RHF-wavefunction data table for lithium, an- notated for easy understanding. Each atomic orbital is a sum of seven Slater-type orbitals. The electron density distributions as a function of distance, according to Eq. 4.35, are shown in Fig. 4.5. As shown, the electron density of the 1s state of the lithium atom has a small radius, almost one half of that of the ground-state hydrogen. The valence electron 2s is loosely attached to the core. With a small ionization energy, the lithium atom can easily loose an electron to become a positive ion with a very small radius, ideal for rechargeable battery.

Fig. 4.5. Electron density distributions of lithium. The electron density distri- butions defined in Eq. 4.35 as a function of r are shown. Being normalized to r, the area below each curve is 1. Not that the 1s electrons are tightly bounded to the nucleus. It can easily loose a 2s electron to become a positive ion with tiny radius. 4.3 The atoms 111

Fig. 4.6. Data for the wavefunctions of carbon. The data set is adapted from open literature, see Section 4.3.2. Each item is annotated for easy understanding.

4.3.4 Carbon Figure 4.6 shows part of the RHF-wavefunction data table for carbon, an- notated for easy understanding. Each carbon atom has two 1s orbitals, two 2s orbitals, and two 2p orbitals. As shown, the two 2s orbitals and the two 2p orbitals have similar ra- dius and energy level. Linear combinations of those orbitals could make four hybridized orbitals. It is the basis of organic chemistry and molecular biology, and the strongest chemical bond in diamond. See Chapter 5. The accuracy of those numerical computations of atoms can be tested directly with experimental observations. First, the energy eigenvalues of each orbitals, including the deepest ones, are the basis of X-ray spectroscopy. Second, using electron diffraction or X-ray diffraction, the charge density distribution in atoms can be measured directly. Both are well-verified. The total electron density distribution function is also the conceptual basis of density functional theory, see Section 4.4.

Fig. 4.7. Electron density distributions of carbon. The electron density distri- butions defined in Eq. 4.35 as a function of r are shown. Being normalized to r, that the area below each curve is 1. The two 2s orbitals and the two 2p orbitals have similar radius and similar energy level. The linear combinations of those orbitals could make four hybirdized orbitals. It is the basis of organic chemistry and molecular biology, and the strongest chemical bond to form diamond. See Chapter 5. 112 Many-Electron Systems

Fig. 4.8. Electron density distribution of argon, theory and experiment. The electron density distribution is defined by Eq. 4.35. The red curve is from electron diffraction experiments, and the dashed curve is from Hartree-Fock computation.

4.3.5 Accuracy of self-consistent computations Because the electron density distribution in atoms is an experimentally measurable quantity, the accuracy of the Hartree-Fock computations can be checked objectively. Figure 4.8 shows a comparison of the electron charge density obtained by electron diffraction experiments and the result of Hartree-Fock computation.2 The red curve in Fig. 4.8 is the electron distribution measured by elec- tron diffraction experiments. The dashed curve is the result of numerical quantum-mechanical computation. The agreement is satisfactory. The to- tal electron density distribution is also the basis of the density functional theory for numerical computations, see the following section.

4.4 Density functional theory

In Section 3.5.4, we show that because the wavefunctions are subject to superposition, the charge density of individual orbitals is not an intrinsic property of the system. However, the sum of the charge density distributions of wavefunctions at the same energy level is an objective reality. From Section 4.3.5, the total electron density distribution n(r) is a measurable quantity, defined as the sum of squares of all single-electron orbitals,

j=M 2 n(r)= |ψj(r)| . (4.42) j=1

2See L. S. Bartell and L. O. Brockway, The Investigation of Electron Distribution in Atoms by Electron Diffraction, Physical Review, 90, 833 (1953). 4.4 Density functional theory 113

The importance of the total electron charge density is apparent by looking at the many-body Schr¨odinger equation, Eq. 4.9. Using Dirac notation, by multiply Eq. 4.9 with ψ| then integrate over the space, one obtains

j=M j=M 1 i=M 1 ψ| − ∇2 |ψ + ψ| v(r )|ψ + ψ| |ψ = E. (4.43) 2 j j r j=1 j=1 i>j>0 ij

Here the first term is the kinetic energy of the electrons,

j=M 1 T = ψ| − ∇2 |ψ . (4.44) 2 j j=1

The last term is the mutual repulsion energy of the electrons,

i=M 1 U = ψ| |ψ . (4.45) r i>j>0 ij

Those two terms are universal to any electron system. The second term in Eq. 4.43 depends on the external potential that defines the specific nature of the system. By writing the many-body wavefunction as a product of single-electron orbitals, see Eq. 4.11, one obtains

j=M 3 V = ψ| v(rj)|ψ = n(r)v(r)d r, (4.46) j=1 where the external potential is the attactive potnetial from all nuclei,

l=N Z v(r)=− l . (4.47) |r − Rl| l=1 The term V contains all information about the specifics of the system. It has a classical meaning in terms of electrostatics in Maxwell’s electro- magnetism. And the electron density function n(r) is an experimentally measurable field quantity, for example by electron diffraction. For a given external potential v(r), following the Schr¨odinger equation, the ground-state electron charge density n(r) is uniquely defined.

4.4.1 The Hohenberg-Kohn theorem In 1964, Hohenberg and Kohn proved a theorem with mathematical rigor that the reverse is true: At least for the ground state, the electron density distribution Eq. 4.11 uniquely defines the external potential v(r). Because the external potential v(r) contains the full information about the system 114 Many-Electron Systems under investigation, the ground-state electron electron density function n(r) contains full information about the system. The proof proceeds by reductio ad absurdum. Assume that a different external potential v(r) gives rise to the same density distribution function n(r). The ground-state energy will become higher, contradicting the defi- nition of the ground state. Therefore, there is only one external potential v(r) corresponding to a given electron distribution function n(r).3

4.4.2 The Kohn-Sham equations The Hohenberg-Kohn theorem paves the way to a new set of self-consistent single-electron wave equations, called Kohn-Sham equations, that are much simpler than the Hartree-Fock equations, and often produces better results. Those new single-electron wave equations become widely used in numerical computations in quantum chemistry and solid-state physics. Similar to Hartree and Hartree-Fock approximations, the Kohn-Sham approximation aims for a series of one-electron orbitals ϕj(r), satisfying differential equations resembling the one-electron Schr¨odinger equation, 1 − ∇2 + v (r) − E ϕ (r)=0, (4.48) 2 eff j j where the index j runs from 1 to the number of electrons in the system M. The electron density distribution n(r) is defined as

j=M 2 n(r)= |ϕj(r)| , (4.49) j=1 similar to Eq. 4.42. There are important differences. First, in each equation, there is only one unknown function, rather than more than one as in the Hartree-Fock approximation, Eqs. 4.22 and 4.24. Therefore, it is significantly simpler than the Hartree-Fock method, and similar to the original Hartree method, Eq. 4.14 and 4.19. Second, the form of the effective potential in Eq. 4.48 is much simpler than in the Hartree and Hartree-Fock approximations. It is defined as n(r) d3r v (r)=v(r)+ + v (r). (4.50) eff |r − r| xc The first term v(r) is the external potential from the nuclei, as in Eq. 4.47. The second term is the repulsive potential from the entire electron density distribution, including the electron under consideration. This is significantly simpler than the Hartree and Hartree-Fock approximation, where the repul- sive potential for each electron must be calculated individually, to exclude

3For mathematical details of the proof, see the 1998 Nobel lecture of Walter Kohn, or any monograph about the density functional theory. 4.4 Density functional theory 115 the electron under consideration. The last term is the exchange and corre- lation potential as a replacement of Eqs. 4.23 and 4.25. It is central to the Kohn-Sham method, deserving more attention. In Hartree-Fock approximation, the exchange potential term Eqs. 4.23 and 4.25 must be computed from all orbitals of the entire system, and acting on other orbitals. It is the most time-consuming computation. In the Kohn- Sham method, the exchange potential is simplified to an effective potential acting only on the electron under consideration. A further simplication is based on the Hohenberg-Kohn theorem, that the electron density distribution function n(r) contains all information on the system, and certainly contains the information about the exchange and correlation potential. Moreover, from a physical point of view, only the values of the electron density distribution function near the location of the electron under consideration is important. This physical consideration is theoriginofthelocal density approximation and generalized gradient ap- proximation widely used in practical computations, see following Sections.

4.4.3 Local density approximation In the local density approximation, abbreviated LDC, the exchange and correlation interaction vxc(r) is assumed to be a universal function of n(r) at the position of the electron. There are many different functional depen- dence in analytic forms or tabulated data, either based on certain theoretical arguments, or determined by semi-empirical trial-and-error methods. As a classical example, the original expression used by Walter Kohn is a sum of the exchange energy and the correlation energy,

vxc(r)=vx(r)+vc(r), (4.51) where the exchange energy is 0.458 vx(r)=− , (4.52) rs the correlation energy is 0.44 vc(r)=− , (4.53) rs +7.8 and the radius of a sphere containing one electron, rs, is defined as 1 4πr3 = s . (4.54) n(r) 3

4.4.4 Generalized gradient approximation Within the spirit of density functional theory, and the physical intuition that the exchange and correlation interaction mainly depends on the values of 116 Many-Electron Systems electron density distribution function near the position of the electron under consideration, an improvement to the LDC is proposed and practiced. By assuming that the exchange and correlation interaction depends both the local value and the average gradient of the electron density distribution,

vxc(r)=f(n(r), |∇n(r)|), (4.55) the agreement with experimental observations can be improved. The method is abbreviated as GGA. The explicit form of the function in Eq. 4.55 is still under intensive research. Chapter 5 The Chemical Bond

Thre are four types of forces between atoms and molecules. The van der Waals force is a ubiquitous, long-ranged but weak force. It can be treated by a stationary perturbation method, and will be discussed in Chapter 7. Some atoms can be ionized, such as Na to become Na+ and Cl to become Cl−, then form ionic molecules and crystals by electrostatic forces, such as rock salt, NaCl. At very short distances, the core-core repulsion due to Pauli exclusion dominates, which makes an ultimate limit of how two atoms can apporach each other. Electrostatic attraction is much stronger than the van der Waals force. However, it cannot explain a basic fact in chemistry: two identical atoms can form a molecule with a bond often much stronger than the ionic bond, such as the nitrogen molecule and the oxygen molecule. The strongest bonds are formed between identical atoms, for example in diamond. Such type of bond is called a covalent bond,orchemical bond, which is the central concept of modern chemistry. Comparing with van der Waals bonds and ionic bonds, covalent bonds have the following features. First, it is short-ranged. Both van der waals and ionic forces follow inverse power laws. The covalent force follows an exponantial law. Second, as van der Waals force, ionic force and Pauli repulsion are omnidirectional, covalent bonds are often directional. Third, the covalent bond can be both attractive and repulsive. The two types of covalent bonds can cancel each other, and become saturated. + The simplest covalent bond is the bond in hydrogen molecular ion, H2 . It is one of the two real systems for which analytic solutions of Schr¨odinger’s equation exist. The mathematics of the analytic solution is quite compli- cated. To achieve a conceptual understanding, simpler analytic models are preferred. The validity of the simple model of the hydrogen molecular ion can be tested against the analytic solution . After such a validation, the simple analytic model can then be applied to more complicated cases.

5.1 The concept of chemical bond

When two atoms come together, the atomic wavefunctions, or the atomic orbitals, abbreviated AO, interact with each other. A molecule is formed. Beause the Schr¨odinger equation is linear, the wavefunctions of the molecule, 118 The Chemical Bond or molecular orbitals, abbreviated MO, are approximately linear combina- tions of atomic orbitals. The process of superposition give rise to a change of energy level, which is the origin of chemical bond energy.

5.1.1 Formation of molecular orbitals The origin of chemical bond can be understood in terms of linear superpo- sition of wavefunctions. Figure 5.1 (A) and (B) show two atoms, each has a wavefunction, or atomic orbital (AO), at the same energy E0. When the two atoms come together to become a diatomic molecule, molecular wave- functions, or molecular orbitals (MO), are formed. Because the Schr¨odinger equation is linear, a molecular orbital can be a linear combination of atomic orbitals (LCAO), see Fig. 5.1(C) and (D). The term orbital refers to the wavefunction of a single electron state. Because the overall sign of a wave- function is irrelevant, to facilitate intuitive understanding, the sign of atomic wavefunctions near the interface is chosen to be positive. There are two types of molecular orbitals. The molecular orbital in Fig. 5.1(C) is the sum of the atomic orbitals, which is called a bonding molecular orbital ψb. In the interface region, the amplitude of the molecular orbital is the sum of the atomic orbitals. The productive linear superposi- tion of atomic wavefunctions lowers the energy level from the atomic orbital E0 to the energy level of the molecule to Eb. It gains a bonding energy,

ΔE = E0 − Eb. (5.1)

In Fig. 5.1(D), the molecular orbital is the difference of the two atomic or-

Fig. 5.1. Concept of chemical bond. When two atoms (A) and (B) approach each other to become a diatomic molecule, the molecular wavefunctions are linear superposi- tions of atomic wavefunctions. Two types of superpositions are possible. The symmetric superposition, (C), makes a bonding wavefunction,orabonding molecular orbital.The antisymmetric superposotion, (D), makes an antibonding wavefunction,oranantibond- ing molecular orbital. The bonding orbital does not have any additional node, and its energy is lower than the energy of individual atomic orbitals. The antibonding orbital creates an additional node, and its energy is higher. 5.1 The concept of chemical bond 119

bitals, which is called an antibonding molecular orbital ψa. The molecular wavefunction changes sign near the interface. The surface where the am- plitude of wavefunction vanishes is called a nodal surface, a nodal plane, or simply a node. Due to the destructive linear superposition, the energy level of the molecule Ea is higher than that of the atoms. The difference, an antibonding energy, approximately equals the bonding energy:

ΔE = Ea − E0. (5.2) Assuming that the overlap of the atomic orbitals is small, a simple and general expression of the bonding energy in terms of the atomic wavefunc- tions near the seperation surface x = 0 can be derived. We begin with the normalized wavefunctions of the atoms, 2 3 2 3 ψ1 d r = ψ2 d r =1. (5.3)

The condition of small overlap means over the entire space, 3 ψ1ψ2 d r ≈ 0. (5.4)

5.1.2 Bonding energy as a surface integral The chemical bond energy can be expressed as a surface integral of the atomic wavefunctions near the separation suface, see Fig. 5.1. The idea was proposed independently by Conyers Herring and Lev Landau in 1961.1 Here is a simple and transparent derivation, see Fig. 5.1. The atomic wavefunction of atom 1 satisfies the Schr¨odinger equation 2 − ∇2 + U ψ = E ψ , (5.5) 2m 1 1 0 1 and the atomic wavefunction of atom 2 satisfies 2 − ∇2 + U ψ = E ψ . (5.6) 2m 2 2 0 2 In the molecule, the bonding wavefunction is 1 ψb = √ (ψ1 + ψ2) . (5.7) 2 Because the overlap is small, as shown in Eqs. 5.3 and 5.4, the molecular wavefunction is approximately normalized: 1 ψ2 d3r = ψ2 d3r +2 ψ ψ d3r + ψ2 d3r ≈ 1. (5.8) b 2 1 1 2 2

1See pages 314-316 of Landau and Lifshitz, Quantum Mechanics, Third edition, re- vised and enlarged, Butteworth Heinemann 1977. 120 The Chemical Bond

It satisfies the Schr¨odinger equation of the molecule, Fig. 5.1(C): 2 − ∇2 + U + U ψ = E ψ . (5.9) 2m 1 2 b b b

Multiply Eq. 5.5 by ψb, integrate over the left region, one obtains 2 2 3 3 ψb − ∇ + U1 ψ1 d r = E0 ψbψ1d r. (5.10) x<0 2m x<0

Multiply Eq. 5.9 by ψ1, integrate over the left region, one obtains 2 2 3 3 ψ1 − ∇ + U1 + U2 ψb d r = Eb ψbψ1d r. (5.11) x<0 2m x<0

Up to this step, everything is exact. To proceed further, two approxi- mations are applied, both can be justified by looking at Fig. 5.1. First, the integrals in the right side of Eqs. 5.10 and 5.11 are 1 1 1 3 √ 2 3 √ 3 ≈ √ ψbψ1d r = ψ1d r + ψ1ψ2d r . (5.12) x<0 2 x<0 2 x<0 2

Second, a term with U2 in Eq. 5.11 is a higher-order infinitesimal, 3 ψ1U2ψb d r ≈ 0. (5.13) x<0 Subtract 5.11 from Eq. 5.10, applying Eqs. 5.12 and 5.13, one obtains 2 1 1 2 2 3 √ [E0 − Eb]= √ ψ2∇ ψ1 − ψ1∇ ψ2 d r. (5.14) 2 2m x<0 2 √ Eliminating the factor 2 and using the identity

2 2 ψ2∇ ψ1 − ψ1∇ ψ2 = ∇·[ψ2∇ψ1 − ψ1∇ψ2] , (5.15) the volume integral can be converted to a surface integral, 2 ΔE = E0 − Eb = [ψ1∇ψ2 − ψ2∇ψ1] · dS. (5.16) 2m x=0 Here the integral is carried out only on the separation surface S, namely x = 0, because on the farther borders of the system, both ψ1 and ψ2 vanish. The antibonding wavefunction is

1 ψa = √ (ψ1 − ψ2) . (5.17) 2 5.2 The hydrogen molecular ion 121

It satiafies 2 − ∇2 + U + U ψ = E ψ . (5.18) 2m 1 2 a a a

Here Ea is the energy eigenvalue of the antiobonding wavefunction ψa.By applying the steps of Eqs. 5.11 to 5.16 on Eqs. 5.6 and 5.18 in the region x>0, similar expressions can be obtained for antibonding case 2 ΔE = Ea − E0 = [ψ1∇ψ2 − ψ2∇ψ1] · dS. (5.19) 2m x=0

+ By applying those formulas to the hydrogen molecular ion H2 ,very simple expressions are obtained. By comparing the simple approximate + expression with the accurate analytic solution of the H2 problem, it shows that the first two terms of the asymptotic expressions are reproduced.

5.2 The hydrogen molecular ion

The hydrogen molecular ion is the simplest molecule. It is also one of two real problems in nature for which exact analytic solutions of Schr¨odinger equation exist. It is no surprise that Slater considered it the cornerstone for the understanding of interatomic forces and even all of condensed-matter physics. Throughout the 20th century, the hydrogen molecular ion problem attracted the attention of many prominent physicists and chemists. Using the old quantum theory of Bohr and Sommerfeld, Wolfgang Pauli did his thesis on the hydrogen molecular ion problem. He concluded that within the framework of the Bohr–Sommerfeld quantum theory, no stable states + can be explained. Yet a stable H2 with a binding energy of 2.7 eV was observed experimentally. After the discovery of the Schr¨odinger equation + in 1926, the H2 problem became a perfect test ground for the validity of quantum mechanics. In 1927, Burrau showed that the Schr¨odinger equation + for the H2 problem is separable. Using numerical integration, he predicted a bonding energy consistent with experimental observations. Eduard Teller + did his thesis on the systematic analytic solution of the H2 problem (not the hydrogen bomb!) and worked out the potential curves for many excited states. This problem is still of considerable current interest, as evidenced by continuing publications up to recent years. The dependence of potential energy on the proton-proton distance is shown in Fig. 5.2. At large distances, especially when R>0.6nm,the system can be considered as a neutral hydrogen atom and a free proton. The electrical field of the proton polarizes the neutral hydrogen atom, and induces a dipole. The attractive force between the proton and the in- duced dipole forms a van der Waals force. As the proton-proton distance is reduced, the effect of wavefunction superposition arises. The bonding 122 The Chemical Bond

Fig. 5.2. Potential curve for the hydrogen molecular ion. At R>0.6 nm, the van der Waals force dominates. For 0.15 nm

wavefunction 1σg generates a strong attactive force and therefore a stable molecule. At a even shorter distance, the Pauli repulsion prevails, the net interaction energy becomes positive, and the force becomes repulsive. For the antibonding wavefunction 1σu, at large distances such as R>0.6nm, the van de Waals force dominates. When R<0.6 nm, the repulsive ef- fect of the antibonding wavefunction dominates. Again, at even shorter proton-proton distance, R<0.15 nm, Pauli repulsion force dominates. The hydrogen molecular ion problem can be treated by the perturbation method presented in Section 5.1.2. Figure 5.3 shows a schematics of the potential curve of the molecule and two componeent atoms. Figure 5.4 the wavefunctions of the bonding orbital and antibonding orbital. Similar to the general concept of chemical bond discussed in Section 5.1.2, when the atoms are isolated, each stays at the ground state 1s. The potential diagrams are shown in Fig. 5.3(b) and (c). The wavefunctions are shown in Fig. 5.4(c) and (d). By bringing the two protons together, a superposition of the atomic wavefunctions takes place. As we discussed in Section 5.1.2, there are two possible molecular wavefunctions. A sym- metric or bonding wavefunction, denoted as 1σg, shown in Fig. 5.4(a), and an antisymmetric or antibonding wavefunction, denoted as 1σu,shownin Fig. 5.4(b). The energy level is split into a bonding level 1σg, and an anti- bonding level 1σu, as shown in Fig. 5.3(a). Because the hydrogen molecular ion problem has an analytic solution, very accurate potantial curves can be derived mathematically. In the fol- lowing, we show that the perturbation method presented in Section 5.1.2 can reproduce the chemical bonding energy with high accuracy, especially in the inermediate region. 5.2 The hydrogen molecular ion 123

Fig. 5.3. Perturbation treatment of the hydrogen molecular ion. (a) The exact potential curve and the exact energy levels of the problem. (b) Solid curve, the left-hand-side potential for a perturbation treatment; dotted curve, the potential for a free hydrogen atom. (c) Solid curve, the right-hand-side potential for a perturbation treatment; dotted curve, the potential for a free hydrogen atom.

5.2.1 Van der Waals force

At large distances, the system can be considered as a neutral hydrogen atom plus a proton. The electrical field of the proton polarizes the hydrogen atom to induce a dipole. The interaction between the proton and the induced dipole generates a van der Waals force. The van der Waals force can be treated by introducing a phenomenological polarizability α:

p = αE, (5.20) where p is the induced dipole of the neutral hydrogen atom, and E is the electrical field of another proton. The coupling energy E between the proton and the neutral hydrogen atom is: 124 The Chemical Bond

Fig. 5.4. Wavefunctions of the hydrogen molecular ion. (a) The exact wave- functions of the ground-state of the hydrogen molecular ion. It is a symmetric (gerade) state. (b) The exact wavefunctions of the first excited state of the hydrogen molecular ion. It is a antisymmetric (ungerade) state. Both (a) and (b) can be approximated as a symmetric or antisymmetric linear combination of the solutions of the left-hand-side and right-hand-side problems, (c) and (d), defined by the potentials shown in Fig. 5.6(b) and Fig. 5.6(c), respectively.

α E = − . (5.21) 2R4 The polarizability of the hydrogen atom α can be calculated accurately using perturbation method, see Section 7.1.2. The result is

9 α = Ry a4, (5.22) 2 0 here 2 a = =0.0529 nm (5.23) 0 me2 is the Bohr radius. And the Rydberg energy is

e2 Ry = =13.6eV. (5.24) 2a0

Defining ρ = R/a0, the van der Waals interaction energy is 9Ry a 4 9Ry E = − 0 = − . (5.25) 2 R 2 ρ4 5.2 The hydrogen molecular ion 125

5.2.2 Evaluation of the surface integral In this subsection, we evaluate the surface integral, Eq. 5.16, to obtain an accurate analytic expression for the asymptotic potential of the hydrogen molecular ion. As shown in Fig. 5.2, the exact Schr¨odinger equation for the electron is 2 1 1 − ∇2ψ − e2 + ψ = Eψ. (5.26) 2m r r where r is the distance between the electron and second proton, r = R2 + r2 − 2Rr cos θ. (5.27) In the absence of the second proton, Eq. 5.26 is the Schr¨odinger equation for the free hydrogen atom. The ground-state wavefunction is:

− a 3/2 ψ = √0 e−r/a0 , (5.28) 0 π where a0 = 0.0529 nm. The energy eigenvalue is:

e2 E = − = −Ry ≈−13.6 eV, (5.29) 2a0 The presence of the second proton induces a perturbation to the wavefunc- tion. The perturbation can be represented by a function g as follows,

−g ψ = e ψ0. (5.30) Because the hydrogen atom wavefunction decays rapidly in x and y,itis sufficient to evaluate g on the z axis. In other words, we should determine the function g = g(z) using Eq. 5.26. To a sufficiently accurate degree of approximation, the energy eigenvalue is

Fig. 5.5. Evaluation of the correction factor. Because the hydrogen atom wavefunction decays quickly with x and y, it is sufficient to evaluate the correction function g on the z axis, with z ≈ R − z. 126 The Chemical Bond

e2 E = −Ry − , (5.31) R which is a sum of the energy eigenvalue of the ground-state hydrogen atom, the Rydberg energy plus a constant potential energy term, −e2/R. Inserting Eqs 5.30 and 5.31 into Eq. 5.26, using the expression of the Bohr radius, 2 2 a0 =  /me , we obtain the equation for g(z): dg 1 1 dg 2 d2g − + = a + . (5.32) dz R z 0 dz dz2 The two terms on the right-hand side of the equation are is much smaller than the terms on the left-hand side. The approximate equation for the correction function g(z)is: dg 1 1 = − , (5.33) dz R R − z here we used the geometrical relation z = R − z, see Fig. 5.2. At z =0, the correction function g should be independent of the sign of z.Inothre words, g(0) = 0. Eq. 5.33 can be integrated immediately to obtain z R − z g(z)= +ln . (5.34) R R We are interested only in the values of the correction function near the medium plane, z = R/2. Therefore, 1 g(R/2) = − ln 2. (5.35) 2 The wavefunction gains a factor

e−g =2e−1/2 ≈ 1.213. (5.36)

On the far side of the molecule, the wavefunction is reduced by a factor 3 e−g = e−1/2 ≈ 0.9098. (5.37) 2

Therefore, the net effect can be considered to be a small pz state, induced by proton B, superimposed on the 1s wavefunction. By evaluating the integral in Eq. 5.16 explicitly, we find

M = −4Ryρe−ρ−1. (5.38)

To simplify notation, a dimensionless radius in the unit of a0 is used, R ρ = . (5.39) a0 5.2 The hydrogen molecular ion 127

Fig. 5.6. Accuracy of the perturbation treatment of hydrogen molecular ion. The exact chemical bond energy is shown as the solid curve. The approximate values represented by Eq. 5.41 are shown as crosses. For the proton-proton distance over the range of 0.25 nm to 0.65 nm, the error is less than 2%.

The bonding energy is the sum of the van der Waals energy, Eq. 5.25, and the covalent bond energy, Eq. 5.38. For the 1σg state, it is: 9Ry ΔE(1σ )=− − 4Ryρe−ρ−1, (5.40) g 2 ρ4 and for the 1σu state, 9Ry ΔE(1σ )=− +4Ryρe−ρ−1. (5.41) u 2 ρ4

5.2.3 Compare with the exact solution Using more sophisticated methods, higher-order terms of the asymptotic expansion of the coupling energy of hydrogen molecular ion were worked out. Up to the third term, it is2 9 15 213 ΔE± =Ry − − − 2ρ4 ρ6 2ρ7 (5.42) 1 25 ±4Ryρe−ρ−1 1+ − . 2ρ 8ρ2

2See R. J. Damburg and R. K. Propin, On asymptotic expansions of electronic terms + of the molecular ion H2 , J. Phys. B. Ser. 2 1 681-691 (1968). 128 The Chemical Bond

Table 5.1: Potential curve of hydrogen molecular ion

R(nm) ΔEapp (eV) ΔEacc (eV) Diff (eV) Err (%) 0.300 0.450936 0.445845 -0.005091 -1.14 0.325 0.307579 0.306605 -0.000974 -0.32 0.350 0.209621 0.210181 0.000560 0.27 0.375 0.142937 0.143922 0.000985 0.68 0.400 0.097668 0.098638 0.000970 0.98 0.425 0.066992 0.067802 0.000810 1.19 0.450 0.046219 0.046846 0.000627 1.34 0.475 0.032144 0.032609 0.000466 1.43 0.500 0.022588 0.022926 0.000338 1.48 0.525 0.016078 0.016321 0.000243 1.49 0.550 0.011621 0.011794 0.000174 1.47 0.575 0.008547 0.008671 0.000124 1.43 0.600 0.006408 0.006497 0.000089 1.38

Table 5.1 shows the accuracy of the surface-integral approximation to the hydrogen molecular ion problem. Here, R is the distance between the protons in nanometers. ΔEapp is the bonding energy from the approximate formula, Eq. 5.40. ΔEacc is the more accurate value, from Eq. 5.40. Diff is the difference in eV, and Err is the error in percentage. As shown by a comparison with the exact solution, the accuracy of the approximate expression is well verified. As shown in Fig. 5.5 and Table 5.1, for the proton-proton distance from 0.3 nm to 0.6 nm, the error of the perturbation treatment versus the more exact solution, Eqs. 5.40, is less than 2%. Therefore, the perturbation treatment of the chemical bond energy is verified quantitatively for hydrogen molecular ion. The perturbation method presented here can be applied to covalent bonds of many-electron atoms.

5.3 Covalent bonds of many-electron atoms

In this Section, we show the perturbation method for the chemical bond based on the surface integral can be applied to systems of many-electron atoms. With transparent methematics and intuitive arguments, its predic- tions fit well to experimental observations. The starting point is the published numerical data of atomic wavefunc- tions, see Section 4.3. In order to obtain simple analytic expressions of the chemical bond energy, an approximation of the valence wavefunctions outside the atomic core is applied, see Fig. 5.7. 5.3 Covalent bonds of many-electron atoms 129

5.3.1 The black-ball model of atoms In order to apply the method of surface integration to many-electron atoms, the folllowing approximation is applied.

1. Each atom centered at R has a core of radius r0, which is slightly smaller than one half of the internuclear distance to avoid overlapping. The potential U(r) is zero outside the core,

U(r)=0, when |r − R| >r0. (5.43)

2. The potential inside the core does not need to be known.

3. The atomic wavefunctions inside the core are assumed to be the results of first-principle computations.

4. The atomic wavefunctions outside the core are expressed as products of spherical harmonics and spherical modified Bessel functions, see Section 3.2.4. The coefficients are obtained by matching with calcu- lated wavefunctions at the spherical surface of the core.

Therefore, such a method is basically an ab initio technique, with a semiemperical parameter, the radius of the atomic core r0. Because the potential inside the core does not need to be known, the core is like a black box. Since the atomic core is spherical, the term black ball is used. For atoms in the first two rows of the periodic table, the valence wave- functions are either s-type or p-type, as schematically shown in Fig. 5.8. As examples, the elements in the first row of the periodic table are explicitly treated. For the elements of primary interest, B, C, N, O, and F, all the valence electrons are of 2p-type. The wavefunctions from ab anitio compu- tations are available from data tables, see Section 4.3. Examples for these

Fig. 5.7. The black-ball approximation for the chemical bond. Similar to the muffin-tin potential approximation, outside the radius of the atomic core r0, the potential is zero, U = 0. Inside the atomic core, there is no limitation about the potential surface. The coefficients of the atomic wavefunctions outside the core is obtained by comparing with the results of first-principle computations. 130 The Chemical Bond elements are shown in Table 5.2. An atomic wavefunction is expressed as a sum of several Slater-type orbitals, defined as

nj −1 −Zj r Sj(r)=Nj r e , (5.44) where the normalization constant Nj is

1 (nj + ) (2Zj) 2 Nj = . (5.45) (2nj)!

The paramneters nj and Zj are as listed in Table 5.2. The radial part of an atomic wavefunction R(r) is a sum of Slater-type orbitals with coefficients Cj, also listed in Table 5.2,

jmax R(r)= Cj Sj(r) , (5.46) j=1 which is also normalized, ∞ R(r) r2dr =1. (5.47) 0

Table 5.2: Examples of atomic wavefunction data. The numerical coefficients in Eqs. 5.43, 5.44, and 5.46. re is the experimental internuclear distances of dimers. The rest of data are publically available, see Section 4.3, converted from atomic units to picometers and nanometers.

−1 Atom re (pm) E (eV) nj Zj(nm ) Cj C 123 -11.7918 2 133.226 0.006977 2 60.9909 0.070877 2 41.4001 0.230802 2 27.2366 0.411931 2 19.3546 0.350701 N 109 -15.4449 2 157.773 0.006323 2 73.3724 0.082938 2 48.9817 0.260147 2 32.0233 0.418361 2 22.5142 0.308272 O 121 -17.195 2 182.304 0.005626 2 81.8686 0.126618 2 51.9713 0.328966 2 33.1175 0.395422 2 23.5706 0.231788 5.3 Covalent bonds of many-electron atoms 131

For wavefunctions in three-dimensional space, a spherical harmonics as in Table 1.3 is multiplied. For example, the s wavefunction is 1 ψ (r)= R(r), (5.48) s 4π the pz wavefunction is 3 z 3 ψ (r)= R(r)= cos θR(r), (5.49) pz 4π r 4π and the px wavefunction is 3 x 3 ψ (r)= R(r)= sin θ cos ϕR(r). (5.50) px 4π r 4π

5.3.2 Wavefunctions outside the atomic core To apply the perturbation method to compute the chemical bond energy, the wavefunctions outside the atomic core are expressed by spherical mod- ified Bessel functions, see Section 3.2.4. The coefficients are obtained by wavefunction matching at r0, which is 0.475 times the internuclear distance re. The parameter κ is related to the energy eigenvalue E in Table 5.2, √ −2mE κ = . (5.51)  According to Eq. 3.43, the s-wavefunction outside the atomic core is 1 ψ (r)=C e−κr. (5.52) s s κr

Following Eq. 3.44, the pz-wavefunction outside the atomic core is 1 1 z ψ (r)=C + e−κr . (5.53) pz p κr (κr)2 r

Similarly, the wavefunctions for px-state and py-state are 1 1 x ψ (r)=C + e−κr (5.54) px p κr (κr)2 r and 1 1 y ψ (r)=C + e−κr . (5.55) py p κr (κr)2 r Thecoefficientsareobtainedbymatchingthevaluesattheinterface r = r0. For a s-state,

1 1 −κr0 Rs(r0)=Cs e . (5.56) 4π κr0 132 The Chemical Bond

Fig. 5.8. Wavefunctions outside the atomic core. The atomic core is repesented by a black sphere. The wavefunctions of valence electrons outside the atomic core are continuations of the wavefunctions inside the core from ab initio computations. (a) an s-wave orbital. (b) a pz-wave orbital. (c) a pz-wave orbital. Because the overall sign of a wavefunction is not observable, the + lobe and the − lobe can be interchanged.

For p-wavefunctions, omitting the angle dependent factor in both sides,

3 1 1 −κr0 Rp(r0)=Cp + 2 e . (5.57) 4π κr0 (κr0)

The radial functions Rs(r)andRp(r) are radial parts of the ab initio atomic wavefunctions Eq. 5.46 computed from the data table for the s-wavefunction and the p-wavefunction, respectively. The coefficients for the pz wavefunc- tion and those for the px and py wavefunctions are identical.

5.3.3 The derivative rule The advantage of expressing the vacuum tails of atomic wavefunctions with spherical modified Bessel functions as in Section 3.2.4 is that extremely simple results of the surface integrals, Eqs. 5.16 and 5.19, can be obtained. To obtain the results, we notice that because in the vacuum where U =0, the wavefunctions of the atoms, ψ1 and ψ2, satisfy the same Schr¨odinger equation. Equations. 5.5 and 5.6 now become 2 − ∇2ψ = E ψ , (5.58) 2m 1 0 1 and 2 − ∇2ψ = E ψ . (5.59) 2m 2 0 2 Therefore, over any arbitrary volume Ω in the space where U =0, 2 2 3 ψ1∇ ψ2 − ψ2∇ ψ1 d r =0. (5.60) Ω Using the identity, Eq. 5.15,

2 2 ψ2∇ ψ1 − ψ1∇ ψ2 = ∇·[ψ2∇ψ1 − ψ1∇ψ2] , (5.61) 5.3 Covalent bonds of many-electron atoms 133

The integration over an arbitrary closed surface Σ enclosing a volume Ω is also zero,

[ψ1∇ψ2 − ψ2∇ψ1] · dS =0. (5.62) Σ Therefore, the surface of integral for bonding energy, Eqs. 5.16 or 5.19, can be executed on any surface between the two atoms. Here we consider two special cases. If the wavefunction of atom 1, ψ1, is an s-wave, the surface integral can be executed on a spherical surface of radius R centered at the nucleus of atom 1. 2 2 ∂ ∂ ΔE = ± dΩ r ψ1 ψ2 − ψ2 ψ1 . (5.63) 2me r=R ∂r ∂r The integration is over the entire solid angle. The ± sign represents bonding and antibonding. Using the expression of ψ1, Eq. 5.52, one obtains 2 −κr −κr Cs re ∂ψ2 −κr e ΔE = ± dΩ + ψ2 re + . (5.64) 2me r=R κ ∂r κ The experssion should be independent of r. Because the experssion of Eq. 5.52 is the only valid function for a s-state in vacuum, Eq. 5.64 should be valid even for R → 0. The only surviving term is the second term in the parenthesis, where the wavefunction of atom 2 takes the value at the center of atom 1, r1. The integration over the solid angle is 4π. Finally,

2 2π Cs ΔE = ± ψ2(r1). (5.65) κme Here we find a very simple result: The absolute value of the chemical bond energy is proportional to the amplitude of the wavefunction of atom 2 at the center of atom 1. The overall sign of the wavefunction is not a physical variable. The value could be either positive or negative. With the ± sign, both the bonding energy and the antibonding energy are obtained. If the wavefunction of atom 1, ψ1,isap-state, a similar treatment can be performed. Take an example of a pz state. the wavefunction of atom 1 is 1 1 z ψ (r)=C + e−κr . (5.66) 1 p κr (κr)2 r By performing an integration over the entire solid angle, the +z hemisphere and the −z hemisphere cause the constant term of the wavefunction of atom 2 to cancel each other. To obtain a non-zero result, the wavefunction of atom 2 has to be expanded into a power series,

∂ψ (r ) ∂ψ (r ) ∂ψ (r ) ψ (r)=ψ (r )+ 2 1 z + 2 1 x + 2 1 y + ... , (5.67) 2 2 1 ∂z ∂x ∂y 134 The Chemical Bond where x, y,andz are the coordinates of a point with respect to the center of atom 1. Following the process of Eqs. 5.64 and 5.65, one finds that the only non-vanishing term must include z2. The chemical bond energy is

2 ±2π Cp ∂ψ2(r1) ΔE = 2 . (5.68) κ me ∂z

The result is again very simple: If atom 1 has a pz-state, then the chemi- cal bond energy is poportional to the z-derirative of the amplitude of the wavefunction of atom 2 taking at the center of atom 1. Similar results can be obtained for px and py states.

5.3.4 Types of chemical bonds Using the preturbation theory of the chemical bond based on the surface integral, especially the derivative rule, intuitive understanding can be ob- tained. As examples, the chemical bonds in dimers of the atoms in the first row of the periotic table are analysed.

Chemical bonds from s-type atomic orbitals The s-type orbitals in a pair of atoms can form a σ bond. According to Tables 3.1 and 3.2, the chemical bond energy as a function of internuclear distance r is 2π2C2 1 ΔE(r)= s e−κr, (5.69) κm κr where the constant Cs is obtained by matching with atomic wavefunctions from ab initio computations, see Eq. 5.56. There are two types of molecular orbitals, the bonding orbital sσ, and the antibonding orbital sσ∗, see Fig. 5.9. Theelectronconfigurationofhydrogenis1s1. By bringing two hydrogen atoms together, sσ molecular orbitals are formed. Because electron has two spin versions, both sσ orbitals are occupied. A stable hydrogen molecule is formed. The electron configuration of the next atom helium is 1s2. Because two helium atoms have four elecrons, two sσ orbitals and two sσ∗ orbitals are occupied. Because the binding energy of the antibonding orbital is equal but in opposite sign to that of the bonding orbital, the net chemical bond energy is zero. There is no stable He2 molecule. The electron configuration of lithium is 1s2 2s1,or[He]2s1. Therefore, it is similar to hydrogen. Because the ionization energy of Li is small, it almost always appears as an ion Li+. Nevertheless, when evaporated at high temperature in a good vacuum, a stable molecule Li2 appears in the gas phase and the spectra were detected. 5.3 Covalent bonds of many-electron atoms 135

Fig. 5.9. Molecular orbitals built from two s-type atomic orbitals. The bonding molecular orbital sσ lowers the energy to form a stable molecule. The antibonding molecular orbital sσ∗ has a node, which raises the energy.

The electron configuration of beryllium is [He] 2s2. Similar to helium, the four 2s electrons form two sσ orbitals and two sσ∗ orbitals. At a first approximation, the net bonding energy is zero. The covalent bonds in Li and Be are not of practicle interest. In the following, the five elements, B, C, N, O, and F are discussed.

Chemical bonds from p-type atomic orbitals The five atoms in the first row of the periodic table as listed in Table 5.2 have something in common: the chemical bonds are formed by 2p atomic orbitals. By defining the the z axis as the line connecting the nuclei, there are two types of chemical bonds. For pz-type atomic orbitals, the molecular orbitals are either pσ type or pσ∗ type. According the derivative rule listed in Table 3.2, the tunneling matrix element, and consequently the chemical bond energy as a function of internuclear distance r is 2 2π Cp ∂ 1 1 −κr z ΔE(r)= Cp + e κm κ∂z κr (κr)2 r z=r (5.70) 2π2C2 1 2 2 = p + + e−κr. κm κr (κr)2 (κr)3

For the pσ bonding molecular orbital, a constructive superposition lowers the energy level of the system. For the pσ∗ antibonding molecular orbital, a node makes the energy level higher. See Fig. 5.10. For px-type or py-type atomic orbitals, the molecular orbitals are either of pπ type or pπ∗ type. According the derivative rule listed in Table 3.2, the chemical bond potential function is 2 2π Cp ∂ 1 1 −κr x ΔE(r)= Cp + e κm κ∂x κr (κr)2 r x=0 (5.71) 2π2C2 1 1 = p + e−κr. κm (κr)2 (κr)3 136 The Chemical Bond

Fig. 5.10. The pσ and pσ∗ molecular orbitals. The bonding molecular orbital pσ lowers the energy to form a stable molecule. The antibonding molecular orbital pσ∗ creates a new node, which raises the energy.

Wavefunctions of pπ bonding molecular orbital and pπ∗ antibonding molec- ular orbital are shown in Fig. 5.11. Comparing Eq. 5.70 with Eq. 5.71 one finds that the bonding energy of the pσ orbital is much greater than the bonding energy of the pπ orbital, as intuitively obvious. Equations 5.69, 5.70, and 5.71 represent the chemical-bond component of the potential func- tion of the internuclear distence r. To form a complete potential function (except the weak van der Waals force at large distances) Pauli repulsion force needs to be included.

5.3.5 Comparing with experimental data

The analysic expressions of the chemical-bond energy, Eqs. 5.69, 5.70, and 5.71 can be evaluated from the results of ab initio calculations of the atomic orbitals, for example, listed in Tables. Here we show the resutls for homonu- clear diatomic molecules from five column-I elements, B, C, N, O, and F, as listed in Table 5.3 and Figure 5.12 [?].

Fig. 5.11. The pπa and pπ∗ molecular orbitals. The bonding molecular orbital pπ lowers the energy to form a stable molecule. The antibonding molecular orbital pπ∗ creates a new node, which raises the energy. 5.3 Covalent bonds of many-electron atoms 137

Boron The electron configuration of boron is 1s2 2s2 2p1. As in the case of beryl- lium, the two 2s electrons form two sσ bonding molecular orbitals and two

Table 5.3: Parameters of homonuclear diatomic molecules. A comparison between theory and experiments for homonuclear diatomic molecules. Column 1 is the electron configuration of the atoms. Column 2, κ, are calculated from the enery eigenvalue from the table through Eq. 5.51. Column 3 is the coefficient for the wavefunctions outside the atomic core in Eq. 5.57 by matching with the atomic wavefunction from ab ini- tio calculations at the border of the atomic core, r0 =0.475re.Column4 is a parameter in the Morse potential, evaluated using a least-squares fit. Column 5 is the experimental dissociation energy from the data in Huber and Herzberg. Column 6 is the theoretical value of the dissociation energy from the Morse potential fit. Column 7 is the reduced mass in atomic unit. Column 8, ωexp, is the experimental vibrational frequency. Column 9, ωthr, theoretical vibrational frequency, from Eq. 5.76.

Atom κCp βDexp Dthr μωexp ωthr − − 3 − − − config. nm 1 nm 2 nm 1 eV eV amu cm 1 cm 1 B..2p1 14.0 32.7 21.5 3.00 2.76 5.50 1051 1063 C..2p2 17.6 38.2 27.0 6.21 5.80 6.00 1961 1852 N..2p3 20.1 49.6 31.2 9.76 9.82 7.00 2358 2579 O..2p4 21.2 68.3 30.4 5.12 7.48 8.00 1580 2051 F ..2p5 22.8 97.2 29.7 1.60 4.12 9.00 916 1402

Fig. 5.12. Covalent bond energy and Morse function. (a) Dotted curve, the covalent energy detemined by Eq. 5.69. (b) Solid curve, fitted with an exponantial function, Eq. 5.74. (c) The repulsive component of the Morse function. (d) Sum of (b) and (c), the Morse curve, Eq. 5.73. 138 The Chemical Bond sσ∗ antibonding molecular orbitals. The net energy gain is zero. The sin- gle 2p atomic orbital forms a pσ bonding molecular orbital. The potential function of the bonding energy is given by Eq. 5.70, using the coefficient Cp by comparing with the ab initio atomic wavefunction. Because there are two spin states, the total chemical bond energy of B2 is 2π2C2 2 4 4 ΔE(r)= p + + e−κr. (5.72) κm κr (κr)2 (κr)3

In order to compare with experimental data, the Pauli repulsion com- ponent should be added. A widely used form for the potential function is the Morse function U(r)=D e−2β(r−re) − 2e−β(r−re) , (5.73) where D is the dissociation energy, and β is a decay constant, see Fig 5.12. The equilibrium internuclear distance re can be adapted from the data table of Huber and Herzberg, see Table 5.2. The first term in Eq. 5.73 is Pauli repulsion, and the second term is the chemical bond potential. Therefore, the second term should be an approximation for the chemical bond energy in Eq. 5.72. To facilitate a least-squares fit calculation, it is convenient to write the negative of the second term in Eq. 5.73 as a lineal function of r,

ln ΔE(r) ≈ α − βr, (5.74) where the parameter α is defined as

α =ln2D + βre. (5.75)

Using standard procedure of linear least-squares fit to the chemical bond potential Eq. 5.72, both pamareters α and β canbeobtainedonaninterval, for example, re

Carbon 2 2 2 Theelectronconfigurationofcarbonis1s 2s 2p .Thetwopz atomic orbitals (with the axis of the wavefunction aligned with the interatomic 5.3 Covalent bonds of many-electron atoms 139

Fig. 5.13. Comparing with experimental data. (a) Dissociation energy. (b) Vibrational frequency. The elements with no dimers, Be and Ne, are also marked for concenience. For numerical data, see Table 5.3. line), form two pσ bonding molecular orbitals. Other p-type atomic orbitals form two pπ molecular orbitals. According to Eqs. 5.70 and 5.71, the total chemical bond energy of dimer C2 is 2π2C2 2 6 6 ΔE(r)= p + + e−κr. (5.77) κm κr (κr)2 (κr)3

Nitrogen The electron configuration of nitrogen is 1s2 2s2 2p3.Asincarbon,thetwo pz atomic orbitals form two pσ bonding molecular orbitals. Other p-type atomic orbitals form four pπ molecular orbitals. According to Eqs. 5.70 and 5.71, the total chemical bond energy of N2 is 2π2C2 2 8 8 ΔE(r)= p + + e−κr. (5.78) κm κr (κr)2 (κr)3

The chemical bonding energy is the strongest among the group.

Oxygen The electron configuration of oxygen is 1s2 2s2 2p4. In addition to the four pπ molecular orbitals, there are two pπ∗ antibonding orbitals, which cancels the energy gain of two pπ bonds. The expression of the total chemical bond energy of O2 is the same as carbon, Eq. 5.77.

Flourine The electron configuration of fluorinen is 1s2 2s2 2p5. Now the four pπ∗ antibonding orbitals neutralize the four pπ bonds. The expression of total chemical bond energy of F2 is the same as boron, Eq. 5.72. 140 The Chemical Bond

Fig. 5.14. Schematics of atomic force microscopy. (a) Dissociation energy. (b) Vibrational frequency. The elements with no dimers, Be and Ne, are also marked for concenience. For numerical data, see Table 5.3.

Neon Theelectronconfigurationofneonis1s2 2s2 2p6.Thepσ∗ antibonds neu- tralize the pσ bonds. There is no chemical bond and no noen dimer.

5.4 Imaging wavefunctions with AFM

There is no difference of imaging a wavefunction with AFM and the observation of Jupiter’s satellites with a telescope or the observation of cells in a living body using a optical microscope. Those are objective realities, that exists independent of observation. Chapter 6 Schr¨odinger’s equation II

Making an analogy to Newtonian mechanics, in Chapter 3 through Chapter 5, we have only presented Newton’s first law of mechanics. The electrons in an atom, a molecule, or a piece of solid are similar to a number of ping-pong balls in a basket. Each sits at a position of lowest possible energy. And those ping-pong balls are mutually exclusive in space. Accord- ing to Pauli’s principle of exclusion, no one ping-pong ball can reoccupy the space already occupied by another ping-pong ball. The system of electrons at rest is is described by the wavefunction, and the charge density of the electrons is described by the squares of the wavefunction. In the absence of external force, the electrons have no motion at all. According to Newton’s second law of mechanics, the ping-pong balls can be set to motion only by external force, for example, a strong wind. Some of the ping-pong balls could be blown into the open air. And their temporary elevated positions are unstable. Once the wind calms up, those ping-pong balls blown into the air will fall down, and the potential energy will convert to other forms of energy, for example, sound waves into the air. Then the ping-pong balls are at rest again. The analogy of wind in quantum mechanics is the electromagnetic waves, which could cause the electrons to be lifted from their ground states (equiv- alent to at rest in Newtonian mechanics) into excited states. The elevated states of electrons are not stable. It can spontaneously fall down to the lower states and emit electromagnetic radiation. In the sixth 1926 paper, Quantization as an Eigenvalue Problem, Part IV,Schr¨odinger introduced his time-dependent wave equation. The external electromagnetic wave is presented as a classical field. A good understand- ing of the interaction of electromagnetic wave and electron system can be achieved. However, to aim at a thorough understanding, the electromag- netic wave should also be quantized. We will present a elementary version of the exciting field of quantum electrodynamics in Chapter 8.

6.1 Time-dependent Schr¨odinger equations

Similar to the derivation of the time-independent Schr¨odinger equation in Chapter 3, the starting point is classical mechanics, the de Broglie’s postu- late, and the Planck-Einstein relation. For a one-electron system in three- 142 Schr¨odinger’s equation II dimensional space, the classical energy integral is p2 + V = E, (6.1) 2me where V is the potential energy function. According to de Broglie, electron is a wave. In a segment of space, a typical wavefunction is

Ψ=C sin(k · r − ωt + φ). (6.2)

Thewavevectork can be obtained from Eq. 6.2,

∇2Ψ=−k2Ψ. (6.3)

According to de Broglie, the momentum p of an electron is associated with thewavevectork as p = k. (6.4) Equation 6.3 now becomes ∇2Ψ p2 = −2 . (6.5) Ψ Insert Eq. 6.5 into Eq. 6.1, multiply both sides by Ψ, one obtains the time- independent wave equation 2 E Ψ= − ∇2 + V Ψ. (6.6) 2me Applying Eq. 6.6 on itself, it becomes 2 2 E2 Ψ= − ∇2 + V Ψ. (6.7) 2me From Eq. 6.2, the circular frequency is ∂2Ψ = −ω2Ψ. (6.8) ∂t2 The energy E is associated with the circular frequency ω through the Planck-Einstein relation E = ω. (6.9) Equation 6.8 becomes ∂2Ψ E2Ψ=−2 . (6.10) ∂t2 Combining Eqs. 6.7 and 6.10, one obtains ∂2Ψ 2 2 2 − − ∇2 2 = + V Ψ. (6.11) ∂t 2me 6.1 Time-dependent Schr¨odinger equations 143

According to Schr¨odinger, Eq. 6.11 is the prototypical time-dependent wave equation. Schr¨odinger made a highlighted comment in his sixth 1926 paper that it is the uniform and general wave equation for the field scalar ψ. However, it includes fourth-order differentiations, which is inconvenient. To reduce the order of differentiation, Schr¨odinger’s proposed to split up Eq. 6.11 into a pair of real differential equations. This can be achieved by introducing another real wavefunction Φ. Equation 6.11 is then split up into a pair of second-order real differential equations, ∂Ψ 2  = − ∇2 + V Φ, (6.12) ∂t 2me and ∂Φ 2  = − − ∇2 + V Ψ. (6.13) ∂t 2me By denoting a Hamiltonian operator as 2 Hˆ = − ∇2 + V, (6.14) 2me Eqs. 6.12 and 6.13 can be written succinctly, ∂Ψ  = Hˆ Φ, (6.15) ∂t and ∂Φ  = −Hˆ Ψ. (6.16) ∂t Equations 6.15 and 6.16 are the time-dependent Schr¨odinger equations for a pair of real time-dependent wavefunctions, Φ and Ψ. Regarding the validity of such arguments, Schr¨odinger made the follow- ing explanations. Strictly speaking, the energy integral Eq. 6.1 is valid only for a time-independent potential function V . Now, using Eq. 6.10, the en- ergy constant E is eliminated. Although Eqs. 6.12 and 6.13 are initially derived for a time-independent potential function V , those equations now do not contain the constant energy E thus could be valid even if the po- tential function V is time dependent. Eventually, the general validity of Eqs. 6.12 and 6.13 can only be verified by comparing its consequences with experiments. Actually this has always been true up to today.

6.1.1 Properties of the real wavefunctions The above differential equations are invariant under a linear transformation with a phase φ, thus the two wavefunctions are identical twins, Ψ cos φ − sin φ Ψ =⇒ . (6.17) Φ sin φ cos φ Φ 144 Schr¨odinger’s equation II

AccordingtoSchr¨odinger, the sum of the squares of the pair of wave- functions represents the density distribution of the particle, and the charge densityofanelectronis

ρ(r)=−e(Ψ2 +Φ2). (6.18)

Because the total charge of an electron is −e, the wavefunctions must be normalized, (Ψ2 +Φ2) d3r =1. (6.19)

Using Eqs. 6.12 and 6.13, the current density can be obtained from the following arguments, ∂ ∂ ∂ ρ(r)=−e Ψ2 + Φ2 ∂t ∂t ∂t ∂Ψ ∂Φ = −2e Ψ +Φ ∂t ∂t (6.20) e = [Ψ∇2Φ − Φ∇2Ψ] m e e = ∇ (Ψ∇Φ − Φ∇Ψ) . me Thus the current density vector is

e j = [Φ∇Ψ − Ψ∇Φ] . (6.21) me The above expression of current density is invariant under the linear trans- formation, Eq. 6.17.

6.1.2 Parallelism to Maxwell’s equations The real Schr¨odinger’s equations, Eqs. 6.15 and 6.16, are similar to the Maxwell’s equations, Eqs. 1.168 and 1.169. Without electrical charge and current, they are 1 ∂E = ∇×B (6.22) c ∂t and 1 ∂B = −∇ × E. (6.23) c ∂t The expression of charge density in the real Schr¨odinger’s equation, Eq. 6.18, is similar to the expression of energy density of the Maxwell’s equations, 1 w = E2 + B2 , (6.24) 8π 6.1 Time-dependent Schr¨odinger equations 145 and the expression of electrical current, Eq. 6.21, is similar to the Pointing vector, 1 S = E × B. (6.25) 8π On the other hand, Maxwell’s two real field vectors can be combined into one complex field vector, similar to the complex wavefunction in con- ventional Schr¨odinger’s equation. Actually, this was done by Schr¨odinger in 1935 as follows. Defining a complex field vector

F = E + iB, (6.26)

Eqs. 6.22 and 6.23 are combined into a single equation i ∂F = ∇×F, (6.27) c ∂t similar to the conventional Schr¨odinger’s equation. The energy density of the electromagnetic fields, as shown in Eq. 6.24, reduces to a compact form in terms of the complex field vector F, 1 w = |F|2 , (6.28) 8π which is similar to the expression of charge density in the complex formu- lation of the wavefunction. The complex Maxwell’s equations, Eq. 6.27, are mathematically equiva- lent to Maxwell’s real equations, Eqs. 6.22 and 6.23. The complex Maxwell equations were known for a long time and may facilitate mathematical anal- ysis. However, the complex version obscures the intuitiveness while adds no additional physical significance. On the other hand, the following complex wavefunction

Θ=Ψ+iΦ (6.29) satisfies the complex Schr¨odinger’s equation defined on his 1926 papers. Re- garding to that differential equation, Dirac stated in an 1929 paper Quantum Mechanics of Many-Electron Systems:

The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, . . . if one neglects relativity variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei.

Consequently, the real Schr¨odinger’s equations, Eqs. 6.15 and 6.16, are capable of handling a large part of physics and whole of chemistry. Never- theless, the real wavefunctions are more intuitive, and the real mathematics is more straightforward than the complex version. 146 Schr¨odinger’s equation II

The similarity of the non-relativistic Schr¨odinger equation with Maxwell’s equations is limited: The wavefunction is a two-component vector, whereas the electromagnetic fields have six components. The spatial differentiation in Schr¨odinger’s equation is in second order, whereas Maxwell’s equations is in first order. Nevertheless, in relativistic quantum mechanics, there are more similarities. Dirac’s spinors have eight real components, and the Dirac equation is also first order in space. There have been a number of publica- tions to reformulate Maxwell’s equations using Dirac’s spinors, which can streamline some mathematical analysis. However, the complex spinor for- mulation of Maxwell’s equations have no advantage.

6.1.3 Time-independent Schr¨odinger’s equation If V only depends on coordinates, a time-independent differential equation can be deduced from Eqs. 6.15 and 6.16. The time dependence of Ψ is sinusoidal Ψ=ψ(r)sin(ωt + φ), (6.30) where the circular frequency ω is determined from E by the Planck-Einstein relation, Eq. 6.9. Following Eqs. 6.15 or 6.16, the companion time-dependent wavefunction is Φ=ψ(r) cos(ωt + φ). (6.31)

The wavefunction of the coordinates only, ψ(r), satisfies the time-independent Schr¨odinger’s equation, Hψˆ (r)= Eψ(r), (6.32) which is Eq. 6.6, defined in the first 1926 papers of Schr¨odinger. According to him, the wavefunction ψ(r) is “everywhere real, single-valued, finite, and continuously differentiable up to the second order”. Here is its relation with the time-dependent wavefunctions: A single spatial wavefunction ψ(r) is shared by two time-dependent wavefunctions Ψ and Φ, with two time- dependent factors differs only by a 90◦ phase. Superficially, here a conventional complex wavefunction is replaced by a pair of real wavefunctions Ψ and Φ. Nevertheless, the mathematics may be even simpler. The two real wavefunctions are identical twins. If the formula for one twin is obtained, the formula for the other twin simply follows, such as in Eqs. 6.30 and 6.31. The relation of time-dependent and time-independent wavefunctions is analogous to those of a vibrating string, see Fig. 1.6. By analogy we show that the global sign of the spatial wavefunction ψ(r) is not observable. As shown in Fig. 1.6, the n-th overtone of the vibrating string is nπx Ψ(x, t)=a sin( )sin(ω t + φ). (6.33) L n 6.2 Electron as a macroscopic particle 147

The spatial wavefunction ψ(r) corresponds to the envelop of the displace- ment of the string, the first factor in Eq. 6.33. Shifting the phase in time

φ =⇒ φ + π (6.34) is equivalent to changing the global sign. Therefore, although the spatial wavefunction ψ(r) often has lobes of opposite signs, the global sign has no physical significance.

6.2 Electron as a macroscopic particle

From the very beginning in 1926 until late life, Erwin Schr¨odinger always insisted that the wavefunction is a physical field. The quantity |ψ(r)|2 is the density distribution of an electron as an extended field in space,and −e|ψ(r)|2 represents its charge density distribution. Also in 1926, Max Born proposed that the quantity |ψ(r)|2 is a probability density for the detectionofanelectronas a material point at location r. In some quantum mechanics textbooks, the two views are described as a dichotomy: they are mutually exclusive. Here we show that both views are indispensible and valid. At a macroscopic scale, Born’s statistical interpretation prevails. At an atomic scale, Schr¨odinger’s continuous field view prevails.

6.2.1 Born’s statistical interpretation Born’s authentic viewpoint is clearly formulated in his 1954 Nobel lecture, entitled The statistical interpretation of quantum mechanics. Here are some key statements:

Wave mechanics enjoyed a very great deal more popularity than the G¨ottingen or Cambridge version of quantum mechanics. It operates with a wave function ψ, which in the case of one particle at least, can be pictured in space, and it uses the mathematical methods of partial differential equations which are in current use by physicists. Schr¨odinger thought that his wave theory made it possible to return to deterministic classical physics. He pro- posed (and he has recently emphasized his proposal anew’s), to dispense with the particle representation entirely, and instead of speaking of electrons as particles, to consider them as a contin- uous density distribution |ψ|2 (or electric density e|ψ|2). To us in G¨ottingen this interpretation seemed unacceptable in face of well established experimental facts. At that time it was already possible to count particles by means of scintillations or with a Geiger counter, and to photograph their tracks with the aid of a Wilson cloud chamber. 148 Schr¨odinger’s equation II

Born’s objections to Schr¨odinger’s field interpretation is based on the experimental observations of individual particles. Nevertheless, all those observations are at a macroscopic scale. In all cases Born referred to, the scale is greater than the wavelength of visible light. By searching the liter- ature, there has never been any report that the position of an electron was observed and measured at a subatomic scale. On the other hand, electrical charge distributions of electrons at a subatomic scale have been repeatedly observed experimentally, by electron diffraction, X-ray diffraction, scanning tunneling microscopy, and atomic force microscopy. In the following, we show that by starting with the concept that electron is a continuous field at a subatomic scale, wave packets of macroscopic scale, for example similar to the wavelengths of visible light, behave exactly like classical Newtonian particles.

6.2.2 Wave packets as macroscopic particles In some quantum mechanics textbooks, the wavefunction spreading is dis- cussed. For example, starting with a Gaussian wave packet

1 2 2 ψ(x)=√ e−x /2a , (6.35) πa

The size of the wave packet varies with time as  2 √a t Δx = 1+ 2 . (6.36) 2 mea

By assuming a =10−10 m, the wave packet would double its size in less than 10−15 second. However, such a discussion assumes that the wave package has a sub- atomic scale and at rest. All observations of electrons as a particle, for example by a scintillators or in a Wilson chamber, are at a macroscopic scale and have non-negligible speed. For example, an electron at a speed of 0.1 times the speed of light has a de Broglie wavelength of

h 6.6 × 10−34 ≈ × −11 λ = = −31 7 2.4 10 m. (6.37) 0.1mec 9.1 × 10 × 3 × 10 The dimension a of the tracks in a Wilson chamber should be greater than the wavelength of visible light, typically a>10−6 m. A typical wave packet of an electron moving in x-direction in real format is

Ψ=f(r,t)sin(kx − ωt), (6.38) and Φ=f(r,t) cos(kx − ωt), (6.39) 6.2 Electron as a macroscopic particle 149 here k =2π/λ. The circular frequency is

k2 ω = . (6.40) 2me The time evolution of the envelop function follows the real Schr¨odinger’s equations, Eqs. 6.12 and 6.13,

∂Ψ  = − ∇2Φ, (6.41) ∂t 2me ∂Φ  = ∇2Ψ. (6.42) ∂t 2me Insert Eqs. 6.38 and 6.39 into Eqs. 6.41 and 6.42, eliminate terms using Eq. 6.40, one obtains ∂f  ∂f sin ξ = − ∇2f cos ξ +2k sin ξ , (6.43) ∂t 2me ∂x ∂f  ∂f cos ξ = − −∇2f sin ξ +2k cos ξ , (6.44) ∂t 2me ∂x here we denote ξ = kx − ωt. (6.45) Multiply Eq. 6.43 by sin ξ and multiply Eq. 6.44 by cos ξ than add together, one obtains ∂f  ∂f = − −∇2f cos 2ξ +2k . (6.46) ∂t 2me ∂x Now, we make an order-of-magnitude estimate of the two terms in the square bracket of Eq. 6.46. Assuming at t = 0, the envelop function is

Fig. 6.1. Spreading of a wave packet. A wave packet of size a and velocity v moving in x-direction. Because a is many orders of magnitude greater than the wavelength λ, the spreading of the wavefunction is negligible. 150 Schr¨odinger’s equation II

Gaussian of size a, 1 x2 + y2 + z2 f(r, 0) = exp − . (6.47) π3/4a3/2 2a2 Because the value of cosine is of order of 1, the first term is 3 x2 + y2 + z2 T = ∇2f = − + f. (6.48) 1 a2 4a4

The second term is ∂f x T =2k = −2k f. (6.49) 2 ∂x a2 Because x ∼ a, the order of magnitude of the ratio is T 1 λ 1 ≈ ≈ ≈ 10−5. (6.50) T2 ka 2πa Therefore, up to a correction term of O(λ/a) ≈ O(10−5), the differential equation of the profile function is ∂f k ∂f ∂f = − = −v , (6.51) ∂t me ∂x ∂x where v is the velocity. The solution is 1 (x − vt)2 + y2 + z2 λ f(r,t)= exp − + O . (6.52) π3/4a3/2 2a2 a Thechargedensityofthewavepacketis e (x − vt)2 + y2 + z2 λ ρ(r,t)= exp − + O . (6.53) π3/2a3 a2 a It is a perfect Newtonian particle of size a moving at a constant speed v, determined by the de Broglie relation v = k/me. Note that Eq. 6.36 is not a correct representation of a particle track. A wave packet of that scale in free space never existed and has never been observed. All experimentally observed individual particles, such as using scintillators, Geiger counters, and tracks in Wilson chambers, are of macro- scopic scale. As shown above, those particle tracks can be accurately de- scribed by wave packets of a size of many thousands of atoms.

6.2.3 Similarity of photons and electrons Photons and electrons share a great similarity. Light is an electromagnetic wave. However, according to Einstein, when light is generated or interacting with atomic systems, energy is quantized. The number of photons observed 6.3 Ehrenfest’s theorem 151 by a macroscopic detector is proportional to the square of the field intensity. The quantization of light is Einstein’s statistical interpretation of electro- magnetic wave. Electron is also a wave. When the electron wave interacts with materials, both charge and mass are quantized. This similarity was explored by Born in his formulation of the statistical interpretation of the wavefunction. As stated in his 1954 Nobel Lecture:

This was left to Schr¨odinger, and I immediately took up his method since it held promise of leading to an interpretation of the ψ-function. Again an idea of Einstein’s gave me the lead. He had tried to make the duality of particles - light quanta or photons - and waves comprehensible by interpreting the square of the optical wave amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the ψ-function: |ψ|2 ought to represent the probability density for electrons (or other particles).

Therefore, historically, Born’s inspiration for the statistical interpreta- tion of wavefunction was an analogy to Einstein’s statistical interpretation of the electromagnetic wave. For an explanation of the similarity of photons and electrons in double-slit experiments with single-particle detection, see Sections 2.1.5 and 2.2.5.

6.3 Ehrenfest’s theorem

In Section 6.2.2, we show that on a macroscopic scale, an electron behaves like a Newtonian particle. In free space, a packet of electron wavefunction moves with a constant velocity v, determined by de Broglie relation

p k v = = . (6.54) me me In this Section, we show that in a potential field, the center of mass of an electron follows Newton’s second law like a material particle. AccordingtoSchr¨odinger, the square of the wavefunction is the density distribution of an electron, with total value of 1, see Eq. 6.18 and 6.19. The average value of x,orthecenterofmassinthex-direction, is x = d3rx(Ψ2 +Φ2). (6.55)

Here we use brackets to represent average value. Using Schr¨odinger’s equations, Eqs. 6.12 and 6.13, the equation of mo- tion of the center of mass can be derived. Not surprisingly, it is Newton’s 152 Schr¨odinger’s equation II second law of mechanics. It is named Ehrenfest’s theorem to honor its dis- coverer. For example, the x-velocity of the center of mass is d d x = d3r x(Ψ2 +Φ2) dt dt ∂Ψ ∂Φ =2 d3r x Ψ +Φ (6.56) ∂t ∂t  = d3r xΦ∇2Ψ − xΨ∇2Φ , me here the potential energy terms are cancelled. Using the identity

∇(xΦ∇Ψ) = ∇(xΦ)∇Ψ+xΦ∇2Ψ (6.57) and a similar one, integrating by part, Eq. 6.56 becomes d x = d3r (∇(xΦ)∇Ψ −∇(xΨ)∇Φ) dt (6.58)  ∂Ψ ∂Φ = d3r Φ − Ψ . me ∂x ∂x The acceleration can be calculated from Eq. 6.58. The result is d2  d ∂Ψ ∂Φ x = d3r Φ − Ψ dt2 m dt ∂x ∂x e 1 ∂ ∂Ψ ∂ ∂Φ = − d3r Φ (V Φ) − V Ψ +Ψ (V Ψ) − V Φ m ∂x ∂x ∂x ∂x e 1 ∂V = − d3r Φ2 +Ψ2 m ∂x e 1 ∂V = − . me ∂x (6.59) Here the kinetic energy terms cancel each other. Similar results can be found for y and z. Finally, Newton’s second law of mechanics, Eq. 1.2, is recovered. An electron is an extended field in space. But its center of mass behaves like a Newtonian material particle,

d2 m r = − ∇V = F . (6.60) e dt2 Therefore, the experimental observations of electrons as macroscopic New- tonian particles in Geiger counters, scintillators, and Wilson chambers are well interpreted by Schr¨odinger’s equations through Ehrenfest’s theorem. No need for hidden variables. As a theory in physics capable of explaining all experimental observations, quantum mechanics is complete. Chapter 7 Perturbation Theories

Only for a handful of real-world systems, the Schr¨odinger’s equation can have analytic solutions. However, for many practical cases, the actual Hamiltonian is only slightly different from the one with exact solutions; the Schr¨odinger’s equations can be resolved using perturbation methods. The additional term in the Hamiltonian is treated as a perturbation,andthe difference from the known solution is expressed as a correction. This is true for time-independent problems, and also for time-dependent problems. For example, the radiation field can be treated as a perturbation, and the evolution of the wavefunction is then expressed by a golden rule.

7.1 Stationary perturbation theory

Assuming the solutions of a system with Hamiltonian Hˆ0 are known

Hˆ0|n = En|n , (7.1) where n =1,2,3,... ∞. The eigenfunctions |n are orthogonal, normalized, and complete, which means that

n|m = δnm, (7.2) and for any wavefunction |ψ , ∞ |ψ = n|ψ |n . (7.3) n=1 Letv ˆ be a perturbation potential. To facilitate reasoning and to indicate that the perturbation is small, a parameter λ is multiplied tov ˆ.Thenwe are looking for solutions of the eigenvalue problem Hˆ0 + λvˆ |ψ = |ψ . (7.4)

By expressing the solution and the energy as a power series of λ,

2 |ψ = |ψ0 + λ|ψ1 + λ |ψ2 ... (7.5) and 2  = 0 + λ1 + λ 2... . (7.6) 154 Perturbation Theories

Insert Eqs. 7.5 and 7.6 into Eq. 7.4, separate terms with different powers of λ, we obtain Hˆ0 − 0 |ψ0 =0, (7.7) Hˆ0 − 0 |ψ1 =(1 − vˆ) |ψ0 , (7.8) Hˆ0 − 0 |ψ2 =(1 − vˆ) |ψ1 + 2|ψ0 . (7.9)

From Eq. 7.7, one finds that |ψ0 must be one of the solutions of the unper- turbed system, say |n .Wehave

|ψ0 = |n , (7.10) and

0 = En. (7.11) Multiply Eq. 7.8 by n|, because of Eq. 7.11, the left-hand side of the equa- tion becomes zero. The right-hand side gives

1 = n|vˆ|n . (7.12)

This is the first-order correction of the energy eigenvalue. By multiply Eq. 7.8 with m|,wherem = n, and complete the bracket, the left-hand side becomes m| Hˆ0 − 0 |ψ1 =(Em − En) m|ψ1 . (7.13)

And because m|n = 0, the right-hand side becomes

m| (1 − vˆ) |ψ0 = − m|vˆ|n . (7.14)

Combining Eqs. 7.13 and 7.14, we have the expression of the first-order perturbation wavefunction,

∞ m|vˆ|n |ψ1 = |m , (7.15) En − Em m= n the term m = n is excluded. Multiplying Eq. 7.9 with n|, because of Eqs. 7.10 and 7.11, the left-hand side of the equation is zero. The right- hand side gives the expression of the second-order energy correction,

∞ | m|vˆ|n |2 2 = . (7.16) En − Em m= n 7.1 Stationary perturbation theory 155

7.1.1 Polarization of hydrogen atom As an application of the stationary perturbation theory, the problem of polarization of a hydrogen atom by an electric field F is treated. The unperturbed wavefunction is the ground state of hydrogen atom, as we presented in Chapter 3 by Eq. 3.86 and shown in Fig. 3.8,

e−r/a0 |ψ = . (7.17) 0 3 πa0 The external potential due to an electric field F is

vˆ = −eF r cos θ. (7.18)

The first-order perturbation energy, Eq. 7.12, is zero. Therefore, follow- ing Eq. 7.8, the differential equation for the first-order perturbation wave- function is 2 2 2 e − ∇ − − E0 |ψ1 = −eF r cos θ |ψ0 . (7.19) 2me r

Because the quantities in the parenthesis and the unperturbed wavefunction is spherically symmetric, one would assume that the only angular depen- dence of the first-order perturbation wavefunction is cos θ, and has a factor similar to the unperturbed wavefunction,

e−r/a0 |ψ = f(r)|ψ cos θ = f(r)cosθ, (7.20) 1 0 3 πa0 where f(r) is the unknown function to be determined. Because the pertur- bation wavefunction depends on θ, the Laplace operator in Eq. 7.19 should have an explicit angular dependent term, 1 ∂ ∂ 1 ∂ ∂ ∇2|ψ = r2 + sin θ |ψ . (7.21) 1 r2 ∂r ∂r r2 sin θ ∂θ ∂θ 1

For the cosine factor, the θ-dependent term generates a constant -2, 1 ∂ ∂ sin θ cos θ = −2cosθ. (7.22) sin θ ∂θ ∂θ

The above result is simply because the form of the perturbative wavefunc- 2 tion |ψ1 is a p-wave, l = 1, which gives an eigenvalue of L of l(l +1)=2, see Section 1.4.2. Using the identities in Section 3.4.1,

2 2 = e a0 (7.23) me 156 Perturbation Theories and e2 E0 = − , (7.24) 2a0 Equation 7.21 is simplified to 2 1 d 1 d − 1 1 − 1 | − Fr| 2 + 2 + 2 f(r) ψ0 = ψ0 . (7.25) 2 dr r dr r a0r 2a0 a0e Using the relation d 1 |ψ0 = − |ψ0 , (7.26) dr a0 Equation 7.25 is reduced to a differential equation of f(r), 2 1 d f(r) 1 − 1 df (r) − f(r) − Fr 2 + 2 = . (7.27) 2 dr r a0 dr 2r a0e To find the solution of Eq. 7.27, we guess that it is a polynomial of r. Try a polynomial with two terms

2 f(r)=c1r + c2r , (7.28) Substitute into Eq. 7.27, one obtains

c1 2c2 Fr 2c2 − − r = − . (7.29) a0 a0 a0e It gives F c1 = , (7.30) a0e and F c = . (7.31) 2 2e

Fig. 7.1. Ground-state and perturbative wavefunctions. Left: the unperturbed ground-state hydrogen wavefunction. Middle: the perturbative wavefunction. Right: The resultant wavefunction. It is equivalent o a shift of the center of the negative electrical charge of the electron, and a electrical dipole is generated. 7.1 Stationary perturbation theory 157

The solution is F 1 f(r)= a r + r2 . (7.32) e 0 2 According to Eq. 7.23, the perturbation wavefunction is F 1 |ψ = e−r/a0 a r + r2 cos θ, (7.33) 1 3 0 2 e πa0 as shown in Fig. 7.1. The first-order perturbation energy, Eq. 7.12, is zero. The second-order perturbation energy can be derived by multiplying ψ0| to the left-hand side of Eq. 7.9. The result is

2 = ψ0|vˆ|ψ1 . (7.34)

Using the expression of |ψ1 in Eq. 7.33, Eq. 7.34 gives F 2 ∞ 1 π 2π  = a r + r2 e−r/a0 r2dr cos2 θdθ dφ 2 πa3 0 2 0 0 0 0 (7.35) 9 = F 2a3. 4 0

+ 7.1.2 The van der Waals force in H2 The effect of perturbation can be understood as the creation of a electrical dipole by the external electrical field F , see Fig. 7.1. The resultant wave- function of the hydrogen atom, the left-side graph of Fig. 7.1, is a sum of the (unperturbed) ground-state wavefunction |ψ0 and the first-orded per- turbation wavefunction |ψ1 . It is equivalent to a shift of the center of the electrical charge of the electron in the hydrogen atom. An electrical dipole is created. The dipole is attracted by the external electrical field to generate an interaction energy 2. The above interaction energy is the source of the van der Waals interac- tion energy in a hydrogen molecular ion. According to Coulomb’s law, the electrical field intensity from a proton at a distance of ρ is e F = . (7.36) ρ2

2 Using the value of the Rydberg constant Ry = e /2a0, the van der Waals interaction energy at that distance is

9Ry ΔE = . (7.37) vdw 2ρ4

We have used this value in Section 5.2. 158 Perturbation Theories

7.2 Interaction with radiation

In this Section, the problem of the interaction of an atomic system with radiation is treated using time-dependent Schr¨odinger’s equations, Eqs. 6.15 and 6.16. For notational brevity, the Dirac bra and ket scheme is applied. To represent a real wavefunction in coordinate space, such as in Eq. 3.71, a ket and a bra are identical. A bracket represents an integration in the entire space, such as Eq. 3.25. A matrix element can be represented succinctly as a Dirac bracket. To avoid ambiguity, a hat is placed on top of an operator, such as in Eqs. 7.45, 3.50, and 3.51.

7.2.1 Time-dependent perturbation theory Denoting the Hamiltonian of an atomic system as 2 2 Hˆ0 = − ∇ + V, (7.38) 2me andthedipoleinteractionoftheradiationfieldas

vˆ =ˆv0 sin ωt. = eE · r sin ωt. (7.39)

The operatorv ˆ0 is time-independent. At t<0, the coordinate wavefunc- tions of the atomic system are eigenstates of Hamiltonian Hˆ0,

Hˆ0 |n = En |n . (7.40) The time-dependent unperturbed wavefunctions are E t Ψ =sin n |n (7.41) n  and E t Φ =cos n |n . (7.42) n  After t>0, the radiation acts on. The Schr¨odinger’s equations, Eqs. 6.15 and 6.16, become ∂Ψ  = Hˆ +ˆv Φ, (7.43) ∂t 0 and ∂Φ  = − Hˆ +ˆv Ψ. (7.44) ∂t 0 Assuming at t<0, the initial spatial wavefunction is |i , and the energy is Ei. After the radiation perturbationv ˆ is turned on, the wavefunction spreads into other states with time-dependent coefficients cn(t), E t E t Ψ=sin i |i + c (t)sin n |n , (7.45)  n  n= i 7.2 Interaction with radiation 159 and E t E t Φ=cos i |i + c (t)cos n |n . (7.46)  n  n= i Insert Eqs. 7.45 and 7.46 into Eqs. 7.43 and 7.44, eliminating equal terms in both sides, the significant terms are

dc (t) E t E t  n sin n |n =cos i vˆ|i , (7.47) dt   n= i and dc (t) E t E t  n cos n |n = − sin i vˆ|i . (7.48) dt   n= i To look for the coefficients for a final state |f , multiplying both sides of Eqs. 7.47 and 7.48 by f|,wehave

dc (t) E t E t  f sin f =cos i f|vˆ|i , (7.49) dt   and dc (t) E t E t  f cos f = − sin i f|vˆ|i . (7.50) dt  

Multiplying Eq. 7.49 by sin(Ef t/) and multiplying Eq. 7.50 by cos(Ef t/) than adding together, we obtain

dc (t) (E − E )t  f =sin f i f|vˆ|i . (7.51) dt  Using the expression of the radiation interaction, Eq. 7.39, Eq. 7.51 splits into two terms,

dc (t) (E − E − ω)t  f = f|vˆ |i cos f i dt 0  (7.52) (E − E + ω)t − f|vˆ |i cos f i . 0 

The first term represents absorption, Ef >Ei, and the second term rep- resents stimulated emission Ef

7.2.2 The golden rule By integrating the first term over time, the coefficeint becomes

sin(Ef − Ei − ω)t cf (t)= f|vˆ|i . (7.53) Ef − Ei − ω 160 Perturbation Theories

Fig. 7.2. Condition of energy conservation. The integrand in Eq. 7.56 approaches a delta function when a approaches infinity. If the time of the experiment is not too short, the condition of energy conservation is valid.

The probability of transition to a final state |f is proportional to the square of the coefficient. 2π sin2 [(E − E − ω)(t/2)] p ≡|c (t)|2 = t | f|vˆ |i |2 f i . (7.54) fi f  0 2 π [Ef − Ei − ω] (t/2)

Denote Ef − Ei − ω = u and t/2 = a; the function in square brackets has a sharp peak near u = 0, as shown in Fig. 7.2. The area under it is 1: ∞ sin2 au 2 du =1. (7.55) −∞ πau As a →∞, it approaches a delta function,

sin2 au lim = δ(u). (7.56) a→∞ πau2 Therefore, when t →∞, the function in square brackets in Eq. ?? ap- proaches a delta-function δ(Ef −Ei−ω). Also, from Eq. ??, the probability of |f is proportional to t, and therefore, the transition rate is p 2π R ≡ fi = | f|vˆ |i |2δ(E − E − ω). (7.57) fi t  0 f i Equation 7.57 is the Golden Rule for atomic systems with discrete energy levels. The Bohr frequency condition,

ω = Ef − Ei, (7.58) comes naturally. The radiation field can only transfer an energy quantum of ω to the atomic system, which is the essence of Einstein’s theory of 7.3 Imaging wavefunctions with STM 161

Fig. 7.3. Schematics of scanning tunneling microscopy. (a) Dissociation energy. (b) Vibrational frequency. The elements with no dimers, Be and Ne, are also marked for concenience. For numerical data, see Table 5.3. photons and provides an explanation of the line spectra, especially the Ritz combination principle. As shown, the Bohr frequency condition comes up naturally as a conse- quence of a trigonometry identity. The two terms correspond to Einstein’s absorption coefficient B12 and stimulated emission coefficient B21 as well as their equality come out naturally. Furthermore, Eq. 7.52 demonstrates that the transition process is not instantaneous. Following Schr¨odinger’s equations, the new state comes up gradually. By waiting long enough, a complete transition takes place, and the transition rate is 2π R = | f|vˆ |i |2 δ(E − E ± ω). (7.59)  0 f i

7.3 Imaging wavefunctions with STM

There is no difference of imaging a wavefunction with STM and the obser- vation of Jupiter’s satellites with a telescope or the observation of cells in a living body using a optical microscope. Those are objective realities, that exists independent of observation. 162 Perturbation Theories Chapter 8 Quantum Theory of Light

In Chapter 7, using the time-dependent Schr¨odinger equation presented in Chapter 6, the problem of the interaction of radiation with atomic systems is treated by a perturbation method. Radiation is treated as a classical electromagnetic wave. The transition probabilities for absorption and stim- ulated emission are derived. However, the quantization of light, as proposed by Einstein in 1905, is not fully explained. Furthermore, the spontaneous emission of light by atomic systems, universally observed experimentally, is not explained. Therefore, a good understanding of the interaction of radiation with atomic systems is still not achieved. To make a better understanding, the electromagnetic waves must be quantized. Then, the concept of photons can be clarified. The mechanism of spontaneous emission of radiation can be understood. In Section 8.1, we discuss some basic concepts of electromagnetic radia- tion first. Then, Einstein’s derivation of the blackbody radiation formula is presented. In Section 2, by treating the electromagnetic radiation as a set of harmonic oscillators, the concept of photons is explained. Finally, using time-dependent perturbation theory, a complete treatment of the interac- tion of the interaction of radiation and atomic systems is presented.1

8.1 Blackbody radiation

It was known for centuries that a hot body emits radiation. At around 700◦C, a body becomes red hot. At even higher temperatures, a body emits much more radiation, and the color changes to orange, yellow, white, and even blue, see Fig 8.1. In the late nineteenth century, in order to understand phenomena related to industry technology such as steel making and incandescent light bulbs, it became a hot subject for physicists. Although all hot bodies emit radiation, blackbodies emit the maximum amount of radiation at a given temperature. At equilibrium, radiation emit- ted must equal radiation absorbed. Therefore, the body that emits the maximum amount also absorbs the maximum amount—which should look black. Practically, a blackbody is constructed by opening a small hole on a

1The current presentation follows W. Heitler’s The Quantum theory of Radiation, Third Edition, Oxford university Press, 1954, and R. Louden’s The Quantum Theory of Light, Second Edition, Clarendon Press, Oxford, 1983. 164 Quantum Theory of Light

Fig. 8.1. Blackbody radiation. At different temperature, a black body radiates different amount of radiation with a well-defined spectrum. large cavity, as shown in Fig. 8.2. Any light ray entering a hole with area A experiences multiple reflections on the internal surface of the cavity. If the material is not absolutely shiny, after several impingements, the light will eventually be completely absorbed by the cavity.

8.1.1 Modes of electromagnetic waves in a cavity The energy density of radiation as a function of its frequency was studied in the late nineteenth century by Lord Rayleigh and then by Sir James Jeans using classical statistical physics. They treated standing electromagnetic waves in a cavity as individual modes, and the modes follow the equal- partition law of Maxwell–Boltzmann statistics. Consider a closed cubic cavity with reflective inner surfaces of sides L. According to Section 1.5.4, a sinusoidal electromagnetic wave with circular frequency ω, or frequency ν, satisfies the following equation:

4π2ν2 ω2 ∇2E = − E = − E. (8.1) c2 c2 Assuming that the cavity is made of metal. On the walls of the cavity, electrical field intensity vanishes. Therefore, the vector potential vanishes. The general solution of Eq. 8.1 satisfying that condition is

E = E0 sin(kxx)sin(kyy)sin(kzz). (8.2) The wavevectors are defined by πn πn πn k = x ,k= y ,k= z , (8.3) x L y L z L 8.1 Blackbody radiation 165

Fig. 8.2. Radiation in a cavity. A large cavity with a small hole is a good blackbody. The light enters the hole will experience multiple reflections, and all be absorbed and thus looks black. A blackbody emits maximum amount of radiation when heated.

where nx,ny, and nz are positive integers. By direct substitution one finds that Eq. 8.2 satisfies differential equation 8.1 and the boundary conditions at the walls. Each set of the integers, nx,ny,nz, represents a pattern of electromagnetic wave in the cavity. Inserting Eq. 8.3 into Eq. 8.1 yields

4π2ν2 k2 + k2 + k2 = , (8.4) x y z c2 and in terms of the numbers nx,ny, and nz, Eq. 8.4 becomes

4ν2L2 n2 + n2 + n2 = . (8.5) x y z c2 Now, we count the number of standing waves with frequencies ν by 2 2 2 considering a sphere of radius nx + ny + nz =2νL/c.ThenumberN of modes with positive nx,ny,andnz up to ν is 1 4 2νL 3 4πν3 V N = π = , (8.6) 8 3 c 3c3 where V = L3 is the volume. For each type of standing wave, there are two polarizations. The number of modes of standing electromagnetic wave is

8πν3 V N = , (8.7) 3c3 and the density of states at frequency ν is d N 8πν2 = . (8.8) dν V c3 166 Quantum Theory of Light

8.1.2 Rayleigh–Jeans law According to Maxwell–Boltzmann statistics, at absolute temperature T , each degree of freedom contributes energy kBT ,wherekB is the Boltzmann constant, and the energy density is d N 8πν2 ρ(ν, T )= k T = k T. (8.9) dν V B c3 B Equation 8.9 is the energy density of radiation per unit frequency interval in a cavity of temperature T . It is not directly observable. The directly observable quantity is the spectral radiance u(ν, T), that is, the energy radiating from a unit area of the hole per unit frequency range. To calculate u(ν, T )fromρ(ν, T), first we consider a simplified situation: If the field has a well-defined direction of radiation with velocity c,wehave

u(ν, T)=cρ(ν, T ). (8.10)

Because the hole is small, the radiation field in a cavity is isotropic. As the radiation only comes through a hole of well-defined direction, u(ν, T) should be a fraction of cρ(ν, T). The value of the fraction can be determined using the following argument. Consider a sphere of radius R. The surface area of the sphere is 4πR2. If the radiation inside the sphere is allowed to emit over all directions, the area is 4πR2. Iftheradiationisallowedtoemitinonly onedirection,theareaisadiscwithradiusR,thatis,πR2. Consequently, the factor is 1/4. Equation 8.10 becomes 1 u(ν, T )= cρ(ν, T ). (8.11) 4 Following is a more detailed proof of the factor 1/4. Consider the ra- diation from a small hole of area A on the cavity; see Fig. 8.2. Because the electromagnetic wave is isotropic and the speed of light is c, the energy radiated through a solid angle dΩatanangleθ is dE c = ρ(ν, T ) A cos θ (8.12) dt dΩ 4π because the area of the hole observed from an angle θ is A cos θ.Integrating over the hemisphere, the total irradiation per unit area is

c π/2 c u(ν, T )= 2π cos θ sin θdθρ(ν, T )= ρ(ν, T ), (8.13) 4π 0 4 confirming Eq. 8.11. Using Eq. 8.9, we finally obtain the Rayleigh–Jeans distribution of blackbody radiation, 2πν2 u(ν, T )= k T. (8.14) c2 B 8.1 Blackbody radiation 167

The Rayleigh–Jeans distribution fits the low-frequency behavior of the ex- perimental energy density very well. However, as the frequency increases, the spectral irradiance increases, the total irradiation energy is infinite. This contradicts the experimental fact that the total blackbody radiation is finite, and the spectral density has a maximum; see Fig. 2.5.

8.1.3 Planck formula and Stefan–Boltzmann’s law

In 1900, Max Planck found an empirical formula that fits accurately the experimental data,

2πν2 hν u(ν, T )= . (8.15) c2 ehν/kBT − 1

The constant h in the formula, Planck’s constant, was initially obtained by fitting with experimental blackbody radiation data. Later, Planck found a mathematical explanation of his formula by assuming that the energy of radiation can only take discrete values. Specifically, he assumed that the energy of radiation with frequency ν can only take integer multiples of a basic value hν,theenergy quantum,

 =0,hν,2hν, 3hν, .... (8.16)

According to Maxwell–Boltzmann statistics, the probability of finding a state with energy nhν is exp(−nhν/kBT ).Theaveragevalueofenergyof a given component of radiation with frequency ν is

Fig. 8.3. Rayleigh-Jeans law and Planck’s law. At different temperature, a black body radiates different amount of radiation with a well-defined spectrum. 168 Quantum Theory of Light

∞ nhν e−nhν/kBT n=0 hν ¯ = ∞ = . (8.17) ehν/kBT − 1 e−nhν/kBT n=0 instead of kBT . By replacing the expression kBT in Eq. 8.9 with Eq. 8.17, we recovered Eq. 8.15. Initially, Max Planck believed that the quantization of energy is only a mathematical trick to reconcile his empirically obtained formula with the knowledge of physics known at that time. The profound significance of the concept of quantization of radiation and the meaning of Planck’s constant were discovered by Albert Einstein in his interpretation of the photoelectric effect, which is the conceptual foundation of solar cells. By integrating the spectral radiance over frequency, the total radiation is found to be ∞ 2πhν3 dν U(T )= 2 hν/kBT 0 c e − 1 2πh k T 4 ∞ x3 dx = B (8.18) 2 x − c h 0 e 1 2 π5k4 = B T 4. 15 c2 h3 Here a mathematical identity is applied, ∞ x3 dx π4 = . (8.19) x − 0 e 1 15 Equation 8.18 is Stefan–Boltzmann’s law, discovered experimentally before the Planck formula and backed by an argument using thermodynamics. The constant in Eq. 8.18,

2 π5k4 π2k4 W σ ≡ B = B =5.67 × 10−8 , (8.20) 15 c2h3 60 c23 m2 · K4 is called the Stefan–Boltzmann’s constant.

8.1.4 Einstein’s derivation of blackbody formula Based on the concept of photons and the interaction of photons with matter, Einstein made a very simple derivation of the blackbody radiation formula. The key of his derivation is the introduction of stimulated emission of ra- diation, which gave birth to the laser, an acronym for light amplification 8.1 Blackbody radiation 169 by stimulated emission of radiation, and provides a better understanding of the interaction of radiation with matter. Einstein studied a simple two-state atomic system; see Fig. 2.8. The radiation field is represented by an energy density σ(ν), where ν is the frequency. The atomic system has two states with energy levels E1 and E2. The photons with energy hν are associated with a transition between the two states. The energy relation is

hν = E2 − E1. (8.21)

According to Maxwell–Boltzmann statistics, the populations of the two states are −E1/kBT N1 ∝ e (8.22) and −E2/kBT N2 ∝ e . (8.23) Using Eq. 8.21, we have N 1 = ehν/kBT . (8.24) N2 Einstein assumed three transition coefficients: the absorption coefficient B12, the spontaneous emission coefficient A, and the stimulated emission coefficient B21. The rate equations are dN 2 = B N σ(ν) − B N ρ(ν) − AN , (8.25) dt 12 1 21 2 2 dN 1 = −B N ρ(ν)+B N ρ(ν)+AN . (8.26) dt 12 1 21 2 2

At equilibrium, both dN1/dt and dN2/dt should vanish. Therefore,

N A + B ρ(ν) 1 = 21 = ehν/kBT . (8.27) N2 B12ρ(ν)

The coefficients are independent on temperature and frequency. At a high temperature, the power density should be high, and the right-hand side of Eq. 8.27 should approach unity. Therefore, the absorption coefficient B12 should equal to the stimulated emission coefficient B21. Both can be represented by one coefficient B:

B12 = B21 = B. (8.28)

Equation 8.27 becomes

A 1 +1=ehν/kBT . (8.29) B ρ(ν) 170 Quantum Theory of Light

Fig. 8.4. Einstein’s derivation of blackbody radiation formula. The radiation field ρ(ν) inter- acts with a two-level atomic system. Three inter- action modes are assumed: absorption, to lift the atomic system from state 1 to state 2; spontaneous emission and stimulated emission, the atomic system decays from state 2 to state 1, giving out energy to the radiation field.

The power density distribution of radiation is then

A 1 ρ(ν)= . (8.30) B ehν/kBT − 1 For radiations of low photon energy, Eq. 8.30 reduces to

A k T ρ(ν) → B . (8.31) B hν It should be identical to the Rayleigh–Jeans formula. Comparing with Eq. 8.9, we find the ratio of coefficients A and B,

A 8πhν3 = . (8.32) B c3 Finally, Planck’s formula is recovered,

8πhν3 1 ρ(ν)= . (8.33) c3 ehν/kBT − 1

8.2 The real quantum electrodynamics

Historically, the quantization of electromagnetic radiation was done first by Planck in 1900 and Einstein in 1905, much earlier than the quantization of the electron wave. Acordingly, the energy of a mode of electromagnetic radiation with circular frequenty ω can only have values of integer multiples of an elementary energy quanta,

 = nω. (8.34)

Nevertheless, the mathematical formulation of quantization of the electro- magnetic waves appears later than the quantization of electrons. It is tradi- tionally considerd as an advanced subject In this Section, we present an ele- mentary theory of quantum electrodynamics using real variables. Quantum electrodynamics is probably the most accurate and most beautiful theory 8.2 The real quantum electrodynamics 171 ever existed. Its predictions have been experimentally verified up to an accu- racy of 12 digits, and never failed. Although it is based on Maxwell’s equa- tions, quantum electrodynamics cannot be derived from Maxwell’s equation only. At a certain point, Planck’s constant must be entered. And every book or introductory article on quantum electrodynamics must do this at a point by certain heuristic arguments. Once a new mathematical equation is in- troduced, the only criterion of its validity hinges on the correctness of its predictions.2 Here we present a simple heuristic way of introducing Planck’s con- stant into electromagnetism based on the real algebraic theory of quantum- mechanical harmonic oscillators in Section 3.3.

8.2.1 Quantization of electromagnetic waves In Section 8.1, we presented standing electromagnetic waves in a cubic cavity of conducting sides. Here we present a slightly generalized case. The electromagnetic waves in vacuum can be decomposed into elemen- tary modes. The electrical field intensity of each individual mode is

E(r,t)=E(t)sin(k · r + φ). (8.35)

Here the wavevector k is defined as 2πn 2πn 2πn k = x ,k= y ,k= z , (8.36) x L y L z L where nx,ny, and nz are positive integers. Instead of a metal cavity as in Section 8.1, a periodic boundary condition on the sides is assumed. For each wave vector k, the electromagnetic wave has two polarizations, see Section 1.5.5. Here we focus on one of the pair. By introducing a unit vector e, which is perpendicular to wave vector k, the electric field vector can be written as the product of a magnitude E(t) and that unit vector,

E(t)=E(t) e. (8.37)

According to Eq. 8.1, the temporal variation of the field is d2E(t) + ω2E(t)=0. (8.38) dt2 It is the differential equation of a harmonic oscillator with circular frequency

ω = c|k|. (8.39)

Each mode of such electromagnetic wave is a harmonic oscillator. According to Section 3.3, the energy level can be quantized. In order to utilize the

2See for example Shanker, Eq. (18.5.87) on page 514. Merzbacher, Quantum Mechan- ics, Second Edition, Eq. (22.10) on page 556. Sakurai, Advanced Quantum Mechanics, Eqs. (2.24a) and (2.24b) on page 24. 172 Quantum Theory of Light arguments in Section 3.3, the relation between the field parameter E(t)and the dimensionless variable q in Eq. 3.48 must be defined. We suggest the following relation, which will be verified later, 8π √ E(t)= ωq. (8.40) V The field intensity of a single mode of electromagnetic wave is 8π √ E(r,t)= ωqe sin(k · r + φ). (8.41) V Where V is the volume. According to Eq. 8.38, the dimensionless coordinate q satisfies the differential equation of a harmonic oscillator,

d2q + ω2q =0. (8.42) dt2 Following Section 3.3, by defining an annihilation operator, 1 d aˆ = √ q + , (8.43) 2 dq and a creation operator 1 d aˆ† = √ q − . (8.44) 2 dq A particle number operator can be established as

Nˆ =ˆa†a.ˆ (8.45)

The state of a quantum-mechanical system is described by an eigenstate of the particle number operator,

Nˆ|n =ˆa†aˆ|n = n |n . (8.46)

The creation operator increases the number of quanta by 1, √ aˆ†|n = n |n +1 , (8.47) and the annihilation operator decrease the number of quanta by 1, √ aˆ|n = n − 1 |n − 1 . (8.48)

The general coordinate q is now an operator, 1 qˆ = √ aˆ† +ˆa . (8.49) 2 8.2 The real quantum electrodynamics 173

The electrical field intensity, Eq. 8.41, is also an operator, 4π √ Eˆ(r,t)= ω aˆ† +ˆa e sin(k · r + φ). (8.50) V In cgs unit system, for an electromagnetic wave, the magnitude of the magnetic field intensity equals the magnitude of the electric field intensity, see Section 1.5.5. The energy density W of electromagnetic wave, Eq. 1.189, can be expressed in electrical field intensity only, 1 1 W = E2 + B2 = E2. (8.51) 8π 4π The energy of a mode of electromagnetic wave is evaluated similar to any dynamic variable. For the n-th state, the energy is 1  = d3r W = d3r n| Eˆ 2(r,t)|n V V 4π 1 = ω n| aˆ†aˆ† +ˆa†aˆ +ˆaaˆ† +ˆaaˆ |n (8.52) 2 1 = n + ω. 2

The correctness of Eq. 8.40 is verified.

8.2.2 Indentical particles: bosons and fermions As a significant feature distinguished from classical mechanics, in quantum mechanics, the particles are indistinguishable. In cases of radiation, in a cavity, for a given mode, the number of photons in that mode can be any non-negative integer. However, every photon in the same mode is equal: they are indistinguishable. Therefore, for each mode of radiation, one can say how many photons are in that mode; but one can never say which photon is in a partical mode. In an atomic system, the electrons are indis- tinguishable. For an atomic system with n energy levels, one can say which level is occupied by an electron, and which level is empty; but one can never say which electeon is in which level. In both cases, the system can be described by a number state. For a system with modes or levels labled 1, 2, 3, ...i, ..., a state of the system can be specified by the occupation number of particles in each mode or level,

|n1,n2,n3, ...ni, ... . (8.53)

For a vacuum state where none of the modes is occupied, the number state is |0, 0, 0, ...0, ... . (8.54) 174 Quantum Theory of Light

For a state of the syetem where only the i−th mode is occupied by one particle, it is |0, 0, 0, ...ni =1,nn+1 =0, ... . (8.55) The is a fundamental difference between photons and electrons. For pho- tons, each mode can have unlimited number of photons. Such a particle is called a Boson. For electrons, each energy level can only be either occupied by an electron or empty. Such a particle is called a Fermion.

8.2.3 The anticommutation relation 9+

8.2.4 Second quantization 8.2.5 Interaction of radiation with atomic systems Chapter 9 Spin and Pauli Equation

In Section 2.4, the Stern-Gerlach experiment is described. Electrons exhibit a phenomenon similar to the polarization of light, which is called spin for historical reasons. In Chapter 4.3, the effect of two spin states of electrons is shown. Because each electron wavefunction has two spin versions, while applying the Pauli exclusion principle, each spatial wavefunction must to be counted twice. However, the nature of the electron spin was not presented. In this Appendix, the nature of the electron spin as its most significant experimental evidence, the Stern-Gerlach experiment, is presented.

9.1 The Pauli equation

Although electron spin was discovered by the Stern-Gerlach experimnet and the spitting of atomic spectral lines in early 1920s, and a phenomenological theory was developed by Pauli in 1927 by adding a term on the Schr¨odinger equation, the true origin of the electron spin was elucidated after Dirac discovered the relativistic wave equation.

9.1.1 The Dirac equation and the electron spin In 1928, while pondering a wave equation to satisfy the requirement of special relativity, Dirac found that two components are not enough. The wavefunction must have four complex components, or eight real compo- nents. However, the meaning of that many components were understood only several years later. In 1930, Dirac published a paper A Theory of Electrons and Positrons, elucidated the meaning of the Dirac equation. The four components are grouped into two pairs. One pair represents the electron, and another pair represents a now elementary particle positron, having the same mass as the electron, but the electrical charge is positive. The two components in each pair represent the spin of the electron, or the positron. The energy of the two spin components split under the in fluence of magnetic field. Dirac obtained analytic solutions of his relativistic wave equation for the hydrogen atom, and received very accurate energy levels including fine struc- tures. In 1932, Carl David Anderson verified the existence of position with definitive experiments. In the entire history of science, the Dirac equation is 176 Spin and Pauli Equation among the most accurate and most beautiful ones. Furthermore, when the speed of electrons is much smaller than the speed of light, the Dirac equa- tion reduces to the Pauli equation, which is Schr¨odinger equation including the electron spin.

9.1.2 The Real Pauli Matrices In the complex Pauli equation, there are three 2×2 complex matrices repre- senting the spin. In the real Pauli equation, Durand introduced the following 4×4 real matrices:1 ⎡ ⎤ ⎡ ⎤ 1 1 ⎢ 1 ⎥ ⎢ 1 ⎥ σ˘ = ⎢ ⎥ , σ˘ = ⎢ ⎥ , (9.1) x ⎣ 1 ⎦ y ⎣ −1 ⎦ 1 −1 ⎡ ⎤ ⎡ ⎤ 1 1 ⎢ −1 ⎥ ⎢ 1 ⎥ σ˘ = ⎢ ⎥ , 1=˘ ⎢ ⎥ , (9.2) z ⎣ 1 ⎦ ⎣ 1 ⎦ −1 1 and a matrix equivalent to the imaginary unit ⎡ ⎤ −1 ⎢ −1 ⎥ ˘i = ⎢ ⎥ (9.3) ⎣ 1 ⎦ 1 with the property ˘i2 = −1˘. (9.4) Notethatherethematricesare4×4.

9.1.3 Pauli equation in a magnetic field The general Pauli equation with arbitrary external fields is rather compli- cated. To study the Stern-Gerlach experiment, we need a Pauli equation in a magnetic field B. We follow the formulation of such a Pauli equation derived by Durand from Dirac equation. The Pauli spinor has four real components: ⎡ ⎤ Ψ↑ ⎢ ⎥ ⎢ Ψ↓ ⎥ u = ⎣ ⎦ (9.5) Φ↑ Φ↓

Here Ψ↑ and Ψ↓ are two spin components, spin up and spin down, split from a single component Ψ in Eqs. 6.12 and 6.13. Furthermore, Φ↑ and Φ↓ are

1See page 291 of E.´ Durand, M´ecanique Quantique, Volume II, Masson, 1970. 9.2 The Stern-Gerlach experiment 177 two spin components, spin up and spin down, split from a single component Φ in Eq. 8.8. Following Durand, the real Pauli equaiton is 2 ∂u 2 ˘i = − ∇ − μBB · σ˘ u, (9.6) ∂t 2me where the Bohr magneton is defined as

e μB = . (9.7) 2mec

In the absence of magnetic field, Eq. 9.6 is identical to Eqs. 6.12 and 6.13, where the spin-up and spin-down components are identical. With the presence of magnetic field, the spin-up component and the spin-down component behaves differently. To analyze the Stern-Gerlach experiment, here we set the magnetic field along the z direction with a gradient,

∂B B = B + z. (9.8) z 0 ∂z Equation 9.6 is expanded to  ∂Φ↑ 2 ∂B = ∇ Ψ↑ + μB B0 + z Ψ↑. (9.9) ∂t 2me ∂z  ∂Φ↓ 2 ∂B = ∇ Ψ↓ − μB B0 + z Ψ↓. (9.10) ∂t 2me ∂z  ∂Ψ↑ 2 ∂B = − ∇ Φ↑ + μB B0 + z Φ↑. (9.11) ∂t 2me ∂z  ∂Ψ↓ 2 ∂B = − ∇ Φ↓ − μB B0 + z Φ↓. (9.12) ∂t 2me ∂z

9.2 The Stern-Gerlach experiment the Stern-Gerlach experiment is described in Section 2.4, and the appa- ratus is shown in Fig. 2.14. A beam of silver atoms is generated by the high-temperature oven and heading the x-direction. The actual subject of research, however, is the valence electrons of the silver atoms. The term with magnetic field deflects the beam according to the sign before μB. It is clear that Eqs. 9.9 and 9.11 are coupled. On the other hand, Eqs. 9.10 and 9.12 are coupled. The two pairs do not mix with each other. Therefore, one can look at the two pairs separately. 178 Spin and Pauli Equation

9.2.1 Similarity to polarization of light The Stern-Gerlach experiment should be understood from a wave point of view. As any electron beam with spin must be understood as a wave with four components, in a non-uniform magnetic field, the spin-up and spin- down components are two different sets of components. It is similar to the electromagnetic wave in a non-isotropic medium, which also has four components. By denoting the speed of light in two perpendicular directions as c1 and c2, for the pair of Ex and By,wehave ∂E ∂B x = −c y , (9.13) ∂t 1 ∂z ∂B ∂E y = −c x . (9.14) ∂t 1 ∂z And for the pair of Ey and Bx,wehave ∂E ∂B y = c x , (9.15) ∂t 2 ∂z ∂B ∂E x = c y . (9.16) ∂t 2 ∂z

9.2.2 Analysis based on Ehrenfest’s theorem An analysis of the Stern-Gerlach experiment using Ehrenfest’s theorem is intuitive and accurate.2 The beam emitted through the slot of the oven can be fairly accurately described as consists of wave packets of the width of the slot, typically 1 mm. Assuming the temperature of the oven is 2000 ◦C, or 2273 K. The atomic weight of silver is 107 a.u. The rms velocity of the silver atoms is 3k T v = B ≈ 500 m/s. (9.17) M The Broglie wavelength for electrons is 1.44 μm, about three orders of mag- nitude smatter than the size of the wave packet. According to the analysis of Section 6.2.2, the beam can be considered as with a constant width. With- out the non-uniform magnetic field, the beam will move with a constant velocity. The term with a non-uniform magnetic field is a potential. The force for the spin-up components is d ∂B ∂B F = μ B + z = μ . (9.18) z dz B 0 ∂z B ∂z And the force for the spin-down components is ∂B F = −μ . (9.19) z B ∂z 2See D. Pratt, A modern analysis of the Stern-Gerlach experiment, American Journal of Physics, 60, 306 (1992). 9.2 The Stern-Gerlach experiment 179

The center of mass of an electron wave packet moves like a Newtonian material particle. For the spin-up component, the trajectory is μ ∂B r = vte + B t2e . (9.20) x 2M ∂z z For the spin-down component, the trajectory is μ ∂B r = vte − B t2e . (9.21) x 2M ∂z z

Here ex and ez are unit vectors in x and z directions, respectively. 180 Spin and Pauli Equation Appendix A Units and Physical Constants

In applied science and engineering, the most widely used unit system is the SI unit (Syst`eme International d’unit´es), or the MKS unit system. Nev- ertheless, in quantum mechnaics, the natural unit system, or the atomic unit system, where several universal physical constants such as the electron mass, reduced Planck’s constant afre set to be 1, is unavoidable. On the other hand, up to late 20th century, the cgs unit system has been used by phjysicists exclusively, because Maxwell’s equations have the cleanest form, and the Schr¨odinger equation has the cleanest form as well. In this book, the cgs unit system and the natural unit systems are used. However, it is easy to convert to SI system.

Table A.1: Units and physical constants

Quantity Symbol Unit and value

speed of light in vacuum c 2.998 ×1010 cm s−1 − reduced Planck’s constant  1.0546 ×10−16 gcm2 s 1 Boltzmann’s constant k 1.3807 ×10−16 gcm2 s−2 K−1 electron charge e 4.8032 ×10−10 g1/2 cm3/2 s−1 −28 electron mass me 9.1094 ×10 g force F dyne = g cm s−2 energy  erg = g cm2 s−2 energy density w gcm−1 s−2 electric field intensity E g1/2 cm−1/2 s−1 magnetic field intensity B gauss = g1/2 cm−1/2 s−1 vector potential A g1/2 cm1/2 s−1 electric charge density ρ g1/2 cm−3/2 s−1 electric current density j g1/2 cm−3/2 s−2 182 Units and Physical Constants Appendix B Vector Analysis

For handling elements in a three-dimensioanl field, especially electromag- netics, vector analysis is a convenient matiematical tool. The central com- ponent is a differential operator usually called a del operator, defined as ∂ ∂ ∂ ∇ = i + j + k , (B.1) ∂x ∂y ∂z where i, j,andk are unit vectors in the x, y,andz directions, respectively. There are three commonly used operations. The gradient of a scalar function ϕ(r), a vector function, is defined as ∂ϕ ∂ϕ ∂ϕ ∇ϕ ≡ i + j + k . (B.2) ∂x ∂y ∂z The divergence of a vector function A(r), a scalar function, is defined as

∂Ax ∂Ay ∂Az ∇ • A ≡ + + . (B.3) ∂x ∂y ∂z And the curl of a vector function A(r), a vector function, is defined as ijk ∂ ∂ ∂ ∇×A ≡ . (B.4) ∂x ∂y ∂z Ax Ay Az The dot product of two del operators is a Laplacian,

∂2 ∂2 ∂2 ∇2 ≡∇• ∇≡ + + . (B.5) ∂x2 ∂y2 ∂z2 There a series of identitites of interest. Here are some examples:

∇(ϕ + ψ) ≡∇ϕ + ∇ψ. (B.6)

∇ • (ϕA) ≡ ϕ(∇ • A)+A∇ • (∇ϕ). (B.7)

∇×(ϕA) ≡ ϕ(∇×A)+(∇ϕ) × A. (B.8) 184 Vector Analysis

∇×(A × B) ≡ A(∇ • B) − B(∇ • A)+B • ∇)A − A • ∇)B. (B.9)

∇ • (A × B) ≡ B • (∇×A) − A • (∇×B). (B.10)

∇ • (∇×A) ≡ 0. (B.11)

∇×(∇ϕ) ≡ 0. (B.12)

∇×(∇×A) ≡∇(∇ • A) −∇2A. (B.13) Problems

Problem B.1. Prove Eq. B.9. Problem B.2. Prove Eq. B.10. Problem B.3. Prove Eq. B.13. Appendix C Bessel Functions

In spherical coordinates, the solutions of the Helmholtz equation for a spher- ical acoustic resonator and the solutions of Schr¨odinger equation in vacuum reduce to an ordinary diffrerential equation, d df (z) z2 =[n(n +1)± z2] f(z). (C.1) dz dz wherethe+signisfortheSchr¨odinger equation in vacuum, and − sign is for the spherical acoustic resonator. In the literature, the solutions are re- ferred to as the spherical Bessel functions and the spherical modified Bessel functions. Although labeled as special functions, those functions are ac- tually elementary functions, that is, simple combinations of trigonimical, exponential functions and power functions. Letusfirstlookatthecaseofn = 0. By making a substitution

g(z) f(z)= , (C.2) z Eqnation C.1 becomes d 1 dg(z) 1 z2 − g(z) dz z dz z2 d dg(z) dg(z) = z − (C.3) dz dz dz d2g(z) = z = ±zg(z) . dz2 for the case of acoustic resonator, with a minus sign, the solutions for g(z) are sin(z) and cos(z). Therefore, the solutions of Eq. C.1 are

sin z j (z)= , (C.4) 0 z and cos z y (z)= . (C.5) 0 z

Because y0(z) approaches infinity near z = 0, it is not a proper solution for the resonator. Only j0(z) is valid. 186 Bessel Functions

Using mathematical induction, one can prove that the solution for arbi- trary n is 1 d sin z j (z)=zn − . (C.6) n z dz z Assuming that for the case of (n +1), 1 d j (z)=−z j (z). (C.7) n+1 z dz n Then, the solution is valid for n,thenfor The radial part of the Schr¨odinger equation in spherical coordinates is This is the modified Bessel equation of order ν = n +1/2. The solutions of Eq. C.1 are modified Bessel functions of the first kind, which is defined through the Bessel function

−2νπi Iν (z) ≡ e Jν (iz), (C.8) and the modified Bessel function of the second kind, which is defined through (1) the Hankel function Hν (x)as π K (z) ≡ e2(ν+1)πiH(1)(iz). (C.9) ν 2 ν The solutions of Eq. ?? are defined through the modified Bessel func- tions. Those are spherical modified Bessel functions of the first kind π i (z)= I (z), (C.10) n 2z n+1/2 and of the second kind, 2 k (z)= K (z); (C.11) n πz n+1/2 see Fig. C.1. These two functions are linearly independent. The function in(z) is the only solution of Eq. ?? that is regular at z =0, and the function kn(z) is the only solution which is regular at z = ∞. These so-called spe- cial functions are actually elementary functions, with the following general expression d n sinh z i (z)=zn , (C.12) n zdz z d n exp(−z) k (z)=(−1)n zn . (C.13) n zdz z The first three pairs of these functions are sinh z i (z)= (C.14) 0 z 187

Fig. C.1. Spherical modified Bessel functions

sinh z cosh z i (z)= + (C.15) 1 z2 z 3 1 3 i (z)= + sinh z − cosh z (C.16) 2 z3 z z2 1 k (z)= e−z (C.17) 0 z 1 1 k (z)= + e−z (C.18) 1 z z2 1 3 3 k (z)= + + e−z. (C.19) 2 z z2 z3 Actually, by starting with Eqs C.12 and C.13 as the definitions, all the properties of the spherical modified Bessel functions can be obtained, without tracing back to the formal definition, Eqs C.10 and C.11. Following the standard series expansion of Bessel functions, the power- series expansion of the function in(z)nearz = 0 has the following form:

∞ 1 z2 k i (z)=zn . (C.20) n k!(2n +2k +1)!! 2 k=0 The first term is proportional to zn:

zn i (z) ≈ . (C.21) n (2n +1)!! 188 Bessel Functions

An in(z)ofevenorderonlyhasevenpowersofz, and an in(z) of odd order only has odd powers of z. These properties are essential in the derivation of the tunneling matrix elements. On the other hand, following the standard asymptotic expansion of Bessel functions, it is easy to prove that the functions kn(z) have the fol- lowing exact general expression,

e−z n (n + k)! 1 k (z)= . (C.22) n z k!(n − k)! (2z)k k=0 Also, following the standard recursion relations of Bessel functions, the recursion relations for both in and kn are:

(2n +1)fn(z)=zfn−1(z) − zfn+1(z), (C.23)

d n f (z)=f (z)+ f (z). (C.24) dz n n+1 z n Finally, the Wronskian of the pair is

1 W (i (z),k(z)) ≡ i (z) k (z) − i (z) k (z)=− . (C.25) n n n n n n z2 Appendix D Statistics of Particles

Statistical physics was developed in late 19th century, especially by the works of Ludwig Boltzmann. The original targets were classical particles, molecules and atoms that follow Newtonian mechanics. The contents and results are well understood and well verified experimentally. In early 20th century, after the advence of quantum mechanics, some new facts were found. First, different types of particles have different statistical behaviors. For classical particles, each state can have many occupations. For Fermions, each state is either occupied or unoccupied. For photons, or Bosons, each state can have any number of occupations. Especially, for photones, the total number of particles is unlimited. First, all paricles . In the microscopic treatment of matter, the particles, for example, molecules and electrons, are distributed in a system of energy levels. How the particles distribute among the energy levels determines the behavior of the system in a significant way. Regarding to applications with solar energy, an understanding of Maxwell–Boltzmann statistics and Fermi–Dirac statistics is essential. Maxwell–Boltzmann statistics is valid for asystemofdistinguishable particles in a system of energy levels allowing unlimited occupancy; whereas Fermi–Dirac statistics is valid for a system of indistinguishable particles in a system of energy levels allowing limited occupancy, that is, systems satisfying the Pauli exclusion principle. The starting point of the derivation is the Boltzmann expression of en- tropy,

S = kB ln W, (D.1) where kB is Boltzmann’s constant and W is the total number of configura- tions of the system. Consider a situation of N particles in a system consisting of a series of energy levels. The occupancy number of the ith level is Ni. The energy of the ith level is Ei, and the total energy of the system is E. Wehavethe conditions N = i Ni, (D.2) E = i Ni Ei. 190 Statistics of Particles

The condition of equilibrium is that the entropy, Eq. D.1, reaches maximum under the two constraints in Eq. D.2. Intuitively, the more uniform the distribution, the greater the random- ness, or the greater the entropy. However, the condition of constant total energy adds another condition: It is preferable to have more particles in the levels of lower energy and less particles in the levels of higher energy. The problem can be resolved using Fermat’s theorem together with the La- grange multiplier method. Introducing two Lagrange multipliers α and β, the condition of equilibrium is

∂ S + α N − N + β E − N E =0. (D.3) ∂N i i i i i i The distribution can be found by combining the solution of Eq. D.3 with the two constraints in Eqs. D.1 and D.2. During the calculation, one needs an approximate value of the derivative of ln N! for large N. This can be simply taken as

d ln N! ≈ ln N! − ln(N − 1)! ≈ ln N. (D.4) dN

D.1 Maxwell–Boltzmann Statistics

In the case of Maxwell–Boltzmann statistics, applicable to classical atomic systems, the particles are distinguishable, and the number of particles per energy level is unlimited. The number of possible configurations W can be determined as follows. Suppose that N particles are placed into containers with occupation numbers N1,N2, ... Ni, ... and so on. Consider now the number of ways to place N1 particles into container 1. First, choose one of the N particles for the first position in container 1, that is, N ways. Next, choose one of the remaining N − 1 particles for the second position in container 1. This generates N − 1 ways. Altogether there are N(N − 1) ways. By continuing the process, the total number of ways is N(N − 1)(N − 2)...(N − N1 − 1). However, the placement of particles in the container is arbitrary. Therefore, there is a N1!-fold redundancy. The net ways of placement are given as

N(N − 1)(N − 2)...(N − N1 − 1) N! W1 = = . (D.5) N1! (N − N1)! N1!

Similarly, the number of ways of placing the remaining N − N1 particles into container 2 generates a factor D.1 Maxwell–Boltzmann Statistics 191

(N − N1)! W2 = . (D.6) (N − N1 − N2)! N2! Similarly,

(N − N1 − N2)! W3 = . (D.7) (N − N1 − N2 − N3)! N3! Continuing the process further, finally we find N! W = W1 W2 ...Wi... = . (D.8) i Ni! The entropy is S = kB ln W = kB ln N! − ln Ni! . (D.9) i The condition of thermal equilibrium gives ∂ S + α N − N + β E − N E =0. (D.10) ∂N i i i i i i The result is

kB ln Ni = −α − βEi, (D.11) The meaning of the parameter β can be interpreted based on thermody- namics. Because the system is under the condition of constant temperature and constant volume, according to Eq. ??,wehave

dE = TdS. (D.12) By treating S and E as variables, comparing Eq. D.3 and Eq. D.12, we find, heuristically, 1 β = . (D.13) T The constant α can be determined by the condition that the total number of particles is N. Equation D.11 can be rewritten as N −Ei ni = exp , (D.14) Z kBT where Z is a constant determined by the condition N −E N = exp i = N, (D.15) i Z k T i i B 192 Statistics of Particles in other words, −E Z = exp i . (D.16) kBT I

By introducing a probability of the ith energy level, pi = Ni/N ,Eq.D.16 can be written as 1 −Ei pi = exp . (D.17) Z kBT Obviously, the sum of all probabilities is 1, pi =1. (D.18) i

D.2 Fermi–Dirac Statistics

Electrons are fermions obeying the Pauli exclusion principle. Each state can only be occupied by one electron. The electrons satisfies Fermi–Dirac statistics. For each energy value, there can be multiple states. For example, each electron can have two spin states that have the same energy level. Let the degeneracy, that is, the number of states at energy Ei,begi.Thenumber of electrons staying at that energy level, Ni, should not exceed gi.The number of different ways of occupation in the gi states is

gi! Wi = , (D.19) (gi − Ni)! Ni! which is a special case of Eq. D.5. Following Eq. D.3, we obtain

kB [ln(gi − Ni) − ln Ni]=α + βEi, (D.20) or g N = i . (D.21) i α + βE exp i +1 kB

Using Eq. D.13 and introducing the probability pi = Ni/gi, 1 p = . (D.22) i E − E exp i F +1 kBT

Equation D.22 is called the Fermi function.AttheFermi level EF ,the probability is 1/2. Apparently, D.3 Bose-Einstein Statistics 193

pi → 1,Ei  EF ; (D.23) pi → 0,Ei  EF .

At low temperature, where kBT  EF , the Fermi–Dirac statistics becomes

pi =1,Ei EF .

At high temperatures, or at high energy levels (Ei − EF )/kBT  1, the Fermi–Dirac statistics reduces to Maxwell–Boltzmann statistics, −Ei pi ∝ exp . (D.25) kBT

D.3 Bose-Einstein Statistics

Photons are indistinguishable, but each state can have unlimited number of occupations. For each energy value, there can be multiple states. For example, each electron can have two spin states that have the same energy level. Let the degeneracy, that is, the number of states at energy Ei,begi.Thenumber of electrons staying at that energy level, Ni, should not exceed gi.The number of different ways of occupation in the gi states is

(gi − 1+Ni)! Wi = , (D.26) (gi − 1)! Ni! which is a special case of Eq. D.5. Following Eq. D.3, we obtain

kB [ln(gi − 1+Ni) − ln Ni]=βEi, (D.27) or g − 1 N = i . (D.28) i βE exp i − 1 kB

Using Eq. D.13 and introducing the probability pi = Ni/gi, 1 p = . (D.29) i E exp i − 1 kBT Equation D.29 is called the Planck function. 194 Statistics of Particles Appendix E Real Matrix Formulation

As we have presented in Chapter 6, the time-dependent wavefunction can be expressed as a two-dimensional vector field. Field quantities with mul- tiple components is very common in physics. For example, in Maxwell’s electromagnetics, both electrical field and magnetic field have three com- ponents. And to represent each field as a three-component vector using a single bold-faced letter is a common practice, see Section 1.5.3. Similarly, the two-component wavefunction can be represented as a single vector field of two dimensions, and the two Schr¨odinger equations can be combined into a single vector equation.

E.1 Schr¨odinger equation in free space

According to Durand, a plane wave in free space is represented by a pair of real wavefunctions, Ψ=a sin(k · r − ωt + φ)(E.1) and Φ=a cos(k · r − ωt + φ), (E.2) satisfying a pair of real Schr¨odinger equations, ∂Ψ 2  = − ∇2Φ(E.3) ∂t 2me and ∂Φ 2  = ∇2Ψ. (E.4) ∂t 2me Define a wavefunction as a two-dimensional vector a sin(k · r − ωt + φ) u = (E.5) a cos(k · r − ωt + φ) and two 2×2 matrices1 10 0 −1 1=˘ , ˘i = . (E.6) 01 10

1Here the concepts of E.´ Durand, M´ecanique Quantique, Volume I, Masson et Cie, 1970, pages V to VIII, pages 64-67, and page 93, are followed. The notation of adding an accent breve on a letter to mark a real matrix follows E. C. G. Strueckberg, Quantum Theory in Real Hilbert Space, Helvetica Physics Acta, 33, 727 (1960). 196 Real Matrix Formulation

There is an obvious relation between those matrices,

˘i2 = −1˘. (E.7)

The two Schr¨odinger equations, Eqs. E.3 and E.4, are combined, ∂u 2 ˘i = − ∇2u. (E.8) ∂t 2me Formally, it is similar to the conventional complex Schr¨odinger equation. Nevertheless, the meaning is different. Here the wavefunction is a two- component real vector, and all operators are 2×2 real matrices. According to de Broglie relation, the components of the momentum are

px = kx,py = ky,pz = kz. (E.9) Using Eqs. E.5 and E.6, it is easy to prove that ∂u ∂u ∂u p u = −˘i ,pu = −˘i ,pu = −˘i . (E.10) x ∂x y ∂y z ∂z Three momentum operators acting on the two-dimensional wavefunction can be defined as ∂ ∂ ∂ pˆ = −˘i , pˆ = −˘i , pˆ = −˘i . (E.11) x ∂x y ∂y z ∂z Or, in vector form, pˆ = −˘i∇. (E.12) The x, y, z components can also be labeled as 1, 2, and 3, respectively. Multiplying the wavefunction by the coordinates x, y, or z are also op- erators, defined as

xˆu = x1˘u, yˆu = y1˘u, zˆu = z1˘u. (E.13)

A straightforward calculation shows that the coordinate operators and the momentum operators are not commutable. They follow the following commutation relations, [ˆpj, xˆk]=˘iδjk, (E.14) where j and k run 1, 2, and 3, respectively.

E.2 Electron in a field

Using the momentum operator, the Schr¨odinger equation in vacuum , Eq. E.8, is of the form ∂u 1 ˘i = pˆ2u. (E.15) ∂t 2me E.3 Angular Momentum 197

According to the general rules in mechanics, the Schr¨odinger equation of an electron in an external electromagnetic field, represented by the vector potential A and scalar potential V is ∂u 1 e 2 ˘i = pˆ − A + V u. (E.16) ∂t 2me c With a potential V , the general time-dependent is ∂u 2 ˘i = − ∇2 + V u. (E.17) ∂t 2me

E.3 Angular Momentum

An important application of the operator formulation is the treatment of angular momentum. Many important properties of angular momentum can be obtained from the algebra of the operators. The definition of angular momentum in quantum mechanics is similar to that in classical mechanics, except that the order of coordinate r and momentum p is fixed,

mˆ = ˆr × pˆ, (E.18) or in tensor notation,

mˆ i = ijk xˆj pˆk, (E.19) where the unit axial tensor ijk is a tensor antisymmetric to all three suffixes, with 123 = 1, and changes sign by exchanging two identical indices, where the value is zero. A sum over j and k is implied. To simplify notation, a dimensionless version of the angular momentum is defined,

ˆ −1 −1 li =  mˆ i =  ijk xˆj pˆk. (E.20) The commutation relations can be obtained from the commutation relations of momentum and coordinate,

[ˆpj, xˆk]=˘iδjk, (E.21) which are ˆ ˆ ˆ ˆ ˆ lxly − lylx =˘ilz, ˆ ˆ ˆ ˆ ˆ lylz − lzly =˘ilx, (E.22) ˆ ˆ ˆ ˆ ˆ lzlx − lxlz =˘ily, 198 Real Matrix Formulation or in tensor form,

ˆ ˆ ˆ [li, lj]=˘iijklk. (E.23) From the components we can form an operator as the square of the modulus of the angular momentum vector,

ˆ2 ˆ2 ˆ2 ˆ2 l = lx + ly + lz. (E.24) As a result of commutation relations E.22, ˆl2 commutes with a component, for example,

ˆ2 ˆ [l , lz]=0. (E.25) Therefore, we can find states |l, m which are simultaneously eigenstate of ˆ2 ˆ l and lz,

ˆl2|l, m = λ|l, m , (E.26) ˆ lz|l, m = μ|l, m . In the following, we introduce a pair of operators which are similar to the creation and annihilation operators in the problem of harmonic oscillators, ˆ ˆ ˆ l+ = lx + ˘ily, (E.27) ˆ ˆ ˆ l− = lx − ˘ily, which have the commutation relations

ˆ ˆ ˆ [lz, l+]=l+, (E.28)

ˆ ˆ ˆ [lz, l−]=−l− (E.29) and the identity

ˆ2 ˆ ˆ ˆ2 ˆ l = l−l+ + lz + lz. (E.30) Using arguments similar to those in the harmonic oscillator problem, we find ˆ ˆ ˆ that l+|l, m is also an eigenstate of lz with eigenvalue μ + 1 and l−|l, m is ˆ an eigenstate of lz with eigenvalue μ − 1. Applying those operators many times, we have

n n ˆ ˆ ˆ lz l+ |l, m =(μ + n) l+ |l, m (E.31) and

n n ˆ ˆ ˆ lz l− |l, m =(μ − n) l− |l, m . (E.32) E.4 Hydrogen Atom 199

ˆ However, because of Eq. E.24, the eigenvalue of lz cannot grow indefinitely. Its absolute value must have a maximum. Because of the symmetry, the ab- solute value of the positive maximum and the absolute value of the negative maximum must be equal. Also, because the difference between the positive maximum and the negative maximum must be an integer, both must be one-half of an integer. Assigning this number as l, the possible eigenvalues ˆ of lz, often assigned as m,mustbe

m = −l, −l +1, −l +2, ...l − 2,l− 1,l. (E.33) with

1 3 5 l =0, 2 , 1, 2 , 2, 2 , 3 ... (E.34) ˆ It is obvious that l, the maximum absolute value of lz, is also a quantum number for the total angular momentum ˆl2. In the following, we will find the eigenvalue of the operator ˆl2. Because m = l is the maximum eigenvalue ˆ of lz, one must have

ˆ l+|l, l =0. (E.35) In view of Eq. E.30, ˆ2| ˆ ˆ ˆ2 ˆ | | l l, l = l−l+ + lz + lz l, l = l(l +1)l, l . (E.36) Because l and m are independent, finally we have the eigenvalues and eigen- states for the angular momentum operator,

mˆ 2 |l, m = l(l +1) |l, m , (E.37) mˆ z |l, m = m |l, m .

E.4 Hydrogen Atom

In classical physics, if the mass of the proton is large, the Hamiltonian of the hydrogen atom is

p2 κ H = − , (E.38) 2me r 2 where κ = e /4π0; see Chapter ??. Because of the spherical symmetry of the problem, the angular momentum is conserved. The vector of angular momentum is always perpendicular to the plane of motion,

L = r × p =const. (E.39) 200 Real Matrix Formulation

In addition to angular momentum, there is another conserved vector related to the fixed orientation of the long axis of the orbital, which is a result of the Coulomb interaction. It is called the Runge–Lenz vector after its discoverers, κr 1 A = − (p × L)=const. (E.40) r me Because the angular momentum vector L is perpendicular to the orbital plane and the Runge–Lenz vector is in the plane, the two vectors are per- pendicular,

L · A =0. (E.41) In 1926, a year before Schr¨odinger discovered his differential equation and solved the hydrogen atom problem, Wolfgang Pauli solved the eigen- value problem by using the algebraic method of Werner Heisenberg based on the two constants of motion. Pauli’s treatment is as follows: In quantum mechanics, the Hamiltonian is an operator,

pˆ2 κ Hˆ = − . (E.42) 2me r The angular momentum, also an operator,

Lˆ = ˆr × pˆ, (E.43) satisfies the commutation relation

[Lˆi, Lˆj]=˘iijkLˆk. (E.44) It also commutes with the Hamiltonian E.42, thus a constant of motion. Because pˆ × Lˆ is not Hermitian, Pauli defined a Hermitian operator equivalent to the classical Runge–Lenz vector κˆr 1 Aˆ = − pˆ × Lˆ − Lˆ × pˆ . (E.45) r 2me With rather tedious but straightforward algebra, this operator is shown to commute with the quantum-mechanical Hamiltonian, equation E.42. Simi- lar algebra results in the commutation relations,

[Lˆi, Aˆj]=˘iijkAˆk, (E.46) −2Hˆ [Aˆi, Aˆj]=˘i ijkLˆk, (E.47) me and the relation E.4 Hydrogen Atom 201

2Hˆ Aˆ 2 = Lˆ 2 + 2 + κ2. (E.48) me Because Aˆ commutes with Hˆ , and there are common eigenstates, a reduced vector can be defined, m 1/2 Bˆ = − e Aˆ . (E.49) 2Hˆ The commutation relations Eqs E.46 and E.47 are reduced to

[Lˆi, Bˆj]=˘iijkBˆk, (E.50)

[Bˆi, Bˆj]=˘iijkLˆk. (E.51) We introduce a pair of operators ˆ 1 ˆ ˆ J = 2 L + B , (E.52) ˆ 1 ˆ − ˆ K = 2 L B . The commutation relations are reduced to those of a pair of independent angular momenta

[Jˆi, Jˆj]=˘iijkJˆk, (E.53)

[Kˆi, Kˆj]=˘iijkKˆk, (E.54)

[Jˆi, Kˆj]=0. (E.55) Using Eqs. E.49 and E.52, Eq. E.48 becomes 1 2 1 ˆ2 ˆ 2 2 − meκ =2 J + K +  . (E.56) 2 Hˆ From Eq. E.52, J2 = K2, and the eigenvalues of Jˆ2 and Kˆ 2 are identical. According to the theory of angular momentum (Eq. E.37), both are j(j+1). Therefore, the solution is

1 2 1 2 2 2 2 − meκ =4j(j +1) +  =(2j +1)  . (E.57) 2 En From Eq. E.34, n =2j + 1 can be any positive integer. Finally, the energy eigenvalues of the hydrogen atom are

m κ2 E = − e . (E.58) n 22n2