Quantum Mechanics for EngineersLeon van Dommelen09/13/10 Version 5.11 alphaCopyrightCopyright 2004, 2007, 2008, 2010 and on, Leon van Dommelen. You areallowed to copy or print out this work for your personal use. You are allowed toattach additional notes, corrections, and additions, as long as they are clearlyidentied as not being part of the original document nor written by its author.Conversions to html of the pdf version of this document are stupid, sincethere is a much better native html version already available, so try not to do it.DedicationTo my parents, Piet and Rietje van Dommelen.iiiContentsPreface xxxiiiTo the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxivComments and Feedback . . . . . . . . . . . . . . . . . . . . . . . . xxxviI Basic Quantum Mechanics 11 Mathematical Prerequisites 31.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Functions as Vectors . . . . . . . . . . . . . . . . . . . . . . . 61.3 The Dot, oops, INNER Product . . . . . . . . . . . . . . . . . 81.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . 131.6 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . 151.7 Additional Points . . . . . . . . . . . . . . . . . . . . . . . . . 171.7.1 Dirac notation . . . . . . . . . . . . . . . . . . . . . . . 181.7.2 Additional independent variables . . . . . . . . . . . . . 182 Basic Ideas of Quantum Mechanics 192.1 The Revised Picture of Nature . . . . . . . . . . . . . . . . . . 212.2 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . 242.3 The Operators of Quantum Mechanics . . . . . . . . . . . . . . 252.4 The Orthodox Statistical Interpretation . . . . . . . . . . . . . 272.4.1 Only eigenvalues . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Statistical selection . . . . . . . . . . . . . . . . . . . . 292.5 A Particle Conned Inside a Pipe . . . . . . . . . . . . . . . . 302.5.1 The physical system . . . . . . . . . . . . . . . . . . . . 302.5.2 Mathematical notations . . . . . . . . . . . . . . . . . . 312.5.3 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 322.5.4 The Hamiltonian eigenvalue problem . . . . . . . . . . . 332.5.5 All solutions of the eigenvalue problem . . . . . . . . . . 33vvi CONTENTS2.5.6 Discussion of the energy values . . . . . . . . . . . . . . 372.5.7 Discussion of the eigenfunctions . . . . . . . . . . . . . 392.5.8 Three-dimensional solution . . . . . . . . . . . . . . . . 412.5.9 Quantum connement . . . . . . . . . . . . . . . . . . . 452.6 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 472.6.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 482.6.2 Solution using separation of variables . . . . . . . . . . 492.6.3 Discussion of the eigenvalues . . . . . . . . . . . . . . . 522.6.4 Discussion of the eigenfunctions . . . . . . . . . . . . . 542.6.5 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . 582.6.6 Non-eigenstates . . . . . . . . . . . . . . . . . . . . . . 603 Single-Particle Systems 633.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 643.1.1 Denition of angular momentum . . . . . . . . . . . . . 643.1.2 Angular momentum in an arbitrary direction . . . . . . 653.1.3 Square angular momentum . . . . . . . . . . . . . . . . 673.1.4 Angular momentum uncertainty . . . . . . . . . . . . . 713.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . 723.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 723.2.2 Solution using separation of variables . . . . . . . . . . 733.2.3 Discussion of the eigenvalues . . . . . . . . . . . . . . . 783.2.4 Discussion of the eigenfunctions . . . . . . . . . . . . . 813.3 Expectation Value and Standard Deviation . . . . . . . . . . . 863.3.1 Statistics of a die . . . . . . . . . . . . . . . . . . . . . 873.3.2 Statistics of quantum operators . . . . . . . . . . . . . . 883.3.3 Simplied expressions . . . . . . . . . . . . . . . . . . . 903.3.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . 913.4 The Commutator . . . . . . . . . . . . . . . . . . . . . . . . . 933.4.1 Commuting operators . . . . . . . . . . . . . . . . . . . 943.4.2 Noncommuting operators and their commutator . . . . 953.4.3 The Heisenberg uncertainty relationship . . . . . . . . . 963.4.4 Commutator reference [Reference] . . . . . . . . . . . . 973.5 The Hydrogen Molecular Ion . . . . . . . . . . . . . . . . . . . 1003.5.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 1013.5.2 Energy when fully dissociated . . . . . . . . . . . . . . . 1013.5.3 Energy when closer together . . . . . . . . . . . . . . . 1023.5.4 States that share the electron . . . . . . . . . . . . . . . 1033.5.5 Comparative energies of the states . . . . . . . . . . . . 1063.5.6 Variational approximation of the ground state . . . . . 1063.5.7 Comparison with the exact ground state . . . . . . . . . 108CONTENTS vii4 Multiple-Particle Systems 1114.1 Wave Function for Multiple Particles . . . . . . . . . . . . . . 1124.2 The Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . . 1144.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 1144.2.2 Initial approximation to the lowest energy state . . . . . 1154.2.3 The probability density . . . . . . . . . . . . . . . . . . 1174.2.4 States that share the electrons . . . . . . . . . . . . . . 1184.2.5 Variational approximation of the ground state . . . . . 1204.2.6 Comparison with the exact ground state . . . . . . . . . 1214.3 Two-State Systems . . . . . . . . . . . . . . . . . . . . . . . . 1224.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.5 Multiple-Particle Systems Including Spin . . . . . . . . . . . . 1294.5.1 Wave function for a single particle with spin . . . . . . 1294.5.2 Inner products including spin . . . . . . . . . . . . . . . 1314.5.3 Commutators including spin . . . . . . . . . . . . . . . 1324.5.4 Wave function for multiple particles with spin . . . . . . 1334.5.5 Example: the hydrogen molecule . . . . . . . . . . . . . 1364.5.6 Triplet and singlet states . . . . . . . . . . . . . . . . . 1364.6 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . 1384.7 Ways to Symmetrize the Wave Function . . . . . . . . . . . . . 1404.8 Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . 1464.9 Heavier Atoms [Descriptive] . . . . . . . . . . . . . . . . . . . 1504.9.1 The Hamiltonian eigenvalue problem . . . . . . . . . . . 1504.9.2 Approximate solution using separation of variables . . . 1514.9.3 Hydrogen and helium . . . . . . . . . . . . . . . . . . . 1534.9.4 Lithium to neon . . . . . . . . . . . . . . . . . . . . . . 1554.9.5 Sodium to argon . . . . . . . . . . . . . . . . . . . . . . 1594.9.6 Potassium to krypton . . . . . . . . . . . . . . . . . . . 1604.9.7 Full periodic table . . . . . . . . . . . . . . . . . . . . . 1614.10 Pauli Repulsion [Descriptive] . . . . . . . . . . . . . . . . . . . 1644.11 Chemical Bonds [Descriptive] . . . . . . . . . . . . . . . . . . . 1654.11.1 Covalent sigma bonds . . . . . . . . . . . . . . . . . . . 1654.11.2 Covalent pi bonds . . . . . . . . . . . . . . . . . . . . . 1664.11.3 Polar covalent bonds and hydrogen bonds . . . . . . . . 1674.11.4 Promotion and hybridization . . . . . . . . . . . . . . . 1694.11.5 Ionic bonds . . . . . . . . . . . . . . . . . . . . . . . . . 1724.11.6 Limitations of valence bond theory . . . . . . . . . . . . 1735 Macroscopic Systems 1755.1 Intro to Particles in a Box . . . . . . . . . . . . . . . . . . . . 1765.2 The Single-Particle States . . . . . . . . . . . . . . . . . . . . 1785.3 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . 180viii CONTENTS5.4 Ground State of a System of Bosons . . . . . . . . . . . . . . . 1835.5 About Temperature . . . . . . . . . . . . . . . . . . . . . . . . 1845.6 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . 1865.6.1 Rough explanation of the condensation . . . . . . . . . 1905.7 Bose-Einstein Distribution . . . . . . . . . . . . . . . . . . . . 1955.8 Blackbody Radiation . . . . . . . . . . . . . . . . . . . . . . . 1975.9 Ground State of a System of Electrons . . . . . . . . . . . . . 2015.10 Fermi Energy of the Free-Electron Gas . . . . . . . . . . . . . 2025.11 Degeneracy Pressure . . . . . . . . . . . . . . . . . . . . . . . 2045.12 Connement and the DOS . . . . . . . . . . . . . . . . . . . . 2065.13 Fermi-Dirac Distribution . . . . . . . . . . . . . . . . . . . . . 2105.14 Maxwell-Boltzmann Distribution . . . . . . . . . . . . . . . . . 2145.15 Thermionic Emission . . . . . . . . . . . . . . . . . . . . . . . 2175.16 Chemical Potential and Diusion . . . . . . . . . . . . . . . . 2195.17 Intro to the Periodic Box . . . . . . . . . . . . . . . . . . . . . 2215.18 Periodic Single-Particle States . . . . . . . . . . . . . . . . . . 2225.19 DOS for a Periodic Box . . . . . . . . . . . . . . . . . . . . . . 2245.20 Intro to Electrical Conduction . . . . . . . . . . . . . . . . . . 2255.21 Intro to Band Structure . . . . . . . . . . . . . . . . . . . . . . 2295.21.1 Metals and insulators . . . . . . . . . . . . . . . . . . . 2305.21.2 Typical metals and insulators . . . . . . . . . . . . . . . 2325.21.3 Semiconductors . . . . . . . . . . . . . . . . . . . . . . 2355.21.4 Semimetals . . . . . . . . . . . . . . . . . . . . . . . . . 2365.21.5 Electronic heat conduction . . . . . . . . . . . . . . . . 2375.21.6 Ionic conductivity . . . . . . . . . . . . . . . . . . . . . 2375.22 Electrons in Crystals . . . . . . . . . . . . . . . . . . . . . . . 2385.22.1 Bloch waves . . . . . . . . . . . . . . . . . . . . . . . . 2395.22.2 Example spectra . . . . . . . . . . . . . . . . . . . . . . 2405.22.3 Eective mass . . . . . . . . . . . . . . . . . . . . . . . 2425.22.4 Crystal momentum . . . . . . . . . . . . . . . . . . . . 2435.22.5 Three-dimensional crystals . . . . . . . . . . . . . . . . 2485.23 Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 2535.24 The P-N Junction . . . . . . . . . . . . . . . . . . . . . . . . . 2605.25 The Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . 2665.26 Zener and Avalanche Diodes . . . . . . . . . . . . . . . . . . . 2685.27 Optical Applications . . . . . . . . . . . . . . . . . . . . . . . 2695.27.1 Atomic spectra . . . . . . . . . . . . . . . . . . . . . . . 2695.27.2 Spectra of solids . . . . . . . . . . . . . . . . . . . . . . 2705.27.3 Band gap eects . . . . . . . . . . . . . . . . . . . . . . 2705.27.4 Eects of crystal imperfections . . . . . . . . . . . . . . 2715.27.5 Photoconductivity . . . . . . . . . . . . . . . . . . . . . 2715.27.6 Photovoltaic cells . . . . . . . . . . . . . . . . . . . . . 272CONTENTS ix5.27.7 Light-emitting diodes . . . . . . . . . . . . . . . . . . . 2725.28 Thermoelectric Applications . . . . . . . . . . . . . . . . . . . 2745.28.1 Peltier eect . . . . . . . . . . . . . . . . . . . . . . . . 2745.28.2 Seebeck eect . . . . . . . . . . . . . . . . . . . . . . . 2795.28.3 Thomson eect . . . . . . . . . . . . . . . . . . . . . . . 2846 Time Evolution 2876.1 The Schrodinger Equation . . . . . . . . . . . . . . . . . . . . 2886.1.1 Intro to the equation . . . . . . . . . . . . . . . . . . . 2896.1.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . 2906.1.3 Energy conservation [Descriptive] . . . . . . . . . . . . . 2936.1.4 Stationary states [Descriptive] . . . . . . . . . . . . . . 2946.1.5 Particle exchange [Descriptive] . . . . . . . . . . . . . . 2956.1.6 Energy-time uncertainty relation [Descriptive] . . . . . . 2976.1.7 Time variation of expectation values [Descriptive] . . . . 2986.1.8 Newtonian motion [Descriptive] . . . . . . . . . . . . . . 2996.1.9 The adiabatic approximation [Descriptive] . . . . . . . . 3016.1.10 Heisenberg picture [Descriptive] . . . . . . . . . . . . . 3026.2 Conservation Laws and Symmetries . . . . . . . . . . . . . . . 3046.3 Unsteady Perturbations of Systems . . . . . . . . . . . . . . . 3096.3.1 Schrodinger equation for a two-state system . . . . . . . 3106.3.2 Spontaneous and stimulated emission . . . . . . . . . . 3116.3.3 Eect of a single wave . . . . . . . . . . . . . . . . . . . 3136.3.4 Forbidden transitions . . . . . . . . . . . . . . . . . . . 3156.3.5 Selection rules . . . . . . . . . . . . . . . . . . . . . . . 3166.3.6 Angular momentum conservation . . . . . . . . . . . . . 3186.3.7 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.3.8 Absorption of a single weak wave . . . . . . . . . . . . . 3236.3.9 Absorption of incoherent radiation . . . . . . . . . . . . 3266.3.10 Spontaneous emission of radiation . . . . . . . . . . . . 3286.4 Position and Linear Momentum . . . . . . . . . . . . . . . . . 3306.4.1 The position eigenfunction . . . . . . . . . . . . . . . . 3316.4.2 The linear momentum eigenfunction . . . . . . . . . . . 3346.5 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3366.5.1 Solution of the Schrodinger equation. . . . . . . . . . . 3366.5.2 Component wave solutions . . . . . . . . . . . . . . . . 3386.5.3 Wave packets . . . . . . . . . . . . . . . . . . . . . . . . 3396.5.4 Group velocity . . . . . . . . . . . . . . . . . . . . . . . 3416.5.5 Electron motion through crystals . . . . . . . . . . . . . 3456.6 Almost Classical Motion [Descriptive] . . . . . . . . . . . . . . 3496.6.1 Motion through free space . . . . . . . . . . . . . . . . . 3496.6.2 Accelerated motion . . . . . . . . . . . . . . . . . . . . 349x CONTENTS6.6.3 Decelerated motion . . . . . . . . . . . . . . . . . . . . 3506.6.4 The harmonic oscillator . . . . . . . . . . . . . . . . . . 3516.7 WKB Theory of Nearly Classical Motion . . . . . . . . . . . . 3526.8 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3566.8.1 Partial reection . . . . . . . . . . . . . . . . . . . . . . 3576.8.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 3576.9 Reection and Transmission Coecients . . . . . . . . . . . . . 359II Gateway Topics 3637 Numerical Procedures 3657.1 The Variational Method . . . . . . . . . . . . . . . . . . . . . 3657.1.1 Basic variational statement . . . . . . . . . . . . . . . . 3657.1.2 Dierential form of the statement . . . . . . . . . . . . 3667.1.3 Example application using Lagrangian multipliers . . . 3677.2 The Born-Oppenheimer Approximation . . . . . . . . . . . . . 3697.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 3707.2.2 The basic Born-Oppenheimer approximation . . . . . . 3717.2.3 Going one better . . . . . . . . . . . . . . . . . . . . . . 3737.3 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . 3767.3.1 Wave function approximation . . . . . . . . . . . . . . . 3767.3.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 3827.3.3 The expectation value of energy . . . . . . . . . . . . . 3847.3.4 The canonical Hartree-Fock equations . . . . . . . . . . 3867.3.5 Additional points . . . . . . . . . . . . . . . . . . . . . 3888 Solids 3978.1 Molecular Solids [Descriptive] . . . . . . . . . . . . . . . . . . 3978.2 Ionic Solids [Descriptive] . . . . . . . . . . . . . . . . . . . . . 4008.3 Metals [Descriptive] . . . . . . . . . . . . . . . . . . . . . . . . 4048.3.1 Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . 4048.3.2 One-dimensional crystals . . . . . . . . . . . . . . . . . 4068.3.3 Wave functions of one-dimensional crystals . . . . . . . 4078.3.4 Analysis of the wave functions . . . . . . . . . . . . . . 4108.3.5 Floquet (Bloch) theory . . . . . . . . . . . . . . . . . . 4118.3.6 Fourier analysis . . . . . . . . . . . . . . . . . . . . . . 4128.3.7 The reciprocal lattice . . . . . . . . . . . . . . . . . . . 4138.3.8 The energy levels . . . . . . . . . . . . . . . . . . . . . 4148.3.9 Merging and splitting bands . . . . . . . . . . . . . . . 4158.3.10 Three-dimensional metals . . . . . . . . . . . . . . . . . 4178.4 Covalent Materials [Descriptive] . . . . . . . . . . . . . . . . . 421CONTENTS xi8.5 Free-Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . 4248.5.1 Lattice for the free electrons . . . . . . . . . . . . . . . 4258.5.2 Occupied states and Brillouin zones . . . . . . . . . . . 4278.6 Nearly-Free Electrons . . . . . . . . . . . . . . . . . . . . . . . 4318.6.1 Energy changes due to a weak lattice potential . . . . . 4338.6.2 Discussion of the energy changes . . . . . . . . . . . . . 4348.7 Additional Points [Descriptive] . . . . . . . . . . . . . . . . . . 4398.7.1 About ferromagnetism . . . . . . . . . . . . . . . . . . . 4408.7.2 X-ray diraction . . . . . . . . . . . . . . . . . . . . . . 4429 Basic and Quantum Thermodynamics 4519.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4529.2 Single-Particle and System Eigenfunctions . . . . . . . . . . . 4539.3 How Many System Eigenfunctions? . . . . . . . . . . . . . . . 4589.4 Particle-Energy Distribution Functions . . . . . . . . . . . . . 4639.5 The Canonical Probability Distribution . . . . . . . . . . . . . 4659.6 Low Temperature Behavior . . . . . . . . . . . . . . . . . . . . 4679.7 The Basic Thermodynamic Variables . . . . . . . . . . . . . . 4709.8 Intro to the Second Law . . . . . . . . . . . . . . . . . . . . . 4749.9 The Reversible Ideal . . . . . . . . . . . . . . . . . . . . . . . . 4759.10 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4819.11 The Big Lie of Distinguishable Particles . . . . . . . . . . . . . 4889.12 The New Variables . . . . . . . . . . . . . . . . . . . . . . . . 4889.13 Microscopic Meaning of the Variables . . . . . . . . . . . . . . 4959.14 Application to Particles in a Box . . . . . . . . . . . . . . . . . 4969.14.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . 4989.14.2 Fermions at low temperatures . . . . . . . . . . . . . . . 4999.14.3 A generalized ideal gas law . . . . . . . . . . . . . . . . 5019.14.4 The ideal gas . . . . . . . . . . . . . . . . . . . . . . . . 5019.14.5 Blackbody radiation . . . . . . . . . . . . . . . . . . . . 5039.14.6 The Debye model . . . . . . . . . . . . . . . . . . . . . 5059.15 Specic Heats . . . . . . . . . . . . . . . . . . . . . . . . . . . 50610 Electromagnetism 51310.1 All About Angular Momentum . . . . . . . . . . . . . . . . . . 51310.1.1 The fundamental commutation relations . . . . . . . . . 51410.1.2 Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . 51510.1.3 Possible values of angular momentum . . . . . . . . . . 51810.1.4 A warning about angular momentum . . . . . . . . . . 52010.1.5 Triplet and singlet states . . . . . . . . . . . . . . . . . 52110.1.6 Clebsch-Gordan coecients . . . . . . . . . . . . . . . . 52310.1.7 Some important results . . . . . . . . . . . . . . . . . . 527xii CONTENTS10.1.8 Momentum of partially lled shells . . . . . . . . . . . . 52910.1.9 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . 53210.1.10 General spin matrices . . . . . . . . . . . . . . . . . . . 53510.2 The Relativistic Dirac Equation . . . . . . . . . . . . . . . . . 53610.3 The Electromagnetic Hamiltonian . . . . . . . . . . . . . . . . 53810.4 Maxwells Equations [Descriptive] . . . . . . . . . . . . . . . . 54110.5 Example Static Electromagnetic Fields . . . . . . . . . . . . . 54810.5.1 Point charge at the origin . . . . . . . . . . . . . . . . . 54910.5.2 Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . 55410.5.3 Arbitrary charge distributions . . . . . . . . . . . . . . 55810.5.4 Solution of the Poisson equation . . . . . . . . . . . . . 56010.5.5 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . 56110.5.6 Principle of the electric motor . . . . . . . . . . . . . . 56310.6 Particles in Magnetic Fields . . . . . . . . . . . . . . . . . . . 56610.7 Stern-Gerlach Apparatus [Descriptive] . . . . . . . . . . . . . . 56910.8 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . 57010.8.1 Description of the method . . . . . . . . . . . . . . . . . 57010.8.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 57110.8.3 The unperturbed system . . . . . . . . . . . . . . . . . 57310.8.4 Eect of the perturbation . . . . . . . . . . . . . . . . . 57511 Nuclei [Unnished Draft] 57911.1 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . 58011.2 The Simplest Nuclei . . . . . . . . . . . . . . . . . . . . . . . . 58011.3 Overview of Nuclei . . . . . . . . . . . . . . . . . . . . . . . . 58211.4 Magic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 58811.5 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 58911.5.1 Decay rate . . . . . . . . . . . . . . . . . . . . . . . . . 58911.5.2 Other denitions . . . . . . . . . . . . . . . . . . . . . . 59011.6 Mass and energy . . . . . . . . . . . . . . . . . . . . . . . . . . 59111.7 Binding energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 59311.8 Nucleon separation energies . . . . . . . . . . . . . . . . . . . . 59511.9 Nuclear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 60011.10 Liquid drop model . . . . . . . . . . . . . . . . . . . . . . . . . 60711.10.1 Nuclear radius . . . . . . . . . . . . . . . . . . . . . . . 60711.10.2 von Weizsacker formula . . . . . . . . . . . . . . . . . . 60811.10.3 Explanation of the formula . . . . . . . . . . . . . . . . 60811.10.4 Accuracy of the formula . . . . . . . . . . . . . . . . . . 60911.11 Alpha Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61111.11.1 Decay mechanism . . . . . . . . . . . . . . . . . . . . . 61111.11.2 Comparison with data . . . . . . . . . . . . . . . . . . . 61411.11.3 Forbidden decays . . . . . . . . . . . . . . . . . . . . . . 616CONTENTS xiii11.11.4 Why alpha decay? . . . . . . . . . . . . . . . . . . . . . 62011.12 Shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62211.12.1 Average potential . . . . . . . . . . . . . . . . . . . . . 62311.12.2 Spin-orbit interaction . . . . . . . . . . . . . . . . . . . 62911.12.3 Example occupation levels . . . . . . . . . . . . . . . . 63311.12.4 Shell model with pairing . . . . . . . . . . . . . . . . . 63711.12.5 Conguration mixing . . . . . . . . . . . . . . . . . . . 64411.12.6 Shell model failures . . . . . . . . . . . . . . . . . . . . 65011.13 Collective Structure . . . . . . . . . . . . . . . . . . . . . . . . 65311.13.1 Classical liquid drop . . . . . . . . . . . . . . . . . . . . 65411.13.2 Nuclear vibrations . . . . . . . . . . . . . . . . . . . . . 65611.13.3 Nonspherical nuclei . . . . . . . . . . . . . . . . . . . . 65811.13.4 Rotational bands . . . . . . . . . . . . . . . . . . . . . . 66011.14 Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67311.14.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . 67311.14.2 Some basic features . . . . . . . . . . . . . . . . . . . . 67411.15 Spin Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67711.15.1 Even-even nuclei . . . . . . . . . . . . . . . . . . . . . . 67711.15.2 Odd mass number nuclei . . . . . . . . . . . . . . . . . 67911.15.3 Odd-odd nuclei . . . . . . . . . . . . . . . . . . . . . . . 68211.16 Parity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68611.16.1 Even-even nuclei . . . . . . . . . . . . . . . . . . . . . . 68611.16.2 Odd mass number nuclei . . . . . . . . . . . . . . . . . 68611.16.3 Odd-odd nuclei . . . . . . . . . . . . . . . . . . . . . . . 69111.16.4 Parity Summary . . . . . . . . . . . . . . . . . . . . . . 69111.17 Electromagnetic Moments . . . . . . . . . . . . . . . . . . . . 69111.17.1 Classical description . . . . . . . . . . . . . . . . . . . . 69411.17.2 Quantum description . . . . . . . . . . . . . . . . . . . 69611.17.3 Magnetic moment data . . . . . . . . . . . . . . . . . . 70311.17.4 Quadrupole moment data . . . . . . . . . . . . . . . . . 70711.18 Isospin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71111.19 Beta decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71611.19.1 Energetics Data . . . . . . . . . . . . . . . . . . . . . . 71611.19.2 Von Weizsacker approximation . . . . . . . . . . . . . . 72311.19.3 Kinetic Energies . . . . . . . . . . . . . . . . . . . . . . 72611.19.4 Forbidden decays . . . . . . . . . . . . . . . . . . . . . . 73011.19.5 Data and Fermi theory . . . . . . . . . . . . . . . . . . 73511.19.6 Parity violation . . . . . . . . . . . . . . . . . . . . . . 74111.20 Gamma Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . 74211.20.1 Energetics . . . . . . . . . . . . . . . . . . . . . . . . . 74311.20.2 Forbidden decays . . . . . . . . . . . . . . . . . . . . . . 74411.20.3 Isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . 747xiv CONTENTS11.20.4 Weisskopf estimates . . . . . . . . . . . . . . . . . . . . 74811.20.5 Internal conversion . . . . . . . . . . . . . . . . . . . . . 75612 Some Additional Topics 75912.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . 75912.1.1 Basic perturbation theory . . . . . . . . . . . . . . . . . 75912.1.2 Ionization energy of helium . . . . . . . . . . . . . . . . 76112.1.3 Degenerate perturbation theory . . . . . . . . . . . . . . 76512.1.4 The Zeeman eect . . . . . . . . . . . . . . . . . . . . . 76712.1.5 The Stark eect . . . . . . . . . . . . . . . . . . . . . . 76812.1.6 The hydrogen atom ne structure . . . . . . . . . . . . 77112.2 Quantum Field Theory in a Nanoshell . . . . . . . . . . . . . . 78412.2.1 Occupation numbers . . . . . . . . . . . . . . . . . . . . 78512.2.2 Annihilation and creation operators . . . . . . . . . . . 79112.2.3 Quantization of radiation . . . . . . . . . . . . . . . . . 79912.2.4 Spontaneous emission . . . . . . . . . . . . . . . . . . . 80612.2.5 Field operators . . . . . . . . . . . . . . . . . . . . . . . 80912.2.6 An example using eld operators . . . . . . . . . . . . . 81013 The Interpretation of Quantum Mechanics 81513.1 Schrodingers Cat . . . . . . . . . . . . . . . . . . . . . . . . . 81613.2 Instantaneous Interactions . . . . . . . . . . . . . . . . . . . . 81713.3 Global Symmetrization . . . . . . . . . . . . . . . . . . . . . . 82213.4 Failure of the Schrodinger Equation? . . . . . . . . . . . . . . 82213.5 The Many-Worlds Interpretation . . . . . . . . . . . . . . . . . 82513.6 The Arrow of Time . . . . . . . . . . . . . . . . . . . . . . . . 831A Notes 835A.1 Why another book on quantum mechanics? . . . . . . . . . . . 835A.2 History and wish list . . . . . . . . . . . . . . . . . . . . . . . 839A.3 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . 844A.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 845A.3.2 Generalized coordinates . . . . . . . . . . . . . . . . . . 845A.3.3 Lagrangian equations of motion . . . . . . . . . . . . . 846A.3.4 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . 850A.4 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 851A.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . 852A.4.2 Overview of relativity . . . . . . . . . . . . . . . . . . . 852A.4.3 Lorentz transformation . . . . . . . . . . . . . . . . . . 855A.4.4 Proper time and distance . . . . . . . . . . . . . . . . . 858A.4.5 Subluminal and superluminal eects . . . . . . . . . . . 859A.4.6 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . 861CONTENTS xvA.4.7 Index notation . . . . . . . . . . . . . . . . . . . . . . . 862A.4.8 Group property . . . . . . . . . . . . . . . . . . . . . . 863A.4.9 Intro to relativistic mechanics . . . . . . . . . . . . . . . 865A.4.10 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . 868A.5 Completeness of Fourier modes . . . . . . . . . . . . . . . . . . 872A.6 Derivation of the Euler formula . . . . . . . . . . . . . . . . . 876A.7 Nature and real eigenvalues . . . . . . . . . . . . . . . . . . . . 876A.8 Are Hermitian operators really like that? . . . . . . . . . . . . 876A.9 Are linear momentum operators Hermitian? . . . . . . . . . . 877A.10 Why boundary conditions are tricky . . . . . . . . . . . . . . . 877A.11 Extension to three-dimensional solutions . . . . . . . . . . . . 878A.12 Derivation of the harmonic oscillator solution . . . . . . . . . . 879A.13 More on the harmonic oscillator and uncertainty . . . . . . . . 883A.14 Derivation of a vector identity . . . . . . . . . . . . . . . . . . 884A.15 Derivation of the spherical harmonics . . . . . . . . . . . . . . 884A.16 The reduced mass . . . . . . . . . . . . . . . . . . . . . . . . . 887A.17 The hydrogen radial wave functions . . . . . . . . . . . . . . . 890A.18 Inner product for the expectation value . . . . . . . . . . . . . 893A.19 Why commuting operators have common eigenvectors . . . . . 893A.20 The generalized uncertainty relationship . . . . . . . . . . . . . 894A.21 Derivation of the commutator rules . . . . . . . . . . . . . . . 895A.22 Is the variational approximation best? . . . . . . . . . . . . . . 897A.23 Solution of the hydrogen molecular ion . . . . . . . . . . . . . 897A.24 Accuracy of the variational method . . . . . . . . . . . . . . . 898A.25 Positive molecular ion wave function . . . . . . . . . . . . . . . 900A.26 Molecular ion wave function symmetries . . . . . . . . . . . . . 900A.27 Solution of the hydrogen molecule . . . . . . . . . . . . . . . . 901A.28 Hydrogen molecule ground state and spin . . . . . . . . . . . . 903A.29 Number of boson states . . . . . . . . . . . . . . . . . . . . . . 904A.30 Shielding approximation limitations . . . . . . . . . . . . . . . 904A.31 Why the s states have the least energy . . . . . . . . . . . . . 905A.32 Density of states . . . . . . . . . . . . . . . . . . . . . . . . . . 905A.33 Radiation from a hole . . . . . . . . . . . . . . . . . . . . . . . 908A.34 Kirchhos law . . . . . . . . . . . . . . . . . . . . . . . . . . . 909A.35 The thermionic emission equation . . . . . . . . . . . . . . . . 910A.36 Explanation of the band gaps . . . . . . . . . . . . . . . . . . 913A.37 Number of conduction band electrons . . . . . . . . . . . . . . 918A.38 Thermoelectric eects . . . . . . . . . . . . . . . . . . . . . . . 919A.38.1 Peltier and Seebeck coecient ballparks . . . . . . . . . 919A.38.2 Figure of merit . . . . . . . . . . . . . . . . . . . . . . . 920A.38.3 Physical Seebeck mechanism . . . . . . . . . . . . . . . 922A.38.4 Full thermoelectric equations . . . . . . . . . . . . . . . 922xvi CONTENTSA.38.5 Charge locations in thermoelectrics . . . . . . . . . . . . 925A.38.6 Kelvin relationships . . . . . . . . . . . . . . . . . . . . 926A.39 Why energy eigenstates are stationary . . . . . . . . . . . . . . 931A.40 Better description of two-state systems . . . . . . . . . . . . . 931A.41 The evolution of expectation values . . . . . . . . . . . . . . . 931A.42 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . 932A.43 The energy-time uncertainty relationship . . . . . . . . . . . . 932A.44 The adiabatic theorem . . . . . . . . . . . . . . . . . . . . . . 933A.44.1 Derivation of the theorem . . . . . . . . . . . . . . . . . 933A.44.2 Some implications . . . . . . . . . . . . . . . . . . . . . 936A.45 Symmetry eigenvalue conservation . . . . . . . . . . . . . . . . 937A.46 The two-state approximation of radiation . . . . . . . . . . . . 937A.47 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 938A.48 About spectral broadening . . . . . . . . . . . . . . . . . . . . 943A.49 Derivation of the Einstein B coecients . . . . . . . . . . . . . 943A.50 Parseval and the Fourier inversion theorem . . . . . . . . . . . 947A.51 Derivation of group velocity . . . . . . . . . . . . . . . . . . . 948A.52 Motion through crystals . . . . . . . . . . . . . . . . . . . . . . 950A.52.1 Propagation speed . . . . . . . . . . . . . . . . . . . . . 950A.52.2 Motion under an external force . . . . . . . . . . . . . . 951A.52.3 Free-electron gas with constant electric eld . . . . . . . 952A.53 Details of the animations . . . . . . . . . . . . . . . . . . . . . 953A.54 Derivation of the WKB approximation . . . . . . . . . . . . . 961A.55 WKB solution near the turning points . . . . . . . . . . . . . . 963A.56 Three-dimensional scattering . . . . . . . . . . . . . . . . . . . 967A.56.1 Partial wave analysis . . . . . . . . . . . . . . . . . . . 969A.56.2 The Born approximation . . . . . . . . . . . . . . . . . 973A.56.3 The Born series . . . . . . . . . . . . . . . . . . . . . . 976A.57 The evolution of probability . . . . . . . . . . . . . . . . . . . 977A.58 A basic description of Lagrangian multipliers . . . . . . . . . . 981A.59 The generalized variational principle . . . . . . . . . . . . . . . 983A.60 Spin degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . 984A.61 Derivation of the approximation . . . . . . . . . . . . . . . . . 985A.62 Why a single Slater determinant is not exact . . . . . . . . . . 989A.63 Simplication of the Hartree-Fock energy . . . . . . . . . . . . 990A.64 Integral constraints . . . . . . . . . . . . . . . . . . . . . . . . 995A.65 Generalized orbitals . . . . . . . . . . . . . . . . . . . . . . . . 996A.66 Derivation of the Hartree-Fock equations . . . . . . . . . . . . 998A.67 Why the Fock operator is Hermitian . . . . . . . . . . . . . . . 1004A.68 Correlation energy . . . . . . . . . . . . . . . . . . . . . . . 1005A.69 Explanation of the London forces . . . . . . . . . . . . . . . . 1008A.70 Ambiguities in the denition of electron anity . . . . . . . . 1012CONTENTS xviiA.71 Why Floquet theory should be called so . . . . . . . . . . . . . 1014A.72 Superuidity versus BEC . . . . . . . . . . . . . . . . . . . . . 1014A.73 Explanation of Hunds rst rule . . . . . . . . . . . . . . . . . 1017A.74 The mechanism of ferromagnetism . . . . . . . . . . . . . . . . 1019A.75 Number of system eigenfunctions . . . . . . . . . . . . . . . . . 1020A.76 The fundamental assumption of quantum statistics . . . . . . . 1023A.77 A problem if the energy is given . . . . . . . . . . . . . . . . . 1025A.78 Derivation of the particle energy distributions . . . . . . . . . 1026A.79 The canonical probability distribution . . . . . . . . . . . . . . 1032A.80 Analysis of the ideal gas Carnot cycle . . . . . . . . . . . . . . 1034A.81 The recipe of life . . . . . . . . . . . . . . . . . . . . . . . . . . 1035A.82 The third law . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036A.83 Checks on the expression for entropy . . . . . . . . . . . . . . 1038A.84 Chemical potential and distribution functions . . . . . . . . . . 1041A.85 Fermi-Dirac integrals at low temperature . . . . . . . . . . . . 1045A.86 Physics of the fundamental commutation relations . . . . . . . 1046A.87 Multiple angular momentum components . . . . . . . . . . . . 1047A.88 Components of vectors are less than the total vector . . . . . . 1048A.89 The spherical harmonics with ladder operators . . . . . . . . . 1048A.90 Why angular momenta components can be added . . . . . . . 1049A.91 Why the Clebsch-Gordan tables are bidirectional . . . . . . . . 1049A.92 How to make Clebsch-Gordan tables . . . . . . . . . . . . . . . 1049A.93 Machine language version of the Clebsch-Gordan tables . . . . 1050A.94 The triangle inequality . . . . . . . . . . . . . . . . . . . . . . 1050A.95 Momentum of shells . . . . . . . . . . . . . . . . . . . . . . . . 1051A.96 Awkward questions about spin . . . . . . . . . . . . . . . . . . 1054A.97 More awkwardness about spin . . . . . . . . . . . . . . . . . . 1055A.98 Emergence of spin from relativity . . . . . . . . . . . . . . . . 1056A.99 Electromagnetic evolution of expectation values . . . . . . . . 1058A.100 Existence of magnetic monopoles . . . . . . . . . . . . . . . . . 1061A.101 More on Maxwells third law . . . . . . . . . . . . . . . . . . . 1061A.102 Various electrostatic derivations. . . . . . . . . . . . . . . . . . 1061A.102.1 Existence of a potential . . . . . . . . . . . . . . . . . . 1061A.102.2 The Laplace equation . . . . . . . . . . . . . . . . . . . 1062A.102.3 Egg-shaped dipole eld lines . . . . . . . . . . . . . . . 1063A.102.4 Ideal charge dipole delta function . . . . . . . . . . . . . 1064A.102.5 Integrals of the current density . . . . . . . . . . . . . . 1064A.102.6 Lorentz forces on a current distribution . . . . . . . . . 1065A.102.7 Field of a current dipole . . . . . . . . . . . . . . . . . . 1067A.102.8 Biot-Savart law . . . . . . . . . . . . . . . . . . . . . . 1069A.103 Energy due to orbital motion in a magnetic eld . . . . . . . . 1070A.104 Energy due to electron spin in a magnetic eld . . . . . . . . . 1071xviii CONTENTSA.105 Setting the record straight on alignment . . . . . . . . . . . . . 1072A.106 Solving the NMR equations . . . . . . . . . . . . . . . . . . . . 1072A.107 Harmonic oscillator revisited . . . . . . . . . . . . . . . . . . . 1073A.108 Impenetrable spherical shell . . . . . . . . . . . . . . . . . . . 1074A.109 Classical vibrating drop . . . . . . . . . . . . . . . . . . . . . . 1075A.109.1 Basic denitions . . . . . . . . . . . . . . . . . . . . . . 1075A.109.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . 1075A.109.3 Energy due to surface tension . . . . . . . . . . . . . . . 1079A.109.4 Energy due to Coulomb repulsion . . . . . . . . . . . . 1081A.109.5 Frequency of vibration . . . . . . . . . . . . . . . . . . . 1083A.110 Shell model quadrupole moment . . . . . . . . . . . . . . . . . 1084A.111 Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085A.111.1 Form of the wave function . . . . . . . . . . . . . . . . . 1085A.111.2 Source of the decay . . . . . . . . . . . . . . . . . . . . 1088A.111.3 Allowed or forbidden . . . . . . . . . . . . . . . . . . . 1091A.111.4 The nuclear operator . . . . . . . . . . . . . . . . . . . 1093A.111.5 Fermis golden rule . . . . . . . . . . . . . . . . . . . . . 1096A.111.6 Mopping up . . . . . . . . . . . . . . . . . . . . . . . . 1100A.111.7 Electron capture . . . . . . . . . . . . . . . . . . . . . . 1105A.112 Weisskopf estimates . . . . . . . . . . . . . . . . . . . . . . . . 1106A.112.1 Very loose derivation . . . . . . . . . . . . . . . . . . . 1106A.112.2 Ocial loose derivation . . . . . . . . . . . . . . . . . . 1110A.113 Auger discovery . . . . . . . . . . . . . . . . . . . . . . . . . . 1111A.114 Derivation of perturbation theory . . . . . . . . . . . . . . . . 1112A.115 Hydrogen ground state Stark eect . . . . . . . . . . . . . . . 1117A.116 Dirac ne structure Hamiltonian . . . . . . . . . . . . . . . . . 1119A.117 Classical spin-orbit derivation . . . . . . . . . . . . . . . . . . 1126A.118 Expectation powers of r for hydrogen . . . . . . . . . . . . . . 1128A.119 A tenth of a googol in universes . . . . . . . . . . . . . . . . . 1132Web Pages 1139Notations 1143List of Figures1.1 The classical picture of a vector. . . . . . . . . . . . . . . . . . . 61.2 Spike diagram of a vector. . . . . . . . . . . . . . . . . . . . . . 71.3 More dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Innite dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 The classical picture of a function. . . . . . . . . . . . . . . . . 81.6 Forming the dot product of two vectors. . . . . . . . . . . . . . 91.7 Forming the inner product of two functions. . . . . . . . . . . . 101.8 Illustration of the eigenfunction concept. Function sin(2x) isshown in black. Its rst derivative 2 cos(2x), shown in red, isnot just a multiple of sin(2x). Therefore sin(2x) is not an eigen-function of the rst derivative operator. However, the secondderivative of sin(2x) is 4 sin(2x), which is shown in green, andthat is indeed a multiple of sin(2x). So sin(2x) is an eigenfunctionof the second derivative operator, and with eigenvalue 4. . . . 142.1 A visualization of an arbitrary wave function. . . . . . . . . . . 212.2 Combined plot of position and momentum components. . . . . . 242.3 The uncertainty principle illustrated. . . . . . . . . . . . . . . . 242.4 Classical picture of a particle in a closed pipe. . . . . . . . . . . 312.5 Quantum mechanics picture of a particle in a closed pipe. . . . . 312.6 Denitions for one-dimensional motion in a pipe. . . . . . . . . 322.7 One-dimensional energy spectrum for a particle in a pipe. . . . . 382.8 One-dimensional ground state of a particle in a pipe. . . . . . . 402.9 Second and third lowest one-dimensional energy states. . . . . . 412.10 Denition of all variables for motion in a pipe. . . . . . . . . . . 422.11 True ground state of a particle in a pipe. . . . . . . . . . . . . . 432.12 True second and third lowest energy states. . . . . . . . . . . . . 442.13 A combination of 111 and 211 seen at some typical times. . . . 462.14 The harmonic oscillator. . . . . . . . . . . . . . . . . . . . . . . 482.15 The energy spectrum of the harmonic oscillator. . . . . . . . . . 532.16 Ground state of the harmonic oscillator . . . . . . . . . . . . . . 552.17 Wave functions 100 and 010. . . . . . . . . . . . . . . . . . . . 56xixxx LIST OF FIGURES2.18 Energy eigenfunction 213. . . . . . . . . . . . . . . . . . . . . . 572.19 Arbitrary wave function (not an energy eigenfunction). . . . . . 603.1 Spherical coordinates of an arbitrary point P. . . . . . . . . . . 653.2 Spectrum of the hydrogen atom. . . . . . . . . . . . . . . . . . . 783.3 Ground state wave function of the hydrogen atom. . . . . . . . . 813.4 Eigenfunction 200. . . . . . . . . . . . . . . . . . . . . . . . . . 823.5 Eigenfunction 210, or 2pz. . . . . . . . . . . . . . . . . . . . . . 833.6 Eigenfunction 211 (and 211). . . . . . . . . . . . . . . . . . . 833.7 Eigenfunctions 2px, left, and 2py, right. . . . . . . . . . . . . . . 843.8 Hydrogen atom plus free proton far apart. . . . . . . . . . . . . 1023.9 Hydrogen atom plus free proton closer together. . . . . . . . . . 1023.10 The electron being anti-symmetrically shared. . . . . . . . . . . 1043.11 The electron being symmetrically shared. . . . . . . . . . . . . . 1054.1 State with two neutral atoms. . . . . . . . . . . . . . . . . . . . 1174.2 Symmetric sharing of the electrons. . . . . . . . . . . . . . . . . 1194.3 Antisymmetric sharing of the electrons. . . . . . . . . . . . . . . 1194.4 Approximate solutions for hydrogen (left) and helium (right)atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1544.5 Abbreviated periodic table of the elements. Boxes below the el-ement names indicate the quantum states being lled with elec-trons in that row. Cell color indicates ionization energy. Thelength of a bar below an atomic number indicates electronega-tivity. A dot pattern indicates that the element is a gas undernormal conditions and wavy lines a liquid. . . . . . . . . . . . . 1564.6 Approximate solutions for lithium (left) and beryllium (right). 1574.7 Example approximate solution for boron. . . . . . . . . . . . . . 1584.8 Periodic table of the elements. . . . . . . . . . . . . . . . . . . 1624.9 Covalent sigma bond consisting of two 2pz states. . . . . . . . . 1664.10 Covalent pi bond consisting of two 2px states. . . . . . . . . . . 1674.11 Covalent sigma bond consisting of a 2pz and a 1s state. . . . . . 1684.12 Shape of an sp3hybrid state. . . . . . . . . . . . . . . . . . . . . 1704.13 Shapes of the sp2(left) and sp (right) hybrids. . . . . . . . . . . 1715.1 Allowed wave number vectors, left, and energy spectrum, right. . 1795.2 Ground state of a system of noninteracting bosons in a box. . . 1835.3 The system of bosons at a very low temperature. . . . . . . . . 1875.4 The system of bosons at a relatively low temperature. . . . . . . 187LIST OF FIGURES xxi5.5 Ground state system energy eigenfunction for a simple modelsystem with only 3 single-particle energy levels, 6 single-particlestates, and 3 distinguishable spinless particles. Left: mathemati-cal form. Right: graphical representation. All three particles arein the single-particle ground state. . . . . . . . . . . . . . . . . . 1905.6 Example system energy eigenfunction with ve times the single-particle ground state energy. . . . . . . . . . . . . . . . . . . . . 1915.7 For distinguishable particles, there are 9 system energy eigen-functions that have energy distribution A. . . . . . . . . . . . . 1925.8 For distinguishable particles, there are 12 system energy eigen-functions that have energy distribution B. . . . . . . . . . . . . 1935.9 For identical bosons, there are only 3 system energy eigenfunc-tions that have energy distribution A. . . . . . . . . . . . . . . . 1935.10 For identical bosons, there are also only 3 system energy eigen-functions that have energy distribution B. . . . . . . . . . . . . 1945.11 Ground state of a system of noninteracting electrons, or otherfermions, in a box. . . . . . . . . . . . . . . . . . . . . . . . . . 2025.12 Severe connement in the y-direction, as in a quantum well. . . 2085.13 Severe connement in both the y- and z-directions, as in a quan-tum wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.14 Severe connement in all three directions, as in a quantum dotor articial atom. . . . . . . . . . . . . . . . . . . . . . . . . . . 2105.15 A system of fermions at a nonzero temperature. . . . . . . . . . 2125.16 Particles at high-enough temperature and low-enough particledensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2155.17 Ground state of a system of noninteracting electrons, or otherfermions, in a periodic box. . . . . . . . . . . . . . . . . . . . . 2245.18 Conduction in the free-electron gas model. . . . . . . . . . . . . 2265.19 Sketch of electron energy spectra in solids at absolute zero tem-perature. (No attempt has been made to picture a density ofstates). Far left: the free-electron gas has a continuous band ofextremely densely spaced energy levels. Far right: lone atomshave only a few discrete electron energy levels. Middle: actualmetals and insulators have energy levels grouped into denselyspaced bands separated by gaps. Insulators completely ll upthe highest occupied band. . . . . . . . . . . . . . . . . . . . . . 2305.20 Sketch of electron energy spectra in solids at a nonzero temper-ature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2355.21 Potential energy seen by an electron along a line of nuclei. Thepotential energy is in green, the nuclei are in red. . . . . . . . . 2385.22 Potential energy seen by an electron in the one-dimensional sim-plied model of Kronig & Penney. . . . . . . . . . . . . . . . . . 238xxii LIST OF FIGURES5.23 Example Kronig & Penney spectra. . . . . . . . . . . . . . . . . 2415.24 Spectrum against wave number in the extended zone scheme. . . 2445.25 Spectrum against wave number in the reduced zone scheme. . . 2455.26 Some one-dimensional energy bands for a few basic semiconduc-tors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.27 Spectrum against wave number in the periodic zone scheme. . . 2485.28 Schematic of the zinc blende (ZnS) crystal relevant to importantsemiconductors including silicon. . . . . . . . . . . . . . . . . . 2495.29 First Brillouin zone of the fcc crystal. . . . . . . . . . . . . . . . 2515.30 Sketch of a more complete spectrum of germanium. (Based onresults of the VASP 5.2 commercial computer code.) . . . . . . . 2525.31 Vicinity of the band gap in the spectra of intrinsic and dopedsemiconductors. The amounts of conduction band electrons andvalence band holes have been vastly exaggerated to make themvisible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2545.32 Relationship between conduction electron density and hole den-sity. Intrinsic semiconductors have neither much conduction elec-trons nor holes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2585.33 The p-n junction in thermal equilibrium. Top: energy spectra.Quantum states with electrons in them are in red. The meanelectrostatic energy of the electrons is in green. Below: Physicalschematic of the junction. The dots are conduction electronsand the small circles holes. The encircled plus signs are donoratoms, and the encircled minus signs acceptor atoms. (Donorsand acceptors are not as regularly distributed, nor as densely, asthis greatly simplied schematic suggests.) . . . . . . . . . . . . 2605.34 Schematic of the operation of an p-n junction. . . . . . . . . . . 2635.35 Schematic of the operation of an n-p-n transistor. . . . . . . . . 2665.36 Vicinity of the band gap in the energy spectrum of an insulator.A photon of light with an energy greater than the band gap cantake an electron from the valence band to the conduction band.The photon is absorbed in the process. . . . . . . . . . . . . . . 2705.37 Peltier cooling. Top: physical device. Bottom: Electron energyspectra of the semiconductor materials. Quantum states lledwith electrons are shown in red. . . . . . . . . . . . . . . . . . . 2755.38 Seebeck voltage generator. . . . . . . . . . . . . . . . . . . . . . 2795.39 The Galvani potential jump over the contact surface does notproduce a usable voltage. . . . . . . . . . . . . . . . . . . . . . . 2815.40 The Seebeck eect is not directly measurable. . . . . . . . . . . 2836.1 The ground state wave function looks the same at all times. . . 2916.2 The rst excited state at all times. . . . . . . . . . . . . . . . . 291LIST OF FIGURES xxiii6.3 A combination of 111 and 211 seen at some typical times. . . . 2926.4 Emission and absorption of radiation by an atom. . . . . . . . . 3126.5 Triangle inequality. . . . . . . . . . . . . . . . . . . . . . . . . . 3196.6 Approximate Dirac delta function (x x) is shown left. Thetrue delta function (xx) is the limit when becomes zero, andis an innitely high, innitely thin spike, shown right. It is theeigenfunction corresponding to a position x. . . . . . . . . . . . 3326.7 The real part (red) and envelope (black) of an example wave. . . 3386.8 The wave moves with the phase speed. . . . . . . . . . . . . . . 3396.9 The real part (red) and magnitude or envelope (black) of a wavepacket. (Schematic). . . . . . . . . . . . . . . . . . . . . . . . . 3406.10 The velocities of wave and envelope are not equal. . . . . . . . . 3416.11 A particle in free space. . . . . . . . . . . . . . . . . . . . . . . 3506.12 An accelerating particle. . . . . . . . . . . . . . . . . . . . . . . 3506.13 An decelerating particle. . . . . . . . . . . . . . . . . . . . . . . 3516.14 Unsteady solution for the harmonic oscillator. The third pictureshows the maximum distance from the nominal position that thewave packet reaches. . . . . . . . . . . . . . . . . . . . . . . . . 3526.15 Harmonic oscillator potential energy V , eigenfunction h50, andits energy E50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3526.16 A partial reection. . . . . . . . . . . . . . . . . . . . . . . . . . 3576.17 An tunneling particle. . . . . . . . . . . . . . . . . . . . . . . . 3586.18 Penetration of an innitely high potential energy barrier. . . . . 3586.19 Schematic of a scattering potential and the asymptotic behaviorof an example energy eigenfunction for a wave packet coming infrom the far left. . . . . . . . . . . . . . . . . . . . . . . . . . . 3608.1 Billiard-ball model of the salt molecule. . . . . . . . . . . . . . . 4008.2 Billiard-ball model of a salt crystal. . . . . . . . . . . . . . . . . 4028.3 The salt crystal disassembled to show its structure. . . . . . . . 4038.4 The lithium atom, scaled more correctly than in chapter 4.9 . . 4058.5 Body-centered-cubic (bcc) structure of lithium. . . . . . . . . . 4068.6 Fully periodic wave function of a two-atom lithium crystal. . . 4078.7 Flip-op wave function of a two-atom lithium crystal. . . . . . 4088.8 Wave functions of a four-atom lithium crystal. The actualpicture is that of the fully periodic mode. . . . . . . . . . . . . . 4098.9 Reciprocal lattice of a one-dimensional crystal. . . . . . . . . . . 4138.10 Schematic of energy bands. . . . . . . . . . . . . . . . . . . . . . 4148.11 Schematic of merging bands. . . . . . . . . . . . . . . . . . . . . 4168.12 A primitive cell and primitive translation vectors of lithium. . . 4178.13 Wigner-Seitz cell of the bcc lattice. . . . . . . . . . . . . . . . . 4188.14 Schematic of crossing bands. . . . . . . . . . . . . . . . . . . . . 422xxiv LIST OF FIGURES8.15 Ball and stick schematic of the diamond crystal. . . . . . . . . . 4238.16 Assumed simple cubic reciprocal lattice, shown as black dots, incross-section. The boundaries of the surrounding primitive cellsare shown as thin red lines. . . . . . . . . . . . . . . . . . . . . 4268.17 Occupied states for one, two, and three free electrons per physicallattice cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4288.18 Redenition of the occupied wave number vectors into Brillouinzones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4298.19 Second, third, and fourth Brillouin zones seen in the periodiczone scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4308.20 The red dot shows the wavenumber vector of a sample free elec-tron wave function. It is to be corrected for the lattice potential. 4328.21 The grid of nonzero Hamiltonian perturbation coecients andthe problem sphere in wave number space. . . . . . . . . . . . . 4348.22 Tearing apart of the wave number space energies. . . . . . . . . 4358.23 Eect of a lattice potential on the energy. The energy is repre-sented by the square distance from the origin, and is relative tothe energy at the origin. . . . . . . . . . . . . . . . . . . . . . . 4368.24 Bragg planes seen in wave number space cross section. . . . . . 4378.25 Occupied states for the energies of gure 8.23 if there are twovalence electrons per lattice cell. Left: energy. Right: wavenumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4378.26 Smaller lattice potential. From top to bottom shows one, twoand three valence electrons per lattice cell. Left: energy. Right:wave numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4388.27 Depiction of an electromagnetic ray. . . . . . . . . . . . . . . . . 4438.28 Law of reection in elastic scattering from a plane. . . . . . . . 4448.29 Scattering from multiple planes of atoms. . . . . . . . . . . . 4458.30 Dierence in travel distance when scattered from P rather thanO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4469.1 Graphical depiction of an arbitrary system energy eigenfunctionfor 36 distinguishable particles. . . . . . . . . . . . . . . . . . . 4559.2 Graphical depiction of an arbitrary system energy eigenfunctionfor 36 identical bosons. . . . . . . . . . . . . . . . . . . . . . . . 4579.3 Graphical depiction of an arbitrary system energy eigenfunctionfor 33 identical fermions. . . . . . . . . . . . . . . . . . . . . . . 4579.4 Illustrative small model system having 4 distinguishable particles.The particular eigenfunction shown is arbitrary. . . . . . . . . . 460LIST OF FIGURES xxv9.5 The number of system energy eigenfunctions for a simple modelsystem with only three energy shelves. Positions of the squaresindicate the numbers of particles on shelves 2 and 3; darknessof the squares indicates the relative number of eigenfunctionswith those shelf numbers. Left: system with 4 distinguishableparticles, middle: 16, right: 64. . . . . . . . . . . . . . . . . . . 4609.6 Number of energy eigenfunctions on the oblique energy line in 9.5.(The curves are mathematically interpolated to allow a continu-ously varying fraction of particles on shelf 2.) Left: 4 particles,middle: 64, right: 1,024. . . . . . . . . . . . . . . . . . . . . . . 4629.7 Probabilities of shelf-number sets for the simple 64 particle modelsystem if there is uncertainty in energy. More probable shelf-number distributions are shown darker. Left: identical bosons,middle: distinguishable particles, right: identical fermions. Thetemperature is the same as in gure 9.5. . . . . . . . . . . . . . 4679.8 Probabilities of shelf-number sets for the simple 64 particle modelsystem if shelf 1 is a non-degenerate ground state. Left: iden-tical bosons, middle: distinguishable particles, right: identicalfermions. The temperature is the same as in gure 9.7. . . . . . 4689.9 Like gure 9.8, but at a lower temperature. . . . . . . . . . . . . 4689.10 Like gure 9.8, but at a still lower temperature. . . . . . . . . . 4699.11 Schematic of the Carnot refrigeration cycle. . . . . . . . . . . . 4769.12 Schematic of the Carnot heat engine. . . . . . . . . . . . . . . . 4799.13 A generic heat pump next to a reversed Carnot one with the sameheat delivery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4809.14 Comparison of two dierent integration paths for nding the en-tropy of a desired state. The two dierent integration paths arein black and the yellow lines are reversible adiabatic process lines. 4829.15 Specic heat at constant volume of gases. Temperatures fromabsolute zero to 1,200 K. Data from NIST-JANAF and AIP. . . 5079.16 Specic heat at constant pressure of solids. Temperatures fromabsolute zero to 1,200 K. Carbon is diamond; graphite is similar.Water is ice and liquid. Data from NIST-JANAF, CRC, AIP,Rohsenow et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50910.1 Example bosonic ladders. . . . . . . . . . . . . . . . . . . . . . . 51710.2 Example fermionic ladders. . . . . . . . . . . . . . . . . . . . . . 51710.3 Triplet and singlet states in terms of ladders . . . . . . . . . . . 52310.4 Clebsch-Gordan coecients of two spin one half particles. . . . . 52410.5 Clebsch-Gordan coecients for a spin-one-half second particle. . 52610.6 Clebsch-Gordan coecients for a spin-one second particle. . . . 52810.7 Relationship of Maxwells rst equation to Coulombs law. . . . 542xxvi LIST OF FIGURES10.8 Maxwells rst equation for a more arbitrary region. The gureto the right includes the eld lines through the selected points. . 54310.9 The net number of eld lines leaving a region is a measure forthe net charge inside that region. . . . . . . . . . . . . . . . . . 54410.10Since magnetic monopoles do not exist, the net number of mag-netic eld lines leaving a region is always zero. . . . . . . . . . . 54510.11Electric power generation. . . . . . . . . . . . . . . . . . . . . . 54610.12Two ways to generate a magnetic eld: using a current (left) orusing a varying electric eld (right). . . . . . . . . . . . . . . . . 54710.13Electric eld and potential of a charge that is distributed uni-formly within a small sphere. The dotted lines indicate the valuesfor a point charge. . . . . . . . . . . . . . . . . . . . . . . . . . 55210.14Electric eld of a two-dimensional line charge. . . . . . . . . . . 55310.15Field lines of a vertical electric dipole. . . . . . . . . . . . . . . 55410.16Electric eld of a two-dimensional dipole. . . . . . . . . . . . . . 55510.17Field of an ideal magnetic dipole. . . . . . . . . . . . . . . . . . 55610.18Electric eld of an almost ideal two-dimensional dipole. . . . . . 55710.19Magnetic eld lines around an innite straight electric wire. . . 56210.20An electromagnet consisting of a single wire loop. The generatedmagnetic eld lines are in blue. . . . . . . . . . . . . . . . . . . 56210.21A current dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . 56310.22Electric motor using a single wire loop. The Lorentz forces (blackvectors) exerted by the external magnetic eld on the electriccurrent carriers in the wire produce a net moment M on the loop.The self-induced magnetic eld of the wire and the correspondingradial forces are not shown. . . . . . . . . . . . . . . . . . . . . 56410.23V