Fundamental Quantum Mechanics for EngineersLeon van Dommelen03/22/09 Version 4.2 alphaCopyrightCopyright 2004, 2007, 2008 and on, Leon van Dommelen. You are allowedto copy or print out this work for your personal use. You are allowed to attachadditional notes, corrections, and additions, as long as they are clearly identiedas 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.iiiContentsPreface xxviiTo the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiiComments and Feedback . . . . . . . . . . . . . . . . . . . . . . . . xxxI 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Additional Points . . . . . . . . . . . . . . . . . . . . . . . . . . 161.7.1 Dirac notation . . . . . . . . . . . . . . . . . . . . . . . . 161.7.2 Additional independent variables . . . . . . . . . . . . . 172 Basic Ideas of Quantum Mechanics 192.1 The Revised Picture of Nature . . . . . . . . . . . . . . . . . . 192.2 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . 222.3 The Operators of Quantum Mechanics . . . . . . . . . . . . . . 232.4 The Orthodox Statistical Interpretation . . . . . . . . . . . . . 252.4.1 Only eigenvalues . . . . . . . . . . . . . . . . . . . . . . 252.4.2 Statistical selection . . . . . . . . . . . . . . . . . . . . . 272.5 A Particle Conned Inside a Pipe . . . . . . . . . . . . . . . . . 282.5.1 The physical system . . . . . . . . . . . . . . . . . . . . 292.5.2 Mathematical notations . . . . . . . . . . . . . . . . . . 302.5.3 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 302.5.4 The Hamiltonian eigenvalue problem . . . . . . . . . . . 312.5.5 All solutions of the eigenvalue problem . . . . . . . . . . 32vvi CONTENTS2.5.6 Discussion of the energy values . . . . . . . . . . . . . . 362.5.7 Discussion of the eigenfunctions . . . . . . . . . . . . . . 372.5.8 Three-dimensional solution . . . . . . . . . . . . . . . . . 402.5.9 Quantum connement . . . . . . . . . . . . . . . . . . . 432.6 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 462.6.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 472.6.2 Solution using separation of variables . . . . . . . . . . . 482.6.3 Discussion of the eigenvalues . . . . . . . . . . . . . . . . 512.6.4 Discussion of the eigenfunctions . . . . . . . . . . . . . . 532.6.5 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . 572.6.6 Non-eigenstates . . . . . . . . . . . . . . . . . . . . . . . 593 Single-Particle Systems 633.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 633.1.1 Denition of angular momentum . . . . . . . . . . . . . 633.1.2 Angular momentum in an arbitrary direction . . . . . . . 643.1.3 Square angular momentum . . . . . . . . . . . . . . . . . 663.1.4 Angular momentum uncertainty . . . . . . . . . . . . . . 693.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . 703.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 703.2.2 Solution using separation of variables . . . . . . . . . . . 713.2.3 Discussion of the eigenvalues . . . . . . . . . . . . . . . . 763.2.4 Discussion of the eigenfunctions . . . . . . . . . . . . . . 793.3 Expectation Value and Standard Deviation . . . . . . . . . . . 843.3.1 Statistics of a die . . . . . . . . . . . . . . . . . . . . . . 853.3.2 Statistics of quantum operators . . . . . . . . . . . . . . 863.3.3 Simplied expressions . . . . . . . . . . . . . . . . . . . . 883.3.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . 893.4 The Commutator . . . . . . . . . . . . . . . . . . . . . . . . . . 913.4.1 Commuting operators . . . . . . . . . . . . . . . . . . . 923.4.2 Noncommuting operators and their commutator . . . . . 933.4.3 The Heisenberg uncertainty relationship . . . . . . . . . 943.4.4 Commutator reference [Reference] . . . . . . . . . . . . . 953.5 The Hydrogen Molecular Ion . . . . . . . . . . . . . . . . . . . 983.5.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 993.5.2 Energy when fully dissociated . . . . . . . . . . . . . . . 993.5.3 Energy when closer together . . . . . . . . . . . . . . . . 1003.5.4 States that share the electron . . . . . . . . . . . . . . . 1013.5.5 Comparative energies of the states . . . . . . . . . . . . 1033.5.6 Variational approximation of the ground state . . . . . . 1043.5.7 Comparison with the exact ground state . . . . . . . . . 106CONTENTS vii4 Multiple-Particle Systems 1094.1 Wave Function for Multiple Particles . . . . . . . . . . . . . . . 1094.2 The Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . . 1114.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 1114.2.2 Initial approximation to the lowest energy state . . . . . 1124.2.3 The probability density . . . . . . . . . . . . . . . . . . . 1134.2.4 States that share the electrons . . . . . . . . . . . . . . . 1144.2.5 Variational approximation of the ground state . . . . . . 1174.2.6 Comparison with the exact ground state . . . . . . . . . 1184.3 Two-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . 1194.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.5 Multiple-Particle Systems Including Spin . . . . . . . . . . . . . 1254.5.1 Wave function for a single particle with spin . . . . . . . 1254.5.2 Inner products including spin . . . . . . . . . . . . . . . 1284.5.3 Commutators including spin . . . . . . . . . . . . . . . . 1294.5.4 Wave function for multiple particles with spin . . . . . . 1304.5.5 Example: the hydrogen molecule . . . . . . . . . . . . . 1324.5.6 Triplet and singlet states . . . . . . . . . . . . . . . . . . 1334.6 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . 1344.7 Ways to Symmetrize the Wave Function . . . . . . . . . . . . . 1374.8 Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . 1434.9 Heavier Atoms [Descriptive] . . . . . . . . . . . . . . . . . . . . 1474.9.1 The Hamiltonian eigenvalue problem . . . . . . . . . . . 1474.9.2 Approximate solution using separation of variables . . . 1484.9.3 Hydrogen and helium . . . . . . . . . . . . . . . . . . . . 1504.9.4 Lithium to neon . . . . . . . . . . . . . . . . . . . . . . . 1524.9.5 Sodium to argon . . . . . . . . . . . . . . . . . . . . . . 1564.9.6 Potassium to krypton . . . . . . . . . . . . . . . . . . . . 1574.10 Pauli Repulsion [Descriptive] . . . . . . . . . . . . . . . . . . . 1574.11 Chemical Bonds [Descriptive] . . . . . . . . . . . . . . . . . . . 1594.11.1 Covalent sigma bonds . . . . . . . . . . . . . . . . . . . 1594.11.2 Covalent pi bonds . . . . . . . . . . . . . . . . . . . . . . 1604.11.3 Polar covalent bonds and hydrogen bonds . . . . . . . . 1614.11.4 Promotion and hybridization . . . . . . . . . . . . . . . . 1624.11.5 Ionic bonds . . . . . . . . . . . . . . . . . . . . . . . . . 1654.11.6 Limitations of valence bond theory . . . . . . . . . . . . 1665 Time Evolution 1695.1 The Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . 1695.1.1 Energy conservation . . . . . . . . . . . . . . . . . . . . 1705.1.2 Stationary states . . . . . . . . . . . . . . . . . . . . . . 1715.1.3 Time variations of symmetric two-state systems . . . . . 172viii CONTENTS5.1.4 Time variation of expectation values . . . . . . . . . . . 1735.1.5 Newtonian motion . . . . . . . . . . . . . . . . . . . . . 1745.1.6 Heisenberg picture . . . . . . . . . . . . . . . . . . . . . 1755.1.7 The adiabatic approximation . . . . . . . . . . . . . . . 1785.2 Unsteady Perturbations of Systems . . . . . . . . . . . . . . . . 1785.2.1 Schrodinger equation for a two-state system . . . . . . . 1795.2.2 Stimulated and spontaneous emission . . . . . . . . . . . 1805.2.3 Eect of a single wave . . . . . . . . . . . . . . . . . . . 1815.2.4 Forbidden transitions . . . . . . . . . . . . . . . . . . . . 1835.2.5 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . 1845.2.6 Angular momentum conservation . . . . . . . . . . . . . 1855.2.7 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.2.8 Absorption of a single weak wave . . . . . . . . . . . . . 1895.2.9 Absorption of incoherent radiation . . . . . . . . . . . . 1915.2.10 Spontaneous emission of radiation . . . . . . . . . . . . . 1935.3 Position and Linear Momentum . . . . . . . . . . . . . . . . . . 1945.3.1 The position eigenfunction . . . . . . . . . . . . . . . . . 1955.3.2 The linear momentum eigenfunction . . . . . . . . . . . 1975.4 Wave Packets in Free Space . . . . . . . . . . . . . . . . . . . . 1995.4.1 Solution of the Schrodinger equation. . . . . . . . . . . . 1995.4.2 Component wave solutions . . . . . . . . . . . . . . . . . 2005.4.3 Wave packets . . . . . . . . . . . . . . . . . . . . . . . . 2015.4.4 Group velocity . . . . . . . . . . . . . . . . . . . . . . . 2035.5 Motion near the Classical Limit . . . . . . . . . . . . . . . . . . 2065.5.1 Motion through free space . . . . . . . . . . . . . . . . . 2065.5.2 Accelerated motion . . . . . . . . . . . . . . . . . . . . . 2075.5.3 Decelerated motion . . . . . . . . . . . . . . . . . . . . . 2075.5.4 The harmonic oscillator . . . . . . . . . . . . . . . . . . 2075.6 WKB Theory of Nearly Classical Motion . . . . . . . . . . . . . 2095.7 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.7.1 Partial reection . . . . . . . . . . . . . . . . . . . . . . 2135.7.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 2145.8 Reection and Transmission Coecients . . . . . . . . . . . . . 2165.9 Alpha Decay of Nuclei . . . . . . . . . . . . . . . . . . . . . . . 217II Advanced Topics 2256 Numerical Procedures 2276.1 The Variational Method . . . . . . . . . . . . . . . . . . . . . . 2276.1.1 Basic variational statement . . . . . . . . . . . . . . . . 2276.1.2 Dierential form of the statement . . . . . . . . . . . . . 228CONTENTS ix6.1.3 Example application using Lagrangian multipliers . . . . 2296.2 The Born-Oppenheimer Approximation . . . . . . . . . . . . . 2316.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 2326.2.2 The basic Born-Oppenheimer approximation . . . . . . . 2336.2.3 Going one better . . . . . . . . . . . . . . . . . . . . . . 2356.3 The Hartree-Fock Approximation . . . . . . . . . . . . . . . . . 2386.3.1 Wave function approximation . . . . . . . . . . . . . . . 2386.3.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 2446.3.3 The expectation value of energy . . . . . . . . . . . . . . 2466.3.4 The canonical Hartree-Fock equations . . . . . . . . . . . 2486.3.5 Additional points . . . . . . . . . . . . . . . . . . . . . . 2507 Solids 2597.1 Molecular Solids [Descriptive] . . . . . . . . . . . . . . . . . . . 2597.2 Ionic Solids [Descriptive] . . . . . . . . . . . . . . . . . . . . . . 2617.3 Introduction to Band Structure [Descriptive] . . . . . . . . . . . 2667.4 Metals [Descriptive] . . . . . . . . . . . . . . . . . . . . . . . . 2687.4.1 Lithium . . . . . . . . . . . . . . . . . . . . . . . . . . . 2687.4.2 One-dimensional crystals . . . . . . . . . . . . . . . . . . 2707.4.3 Wave functions of one-dimensional crystals . . . . . . . . 2717.4.4 Analysis of the wave functions . . . . . . . . . . . . . . . 2747.4.5 Floquet (Bloch) theory . . . . . . . . . . . . . . . . . . . 2757.4.6 Fourier analysis . . . . . . . . . . . . . . . . . . . . . . . 2767.4.7 The reciprocal lattice . . . . . . . . . . . . . . . . . . . . 2777.4.8 The energy levels . . . . . . . . . . . . . . . . . . . . . . 2787.4.9 Electrical conduction . . . . . . . . . . . . . . . . . . . . 2797.4.10 Merging and splitting bands . . . . . . . . . . . . . . . . 2807.4.11 Three-dimensional metals . . . . . . . . . . . . . . . . . 2827.5 Covalent Materials [Descriptive] . . . . . . . . . . . . . . . . . . 2867.6 Conned Free Electrons . . . . . . . . . . . . . . . . . . . . . . 2897.6.1 The Hamiltonian eigenvalue problem . . . . . . . . . . . 2907.6.2 Solution by separation of variables . . . . . . . . . . . . 2907.6.3 Discussion of the solution . . . . . . . . . . . . . . . . . 2927.6.4 A numerical example . . . . . . . . . . . . . . . . . . . . 2947.6.5 The density of states and connement . . . . . . . . . . 2957.6.6 Relation to Bloch functions . . . . . . . . . . . . . . . . 3017.7 Free Electrons in a Lattice . . . . . . . . . . . . . . . . . . . . . 3017.7.1 The lattice structure . . . . . . . . . . . . . . . . . . . . 3037.7.2 Occupied states and Brillouin zones . . . . . . . . . . . . 3057.8 Nearly-Free Electrons . . . . . . . . . . . . . . . . . . . . . . . 3097.8.1 Energy changes due to a weak lattice potential . . . . . . 3107.8.2 Discussion of the energy changes . . . . . . . . . . . . . 312x CONTENTS7.9 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . 3177.10 Additional Points [Descriptive] . . . . . . . . . . . . . . . . . . 3227.10.1 Thermal properties . . . . . . . . . . . . . . . . . . . . . 3227.10.2 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . 3277.10.3 X-ray diraction . . . . . . . . . . . . . . . . . . . . . . 3308 Basic and Quantum Thermodynamics 3378.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3388.2 Single-Particle and System Eigenfunctions . . . . . . . . . . . . 3398.3 How Many System Eigenfunctions? . . . . . . . . . . . . . . . . 3448.4 Particle-Energy Distribution Functions . . . . . . . . . . . . . . 3498.5 The Canonical Probability Distribution . . . . . . . . . . . . . 3518.6 Low Temperature Behavior . . . . . . . . . . . . . . . . . . . . 3538.7 The Basic Thermodynamic Variables . . . . . . . . . . . . . . . 3568.8 Introduction to the Second Law . . . . . . . . . . . . . . . . . . 3608.9 The Reversible Ideal . . . . . . . . . . . . . . . . . . . . . . . . 3618.10 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3678.11 The Big Lie of Distinguishable Particles . . . . . . . . . . . . . 3748.12 The New Variables . . . . . . . . . . . . . . . . . . . . . . . . . 3748.13 Microscopic Meaning of the Variables . . . . . . . . . . . . . . . 3818.14 Application to Particles in a Box . . . . . . . . . . . . . . . . . 3828.14.1 Bose-Einstein condensation . . . . . . . . . . . . . . . . 3848.14.2 Fermions at low temperatures . . . . . . . . . . . . . . . 3858.14.3 A generalized ideal gas law . . . . . . . . . . . . . . . . . 3878.14.4 The ideal gas . . . . . . . . . . . . . . . . . . . . . . . . 3878.14.5 Blackbody radiation . . . . . . . . . . . . . . . . . . . . 3898.14.6 The Debye model . . . . . . . . . . . . . . . . . . . . . . 3929 Electromagnetism 3959.1 All About Angular Momentum . . . . . . . . . . . . . . . . . . 3959.1.1 The fundamental commutation relations . . . . . . . . . 3969.1.2 Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . 3979.1.3 Possible values of angular momentum . . . . . . . . . . . 4009.1.4 A warning about angular momentum . . . . . . . . . . . 4029.1.5 Triplet and singlet states . . . . . . . . . . . . . . . . . . 4039.1.6 Clebsch-Gordan coecients . . . . . . . . . . . . . . . . 4059.1.7 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . 4099.1.8 General spin matrices . . . . . . . . . . . . . . . . . . . . 4119.2 The Relativistic Dirac Equation . . . . . . . . . . . . . . . . . . 4129.3 The Electromagnetic Hamiltonian . . . . . . . . . . . . . . . . 4149.4 Maxwells Equations [Descriptive] . . . . . . . . . . . . . . . . . 4179.5 Example Static Electromagnetic Fields . . . . . . . . . . . . . . 424CONTENTS xi9.5.1 Point charge at the origin . . . . . . . . . . . . . . . . . 4259.5.2 Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4299.5.3 Arbitrary charge distributions . . . . . . . . . . . . . . . 4339.5.4 Solution of the Poisson equation . . . . . . . . . . . . . . 4369.5.5 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 4379.5.6 Principle of the electric motor . . . . . . . . . . . . . . . 4399.6 Particles in Magnetic Fields . . . . . . . . . . . . . . . . . . . . 4429.7 Stern-Gerlach Apparatus [Descriptive] . . . . . . . . . . . . . . 4459.8 Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . 4469.8.1 Description of the method . . . . . . . . . . . . . . . . . 4469.8.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 4479.8.3 The unperturbed system . . . . . . . . . . . . . . . . . . 4499.8.4 Eect of the perturbation . . . . . . . . . . . . . . . . . 45110 Some Additional Topics 45510.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . 45510.1.1 Basic perturbation theory . . . . . . . . . . . . . . . . . 45510.1.2 Ionization energy of helium . . . . . . . . . . . . . . . . 45710.1.3 Degenerate perturbation theory . . . . . . . . . . . . . . 46110.1.4 The Zeeman eect . . . . . . . . . . . . . . . . . . . . . 46310.1.5 The Stark eect . . . . . . . . . . . . . . . . . . . . . . . 46410.1.6 The hydrogen atom ne structure . . . . . . . . . . . . . 46710.2 Quantum Field Theory in a Nanoshell . . . . . . . . . . . . . . 48010.2.1 Occupation numbers . . . . . . . . . . . . . . . . . . . . 48110.2.2 Annihilation and creation operators . . . . . . . . . . . . 48710.2.3 Quantization of radiation . . . . . . . . . . . . . . . . . . 49510.2.4 Spontaneous emission . . . . . . . . . . . . . . . . . . . . 50210.2.5 Field operators . . . . . . . . . . . . . . . . . . . . . . . 50510.2.6 An example using eld operators . . . . . . . . . . . . . 50611 The Interpretation of Quantum Mechanics 51111.1 Schrodingers Cat . . . . . . . . . . . . . . . . . . . . . . . . . . 51211.2 Instantaneous Interactions . . . . . . . . . . . . . . . . . . . . . 51311.3 Global Symmetrization . . . . . . . . . . . . . . . . . . . . . . 51811.4 Conservation Laws and Symmetries . . . . . . . . . . . . . . . . 51811.5 Failure of the Schrodinger Equation? . . . . . . . . . . . . . . . 52211.6 The Many-Worlds Interpretation . . . . . . . . . . . . . . . . . 525A Notes 533A.1 Why another book on quantum mechanics? . . . . . . . . . . . 533A.2 History and wishlist . . . . . . . . . . . . . . . . . . . . . . . . 537A.3 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . 541xii CONTENTSA.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 541A.3.2 Generalized coordinates . . . . . . . . . . . . . . . . . . 542A.3.3 Lagrangian equations of motion . . . . . . . . . . . . . . 543A.3.4 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . 546A.4 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 548A.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548A.4.2 Overview of relativity . . . . . . . . . . . . . . . . . . . . 549A.4.3 Lorentz transformation . . . . . . . . . . . . . . . . . . . 552A.4.4 Proper time and distance . . . . . . . . . . . . . . . . . . 555A.4.5 Subluminal and superluminal eects . . . . . . . . . . . 556A.4.6 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . 558A.4.7 Index notation . . . . . . . . . . . . . . . . . . . . . . . 559A.4.8 Group property . . . . . . . . . . . . . . . . . . . . . . . 560A.4.9 Intro to relativistic mechanics . . . . . . . . . . . . . . . 562A.4.10 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . 565A.5 Completeness of Fourier modes . . . . . . . . . . . . . . . . . . 569A.6 Derivation of the Euler formula . . . . . . . . . . . . . . . . . . 573A.7 Nature and real eigenvalues . . . . . . . . . . . . . . . . . . . . 573A.8 Are Hermitian operators really like that? . . . . . . . . . . . . . 573A.9 Are linear momentum operators Hermitian? . . . . . . . . . . . 574A.10 Why boundary conditions are tricky . . . . . . . . . . . . . . . 574A.11 Extension to three-dimensional solutions . . . . . . . . . . . . . 575A.12 Derivation of the harmonic oscillator solution . . . . . . . . . . 576A.13 More on the harmonic oscillator and uncertainty . . . . . . . . 580A.14 Derivation of a vector identity . . . . . . . . . . . . . . . . . . . 581A.15 Derivation of the spherical harmonics . . . . . . . . . . . . . . . 581A.16 The eective mass . . . . . . . . . . . . . . . . . . . . . . . . . 584A.17 The hydrogen radial wave functions . . . . . . . . . . . . . . . . 587A.18 Inner product for the expectation value . . . . . . . . . . . . . 590A.19 Why commuting operators have a common set of eigenvectors . 590A.20 The generalized uncertainty relationship . . . . . . . . . . . . . 591A.21 Derivation of the commutator rules . . . . . . . . . . . . . . . . 592A.22 Is the variational approximation best? . . . . . . . . . . . . . . 594A.23 Solution of the hydrogen molecular ion . . . . . . . . . . . . . . 594A.24 Accuracy of the variational method . . . . . . . . . . . . . . . . 595A.25 Positive molecular ion wave function . . . . . . . . . . . . . . . 597A.26 Molecular ion wave function symmetries . . . . . . . . . . . . . 597A.27 Solution of the hydrogen molecule . . . . . . . . . . . . . . . . 598A.28 Hydrogen molecule ground state and spin . . . . . . . . . . . . 600A.29 Shielding approximation limitations . . . . . . . . . . . . . . . 601A.30 Why the s states have the least energy . . . . . . . . . . . . . . 601A.31 Why energy eigenstates are stationary . . . . . . . . . . . . . . 602CONTENTS xiiiA.32 Better description of two-state systems . . . . . . . . . . . . . . 602A.33 The evolution of expectation values . . . . . . . . . . . . . . . . 602A.34 The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . 603A.35 The energy-time uncertainty relationship . . . . . . . . . . . . . 603A.36 The adiabatic theorem . . . . . . . . . . . . . . . . . . . . . . . 604A.36.1 Derivation of the theorem . . . . . . . . . . . . . . . . . 604A.36.2 Some implications . . . . . . . . . . . . . . . . . . . . . . 607A.37 The two-state approximation of radiation . . . . . . . . . . . . 608A.38 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 609A.39 About spectral broadening . . . . . . . . . . . . . . . . . . . . . 613A.40 Derivation of the Einstein B coecients . . . . . . . . . . . . . 614A.41 Parseval and the Fourier inversion theorem . . . . . . . . . . . 617A.42 Derivation of group velocity . . . . . . . . . . . . . . . . . . . . 618A.43 Details of the animations . . . . . . . . . . . . . . . . . . . . . 621A.44 Derivation of the WKB approximation . . . . . . . . . . . . . . 629A.45 WKB solution near the turning points . . . . . . . . . . . . . . 630A.46 Three-dimensional scattering . . . . . . . . . . . . . . . . . . . 634A.46.1 Partial wave analysis . . . . . . . . . . . . . . . . . . . . 637A.46.2 The Born approximation . . . . . . . . . . . . . . . . . . 641A.46.3 The Born series . . . . . . . . . . . . . . . . . . . . . . . 643A.47 The evolution of probability . . . . . . . . . . . . . . . . . . . . 644A.48 A basic description of Lagrangian multipliers . . . . . . . . . . 648A.49 The generalized variational principle . . . . . . . . . . . . . . . 650A.50 Spin degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . 651A.51 Derivation of the approximation . . . . . . . . . . . . . . . . . 652A.52 Why a single Slater determinant is not exact . . . . . . . . . . 657A.53 Simplication of the Hartree-Fock energy . . . . . . . . . . . . 658A.54 Integral constraints . . . . . . . . . . . . . . . . . . . . . . . . . 662A.55 Generalized orbitals . . . . . . . . . . . . . . . . . . . . . . . . 663A.56 Derivation of the Hartree-Fock equations . . . . . . . . . . . . . 665A.57 Why the Fock operator is Hermitian . . . . . . . . . . . . . . . 671A.58 Correlation energy . . . . . . . . . . . . . . . . . . . . . . . . 672A.59 Explanation of the London forces . . . . . . . . . . . . . . . . . 675A.60 Ambiguities in the denition of electron anity . . . . . . . . . 679A.61 Why Floquet theory should be called so . . . . . . . . . . . . . 681A.62 Superuidity versus BEC . . . . . . . . . . . . . . . . . . . . . 681A.63 Explanation of Hunds rst rule . . . . . . . . . . . . . . . . . . 684A.64 The mechanism of ferromagnetism . . . . . . . . . . . . . . . . 686A.65 Number of system eigenfunctions . . . . . . . . . . . . . . . . . 687A.66 The fundamental assumption of quantum statistics . . . . . . . 690A.67 A problem if the energy is given . . . . . . . . . . . . . . . . . 692A.68 Derivation of the particle energy distributions . . . . . . . . . . 693xiv CONTENTSA.69 The canonical probability distribution . . . . . . . . . . . . . . 699A.70 Analysis of the ideal gas Carnot cycle . . . . . . . . . . . . . . 701A.71 The recipe of life . . . . . . . . . . . . . . . . . . . . . . . . . . 702A.72 Checks on the expression for entropy . . . . . . . . . . . . . . . 703A.73 Chemical potential and distribution functions . . . . . . . . . . 706A.74 The Fermi-Dirac integral for small temperature . . . . . . . . . 710A.75 Physics of the fundamental commutation relations . . . . . . . 710A.76 Multiple angular momentum components . . . . . . . . . . . . 711A.77 Components of vectors are less than the total vector . . . . . . 711A.78 The spherical harmonics with ladder operators . . . . . . . . . 712A.79 Why angular momenta components can be added . . . . . . . . 712A.80 Why the Clebsch-Gordan tables are bidirectional . . . . . . . . 713A.81 How to make Clebsch-Gordan tables . . . . . . . . . . . . . . . 713A.82 Machine language version of the Clebsch-Gordan tables . . . . . 713A.83 The triangle inequality . . . . . . . . . . . . . . . . . . . . . . . 714A.84 Awkward questions about spin . . . . . . . . . . . . . . . . . . 715A.85 More awkwardness about spin . . . . . . . . . . . . . . . . . . . 716A.86 Emergence of spin from relativity . . . . . . . . . . . . . . . . . 717A.87 Electromagnetic evolution of expectation values . . . . . . . . . 719A.88 Existence of magnetic monopoles . . . . . . . . . . . . . . . . . 721A.89 More on Maxwells third law . . . . . . . . . . . . . . . . . . . 721A.90 Various electrostatic derivations. . . . . . . . . . . . . . . . . . 722A.90.1 Existence of a potential . . . . . . . . . . . . . . . . . . 722A.90.2 The Laplace equation . . . . . . . . . . . . . . . . . . . . 723A.90.3 Egg-shaped dipole eld lines . . . . . . . . . . . . . . . . 724A.90.4 Ideal charge dipole delta function . . . . . . . . . . . . . 724A.90.5 Integrals of the current density . . . . . . . . . . . . . . 725A.90.6 Lorentz forces on a current distribution . . . . . . . . . . 726A.90.7 Field of a current dipole . . . . . . . . . . . . . . . . . . 727A.90.8 Biot-Savart law . . . . . . . . . . . . . . . . . . . . . . . 729A.91 Energy due to orbital motion in a magnetic eld . . . . . . . . 730A.92 Energy due to electron spin in a magnetic eld . . . . . . . . . 731A.93 Setting the record straight on alignment . . . . . . . . . . . . . 733A.94 Solving the NMR equations . . . . . . . . . . . . . . . . . . . . 733A.95 Derivation of perturbation theory . . . . . . . . . . . . . . . . . 733A.96 Stark eect on the hydrogen ground state . . . . . . . . . . . . 739A.97 Dirac ne structure Hamiltonian . . . . . . . . . . . . . . . . . 740A.98 Classical spin-orbit derivation . . . . . . . . . . . . . . . . . . . 747A.99 Expectation powers of r for hydrogen . . . . . . . . . . . . . . . 750A.100Symmetry eigenvalue preservation . . . . . . . . . . . . . . . . 754A.101Everetts theory and vacuum energy . . . . . . . . . . . . . . . 755A.102A tenth of a googol in universes . . . . . . . . . . . . . . . . . . 755CONTENTS xvWeb Pages 759Notations 761xvi CONTENTSList of Figures1.1 The classical picture of a vector. . . . . . . . . . . . . . . . . . . 61.2 Spike diagram of a vector. . . . . . . . . . . . . . . . . . . . . . 61.3 More dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Innite dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 The classical picture of a function. . . . . . . . . . . . . . . . . 71.6 Forming the dot product of two vectors. . . . . . . . . . . . . . 81.7 Forming the inner product of two functions. . . . . . . . . . . . 92.1 A visualization of an arbitrary wave function. . . . . . . . . . . 202.2 Combined plot of position and momentum components. . . . . . 222.3 The uncertainty principle illustrated. . . . . . . . . . . . . . . . 232.4 Classical picture of a particle in a closed pipe. . . . . . . . . . . 292.5 Quantum mechanics picture of a particle in a closed pipe. . . . . 292.6 Denitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7 One-dimensional energy spectrum for a particle in a pipe. . . . . 362.8 One-dimensional ground state of a particle in a pipe. . . . . . . 382.9 Second and third lowest one-dimensional energy states. . . . . . 392.10 Denition of all variables. . . . . . . . . . . . . . . . . . . . . . 402.11 True ground state of a particle in a pipe. . . . . . . . . . . . . . 422.12 True second and third lowest energy states. . . . . . . . . . . . . 432.13 A combination of 111 and 211 seen at some typical times. . . . 452.14 The harmonic oscillator. . . . . . . . . . . . . . . . . . . . . . . 472.15 The energy spectrum of the harmonic oscillator. . . . . . . . . . 522.16 Ground state 000 of the harmonic oscillator . . . . . . . . . . . 542.17 Wave functions 100 and 010. . . . . . . . . . . . . . . . . . . . 552.18 Energy eigenfunction 213. . . . . . . . . . . . . . . . . . . . . . 562.19 Arbitrary wave function (not an energy eigenfunction). . . . . . 593.1 Spherical coordinates of an arbitrary point P. . . . . . . . . . . 643.2 Spectrum of the hydrogen atom. . . . . . . . . . . . . . . . . . . 763.3 Ground state wave function 100 of the hydrogen atom. . . . . . 793.4 Eigenfunction 200. . . . . . . . . . . . . . . . . . . . . . . . . . 80xviixviii LIST OF FIGURES3.5 Eigenfunction 210, or 2pz. . . . . . . . . . . . . . . . . . . . . . 813.6 Eigenfunction 211 (and 211). . . . . . . . . . . . . . . . . . . 813.7 Eigenfunctions 2px, left, and 2py, right. . . . . . . . . . . . . . . 823.8 Hydrogen atom plus free proton far apart. . . . . . . . . . . . . 1003.9 Hydrogen atom plus free proton closer together. . . . . . . . . . 1003.10 The electron being anti-symmetrically shared. . . . . . . . . . . 1023.11 The electron being symmetrically shared. . . . . . . . . . . . . . 1034.1 State with two neutral atoms. . . . . . . . . . . . . . . . . . . . 1144.2 Symmetric sharing of the electrons. . . . . . . . . . . . . . . . . 1164.3 Antisymmetric sharing of the electrons. . . . . . . . . . . . . . . 1164.4 Approximate solutions for hydrogen (left) and helium (right). . 1514.5 Approximate solutions for lithium (left) and beryllium (right). 1534.6 Example approximate solution for boron. . . . . . . . . . . . . . 1554.7 Covalent sigma bond consisting of two 2pz states. . . . . . . . . 1594.8 Covalent pi bond consisting of two 2px states. . . . . . . . . . . 1604.9 Covalent sigma bond consisting of a 2pz and a 1s state. . . . . . 1614.10 Shape of an sp3hybrid state. . . . . . . . . . . . . . . . . . . . . 1634.11 Shapes of the sp2(left) and sp (right) hybrids. . . . . . . . . . . 1645.1 Emission and absorption of radiation by an atom. . . . . . . . . 1805.2 Triangle inequality. . . . . . . . . . . . . . . . . . . . . . . . . . 1875.3 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. . . . . . . . . . . . 1965.4 The real part (red) and envelope (black) of an example wave. . . 2015.5 The wave moves with the phase speed. . . . . . . . . . . . . . . 2015.6 The real part (red) and magnitude or envelope (black) of a wavepacket. (Schematic) . . . . . . . . . . . . . . . . . . . . . . . . . 2025.7 The velocities of wave and envelope are not equal. . . . . . . . . 2035.8 A particle in free space. . . . . . . . . . . . . . . . . . . . . . . 2065.9 An accelerating particle. . . . . . . . . . . . . . . . . . . . . . . 2075.10 An decelerating particle. . . . . . . . . . . . . . . . . . . . . . . 2085.11 Unsteady solution for the harmonic oscillator. The third pictureshows the maximum distance from the nominal position that thewave packet reaches. . . . . . . . . . . . . . . . . . . . . . . . . 2085.12 Harmonic oscillator potential energy V , eigenfunction h50, andits energy E50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2095.13 A partial reection. . . . . . . . . . . . . . . . . . . . . . . . . . 2145.14 An tunneling particle. . . . . . . . . . . . . . . . . . . . . . . . 2145.15 Penetration of an innitely high potential energy barrier. . . . . 215LIST OF FIGURES xix5.16 Schematic of a scattering potential and the asymptotic behaviorof an example energy eigenfunction for a wave packet coming infrom the far left. . . . . . . . . . . . . . . . . . . . . . . . . . . 2165.17 Half-life versus energy release for the atomic nuclei marked inNUBASE 2003 as showing pure alpha decay with unqualied en-ergies. Top: only the even values of the mass and atomic numberscherry-picked. Inset: really cherry-picking, only a few even massnumbers for thorium and uranium! Bottom: all the nuclei exceptone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2205.18 Schematic potential for an alpha particle that tunnels out of anucleus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.19 Half-life predicted by the Gamow / Gurney & Condon theoryversus the true value. . . . . . . . . . . . . . . . . . . . . . . . . 2247.1 Billiard-ball model of the salt molecule. . . . . . . . . . . . . . . 2627.2 Billiard-ball model of a salt crystal. . . . . . . . . . . . . . . . . 2637.3 The salt crystal disassembled to show its structure. . . . . . . . 2657.4 Sketch of electron energy spectra in solids. . . . . . . . . . . . . 2667.5 The lithium atom, scaled more correctly than in chapter 4.9 . . 2687.6 Body-centered-cubic (bcc) structure of lithium. . . . . . . . . . 2697.7 Fully periodic wave function of a two-atom lithium crystal. . . 2717.8 Flip-op wave function of a two-atom lithium crystal. . . . . . 2727.9 Wave functions of a four-atom lithium crystal. The actualpicture is that of the fully periodic mode. . . . . . . . . . . . . . 2737.10 Reciprocal lattice of a one-dimensional crystal. . . . . . . . . . . 2777.11 Schematic of energy bands. . . . . . . . . . . . . . . . . . . . . . 2787.12 Energy versus linear momentum. . . . . . . . . . . . . . . . . . 2807.13 Schematic of merging bands. . . . . . . . . . . . . . . . . . . . . 2817.14 A primitive cell and primitive translation vectors of lithium. . . 2827.15 Wigner-Seitz cell of the bcc lattice. . . . . . . . . . . . . . . . . 2837.16 Schematic of crossing bands. . . . . . . . . . . . . . . . . . . . . 2877.17 Ball and stick schematic of the diamond crystal. . . . . . . . . . 2887.18 Allowed wave number vectors. . . . . . . . . . . . . . . . . . . . 2927.19 Schematic energy spectrum of the free electron gas. . . . . . . . 2937.20 Occupied wave number states and Fermi surface in the groundstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947.21 Density of states for the free electron gas. . . . . . . . . . . . . . 2967.22 Energy states, top, and density of states, bottom, when there issevere connement in the y-direction, as in a quantum well. . . . 2977.23 Energy states, top, and density of states, bottom, when thereis severe connement in both the y- and z-directions, as in aquantum wire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299xx LIST OF FIGURES7.24 Energy states, top, and density of states, bottom, when there issevere connement in all three directions, as in a quantum dot orarticial atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3007.25 Wave number vectors seen in a cross section of constant kz. Top:sinusoidal solutions. Bottom: exponential solutions. . . . . . . . 3027.26 Assumed simple cubic reciprocal lattice, shown as black dots, incross-section. The boundaries of the surrounding primitive cellsare shown as thin red lines. . . . . . . . . . . . . . . . . . . . . 3047.27 Occupied states for one, two, and three free electrons per physicallattice cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3067.28 Redenition of the occupied wave number vectors into Brillouinzones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3077.29 Second, third, and fourth Brillouin zones seen in the periodiczone scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3087.30 The red dot shows the wavenumber vector of a sample free elec-tron wave function. It is to be corrected for the lattice potential. 3107.31 The grid of nonzero Hamiltonian perturbation coecients andthe problem sphere in wave number space. . . . . . . . . . . . . 3117.32 Tearing apart of the wave number space energies. . . . . . . . . 3137.33 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. . . . . . . . . . . . . . . . . . . . . . . 3147.34 Bragg planes seen in wave number space cross section. . . . . . 3157.35 Occupied states for the energies of gure 7.33 if there are twovalence electrons per lattice cell. Left: energy. Right: wavenumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3157.36 Smaller lattice potential. From top to bottom shows one, twoand three valence electrons per lattice cell. Left: energy. Right:wave numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3167.37 Sketch of electron energy spectra in solids at absolute zero tem-perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3187.38 Sketch of electron energy spectra in solids at a nonzero temperature.3187.39 Specic heat at constant volume of gases. Temperatures fromabsolute zero to 1200 K. Data from NIST-JANAF and AIP. . . 3237.40 Specic heat at constant pressure of solids. Temperatures fromabsolute zero to 1200 K. Carbon is diamond; graphite is similar.Water is ice and liquid. Data from NIST-JANAF, CRC, AIP,Rohsenow et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3257.41 Depiction of an electromagnetic ray. . . . . . . . . . . . . . . . . 3317.42 Law of reection in elastic scattering from a plane. . . . . . . . 3317.43 Scattering from multiple planes of atoms. . . . . . . . . . . . 3327.44 Dierence in travel distance when scattered from P rather than O.334LIST OF FIGURES xxi8.1 Graphical depiction of an arbitrary system energy eigenfunctionfor 95 distinguishable particles. . . . . . . . . . . . . . . . . . . 3418.2 Graphical depiction of an arbitrary system energy eigenfunctionfor 95 identical bosons. . . . . . . . . . . . . . . . . . . . . . . . 3428.3 Graphical depiction of an arbitrary system energy eigenfunctionfor 31 identical fermions. . . . . . . . . . . . . . . . . . . . . . . 3438.4 Illustrative small model system having 4 distinguishable particles.The particular eigenfunction shown is arbitrary. . . . . . . . . . 3468.5 The number of system energy eigenfunctions for a simple modelsystem with only three energy buckets. Positions of the squaresindicate the numbers of particles in buckets 2 and 3; darknessof the squares indicates the relative number of eigenfunctionswith those bucket numbers. Left: system with 4 distinguishableparticles, middle: 16, right: 64. . . . . . . . . . . . . . . . . . . 3468.6 Number of energy eigenfunctions on the oblique energy line in 8.5.(The curves are mathematically interpolated to allow a continu-ously varying fraction of particles in bucket 2.) Left: 4 particles,middle: 64, right: 1024. . . . . . . . . . . . . . . . . . . . . . . . 3488.7 Probabilities for bucket-number sets for the simple 64 particlemodel system if there is uncertainty in energy. More proba-ble bucket-number distributions are shown darker. Left: iden-tical bosons, middle: distinguishable particles, right: identicalfermions. The temperature is the same as in gure 8.5. . . . . . 3538.8 Probabilities of bucket-number sets for the simple 64 particlemodel system if bucket 1 is a non-degenerate ground state. Left:identical bosons, middle: distinguishable particles, right: identi-cal fermions. The temperature is the same as in gure 8.7. . . . 3548.9 Like gure 8.8, but at a lower temperature. . . . . . . . . . . . . 3548.10 Like gure 8.8, but at a still lower temperature. . . . . . . . . . 3558.11 Schematic of the Carnot refrigeration cycle. . . . . . . . . . . . 3628.12 Schematic of the Carnot heat engine. . . . . . . . . . . . . . . . 3658.13 A generic heat pump next to a reversed Carnot one with the sameheat delivery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3668.14 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. 3689.1 Example bosonic ladders. . . . . . . . . . . . . . . . . . . . . . . 3999.2 Example fermionic ladders. . . . . . . . . . . . . . . . . . . . . . 3999.3 Triplet and singlet states in terms of ladders . . . . . . . . . . . 4059.4 Clebsch-Gordan coecients of two spin one half particles. . . . . 4069.5 Clebsch-Gordan coecients for lb equal to one half. . . . . . . . 407xxii LIST OF FIGURES9.6 Clebsch-Gordan coecients for lb equal to one. . . . . . . . . . . 4089.7 Relationship of Maxwells rst equation to Coulombs law. . . . 4189.8 Maxwells rst equation for a more arbitrary region. The gureto the right includes the eld lines through the selected points. . 4199.9 The net number of eld lines leaving a region is a measure forthe net charge inside that region. . . . . . . . . . . . . . . . . . 4209.10 Since magnetic monopoles do not exist, the net number of mag-netic eld lines leaving a region is always zero. . . . . . . . . . . 4219.11 Electric power generation. . . . . . . . . . . . . . . . . . . . . . 4229.12 Two ways to generate a magnetic eld: using a current (left) orusing a varying electric eld (right). . . . . . . . . . . . . . . . . 4239.13 Electric eld and potential of a charge that is distributed uni-formly within a small sphere. The dotted lines indicate the valuesfor a point charge. . . . . . . . . . . . . . . . . . . . . . . . . . 4289.14 Electric eld of a two-dimensional line charge. . . . . . . . . . . 4299.15 Field lines of a vertical electric dipole. . . . . . . . . . . . . . . 4309.16 Electric eld of a two-dimensional dipole. . . . . . . . . . . . . . 4319.17 Field of an ideal magnetic dipole. . . . . . . . . . . . . . . . . . 4329.18 Electric eld of an almost ideal two-dimensional dipole. . . . . . 4339.19 Magnetic eld lines around an innite straight electric wire. . . 4379.20 An electromagnet consisting of a single wire loop. The generatedmagnetic eld lines are in blue. . . . . . . . . . . . . . . . . . . 4389.21 A current dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . 4399.22 Electric 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. . . . . . . . . . . . . . . . . . . . . 4409.23 Variables for the computation of the moment on a wire loop in amagnetic eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.24 Larmor precession of the expectation spin (or magnetic moment)vector around the magnetic eld. . . . . . . . . . . . . . . . . . 4509.25 Probability of being able to nd the nuclei at elevated energyversus time for a given perturbation frequency . . . . . . . . . 4529.26 Maximum probability of nding the nuclei at elevated energy. . 4529.27 A perturbing magnetic eld, rotating at precisely the Larmorfrequency, causes the expectation spin vector to come cascadingdown out of the ground state. . . . . . . . . . . . . . . . . . . . 45310.1 Graphical depiction of an arbitrary system energy eigenfunctionfor 95 distinguishable particles. . . . . . . . . . . . . . . . . . . 482LIST OF FIGURES xxiii10.2 Graphical depiction of an arbitrary system energy eigenfunctionfor 95 identical bosons. . . . . . . . . . . . . . . . . . . . . . . . 48310.3 Graphical depiction of an arbitrary system energy eigenfunctionfor 31 identical fermions. . . . . . . . . . . . . . . . . . . . . . . 48410.4 Example wave functions for a system with just one type of singleparticle state. Left: identical bosons; right: identical fermions. . 48510.5 Annihilation and creation operators for a system with just onetype of single particle state. Left: identical bosons; right: identi-cal fermions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48811.1 Separating the hydrogen ion. . . . . . . . . . . . . . . . . . . . . 51311.2 The Bohm experiment before the Venus measurement (left), andimmediately after it (right). . . . . . . . . . . . . . . . . . . . . 51411.3 Spin measurement directions. . . . . . . . . . . . . . . . . . . . 51511.4 Earths view of events (left), and that of a moving observer (right).51711.5 Bohms version of the Einstein, Podolski, Rosen Paradox . . . . 52511.6 Non entangled positron and electron spins; up and down. . . . . 52611.7 Non entangled positron and electron spins; down and up. . . . . 52611.8 The wave functions of two universes combined . . . . . . . . . . 52611.9 The Bohm experiment repeated. . . . . . . . . . . . . . . . . . . 52911.10Repeated experiments on the same electron. . . . . . . . . . . . 530A.1 Coordinate systems for the Lorentz transformation. . . . . . . . 552A.2 Example elastic collision seen by dierent observers. . . . . . . . 563A.3 A completely inelastic collision. . . . . . . . . . . . . . . . . . . 565A.4 Example energy eigenfunction for the particle in free space. . . . 621A.5 Example energy eigenfunction for a particle entering a constantaccelerating force eld. . . . . . . . . . . . . . . . . . . . . . . . 622A.6 Example energy eigenfunction for a particle entering a constantdecelerating force eld. . . . . . . . . . . . . . . . . . . . . . . . 623A.7 Example energy eigenfunction for the harmonic oscillator. . . . . 624A.8 Example energy eigenfunction for a particle encountering a briefaccelerating force. . . . . . . . . . . . . . . . . . . . . . . . . . . 625A.9 Example energy eigenfunction for a particle tunneling through abarrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626A.10 Example energy eigenfunction for tunneling through a delta func-tion barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626A.11 The Airy Ai and Bi functions that solve the Hamiltonian eigen-value problem for a linearly varying potential energy. Bi veryquickly becomes too large to plot for positive values of its argu-ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631A.12 Connection formulae for a turning point from classical to tunneling.632xxiv LIST OF FIGURESA.13 Connection formulae for a turning point from tunneling to classical.632A.14 WKB approximation of tunneling. . . . . . . . . . . . . . . . . . 633A.15 Scattering of a beam o a target. . . . . . . . . . . . . . . . . . 635A.16 Graphical interpretation of the Born series. . . . . . . . . . . . . 644A.17 Possible polarizations of a pair of hydrogen atoms. . . . . . . . . 676A.18 Schematic of an example boson distribution in a bucket. . . . . 690A.19 Schematic of the Carnot refrigeration cycle. . . . . . . . . . . . 702List of Tables2.1 First few one-dimensional eigenfunctions of the harmonic oscillator. 503.1 The rst few spherical harmonics. . . . . . . . . . . . . . . . . . 673.2 The rst few radial wave functions for hydrogen. . . . . . . . . . 744.1 Abbreviated periodic table of the elements, showing element sym-bol, atomic number, ionization energy, and electronegativity. . . 1529.1 Elecromagnetics I: Fundamental equations and basic solutions. . 4269.2 Elecromagnetics II: Electromagnetostatic solutions. . . . . . . . 427xxvxxvi LIST OF TABLESPrefaceTo the StudentThis book was primarily written for engineering graduate students who ndthemselves caught up in nano technology. It is a simple fact that the typicalengineering education does not provide anywhere close to the amount of physicsyou will need to make sense out of the literature of your eld. You can startfrom scratch as an undergraduate in the physics department, or you can readthis book.This book covers the real quantum mechanics; it is not just a summaryof selected results as you can nd elsewhere. The rst part of this book pro-vides a solid introduction to classical (i.e. nonrelativistic) quantum mechanics.It is intended to explain the ideas both rigorously and clearly. It follows ajust-in-time learning approach. The mathematics is fully explained, but notemphasized. The intention is not to practice clever mathematics, but to under-stand quantum mechanics. The coverage is at the normal calculus and physicslevel of undergraduate engineering students. If you did well in these courses, youshould be able to understand the discussion, assuming that you start readingfrom the beginning. There are some hints in the notations section, if you forgotsome calculus. If you forgot some physics, just dont worry too much about it:quantum physics is so much dierent that even the most basic concepts need tobe covered from scratch.Derivations are usually banned to notes at the end of this book, in caseyou need them for one reason or the other. They correct a considerable numberof mistakes that you will nd in other mainstream books. No doubt they addsome new ones. Let me know and I will x them in a jiy; that is the advantageof a web book.Some sections are marked [descriptive]. These sections provide an a nonmathematical introduction to material that you will almost certainly run into ifyou stay in nano technology. Read through them, more than once, so that youhave a general idea of what they are all about. If you need to know more aboutthese topics, the introductions in this book should make dedicated textbooksmore easily accessible.xxviixxviii LIST OF TABLESThe second part of this book discusses more advanced topics. It starts withnumerical methods, since engineering graduate students are typically supportedby a research grant, and the quicker you can produce some results, the better.A description of density functional theory is still missing, unfortunately.The remaining chapters of the second part are intended to provide a crashcourse on many topics that nano literature would consider elementary physics,but that nobody has ever told you about. Most of it is not really part of whatis normally understood to be a quantum mechanics course. Reading, rereading,and understanding it is highly recommended anyway.The purpose is not just to provide basic literacy in those topics, althoughthat is very important. But the purpose is also explain enough of their funda-mentals, in terms that an engineer can understand, so that you can make senseof the literature in those elds if you do need to know more than can be coveredhere. Consider these chapters gateways into their topic areas.There is a nal chapter on how to interpret quantum mechanics philosoph-ically. Read it if you are interested; it will probably not help you do quantummechanics any better. But as a matter of basic literacy, it is good to know howtruly weird quantum mechanics really is.The usual Why this book? blah-blah can be found in a note at the backof this book, A.1 A version history is in note A.2.AcknowledgmentsThis book is for a large part based on my reading of the excellent book byGriths, [10]. It now contains essentially all material in that book in one wayor the other. (But you may need to look in the notes for some of it.) This bookalso evolved to include a lot of additional material that I thought would beappropriate for a physically-literate engineer. There are chapters on numericalmethods, thermodynamics, solid mechanics, and electromagnetism.Somewhat to my surprise, I nd that my coverage actually tends to be closerto Yarivs book, [20]. I still think Griths is more readable for an engineer,though Yariv has some very good items Griths does not.The discussions on two-state systems are mainly based on Feynmans notes,[9, chapters 8-11]. Since it is hard to determine the precise statements beingmade, much of that has been augmented by data from web sources, mainly thosereferenced.The nanomaterials lectures of colleague Anter El-Azab that I audited in-spired me to add a bit on simple quantum connement to the rst systemstudied, the particle in the box. That does add a bit to a section that I wantedto keep as simple as possible, but then I gure it also adds a sense that this isreally relevant stu for future engineers. I also added a discussion of the eectsLIST OF TABLES xxixof connement on the density of states to the section on the free electron gas.I thank Swapnil Jain for pointing out that the initial subsection on quantumconnement in the pipe was denitely unclear and is hopefully better now.I thank Johann Joss for pointing out a mistake in the formula for the aver-aged energy of two-state systems. Harald Kirsch reported various problems inthe sections on conservation laws and on position eigenfunctions.The note on the derivation of the selection rules is from [10] and lecture notesfrom a University of Tennessee quantum course taught by Marianne Breinig.The subsection on conservation laws and selection rules is mainly from Ellis,[4].The section on the Born-Oppenheimer approximation comes from Wikipe-dia, [[9]], with modications including the inclusion of spin.The section on the Hartree-Fock method is mainly based on Szabo andOstlund [18], a well-written book, with some Parr and Yang [13] thrown in.The section on solids is mainly based on Sproull, [16], a good source forpractical knowledge about application of the concepts. It is surprisingly up todate, considering it was written half a century ago. Various items, however,come from Kittel [11]. The discussion of ionic solids really comes straight fromhyperphysics [[4]]. I prefer hyperphysics example of NaCl, instead of Sproullsequivalent discussion of KCl. My colleague Steve Van Sciver helped me getsome handle on what to say about helium and Bose-Einstein condensation.The thermodynamics section started from Griths discussion, [10], whichfollows Yarivs, [20]. However, it expanded greatly during writing. It now comesmostly from Baierlein [3], with some help from Feynman, [7], and some of thebooks I use in undergraduate thermo.The derivation of the classical energy of a spinning particle in a magneticeld is from Yariv, [20].The section on nuclear models is pieced together from the Handbook ofPhysics, Hyperphysics, Mayers Nobel prize lecture, and various web sources.The brief description of quantum eld theory is mostly from Wikipedia, witha bit of ll-in from Feynman [7] and Kittel [11]. The example on eld operatorsis an exercise from Srednicki [17], whose solution was posted online by a TA ofJoe Polchinski from UCSB.The many-worlds discussion is based on Everetts exposition, [5]. It is bril-liant but quite impenetrable.The idea of using the Lagrangian for the derivations of relativistic mechanicsis from A. Kompanayets, theoretical physics, an excellent book.Acknowledgements for specic items are not listed here if a citation is givenin the text, or if, as far as I know, the argument is standard theory. This is a textbook, not a research paper or historical note. But if a reference is appropriatesomewhere, let me know.xxx LIST OF TABLESComments and FeedbackIf you nd an error, please let me know. There seems to be an unending supplyof them. As one author described it brilliantly, the hand is still writing thoughthe brain has long since disengaged.Also let me know if you nd points that are unclear to the intended reader-ship, ME graduate students with a typical exposure to mathematics and physics,or equivalent. Every section, except a few explicitly marked as requiring ad-vanced linear algebra, should be understandable by anyone with a good knowl-edge of calculus and undergraduate physics.The same for sections that cannot be understood without delving back intoearlier material. All within reason of course. If you pick a random starting wordamong the half million or so and start reading from there, you most likely willbe completely lost. But sections are intended to be fairly self-contained, andyou should be able read one without backing up through all of the text.General editorial comments are also welcome. Ill skip the philosophicaldiscussions. I am an engineer.Feedback can be e-mailed to me at quantum@dommelen.net.This is a living document. I am still adding things here and there, and xingvarious mistakes and doubtful phrasing. Even before every comma is perfect,I think the document can be of value to people looking for an easy-to-readintroduction to quantum mechanics at a calculus level. So I am treating it assoftware, with version numbers indicating the level of condence I have in it all.Part IBasic Quantum Mechanics1Chapter 1Mathematical PrerequisitesQuantum mechanics is based on a number of advanced mathematical ideas thatare described in this chapter.1.1 Complex NumbersQuantum mechanics is full of complex numbers, numbers involvingi =1.Note that1 is not an ordinary, real, number, since there is no real numberwhose square is 1; the square of a real number is always positive. This sectionsummarizes the most important properties of complex numbers.First, any complex number, call it c, can by denition always be written inthe formc = cr + ici (1.1)where both cr and ci are ordinary real numbers, not involving1. The numbercr is called the real part of c and ci the imaginary part.You can think of the real and imaginary parts of a complex number as thecomponents of a two-dimensional vector:crci.......cThe length of that vector is called the magnitude, or absolute value [c[ ofthe complex number. It equals[c[ =

c2r +c2i.34 CHAPTER 1. MATHEMATICAL PREREQUISITESComplex numbers can be manipulated pretty much in the same way asordinary numbers can. A relation to remember is:1i = i (1.2)which can be veried by multiplying top and bottom of the fraction by i andnoting that by denition i2= 1 in the bottom.The complex conjugate of a complex number c, denoted by c, is found byreplacing i everywhere by i. In particular, if c = cr + ici, where cr and ci arereal numbers, the complex conjugate isc = crici (1.3)The following picture shows that graphically, you get the complex conjugate ofa complex number by ipping it over around the horizontal axis:crcici.......ccYou can get the magnitude of a complex number c by multiplying c with itscomplex conjugate c and taking a square root:[c[ =cc (1.4)If c = cr + icr, where cr and ci are real numbers, multiplying out cc shows themagnitude of c to be[c[ =

c2r + c2iwhich is indeed the same as before.From the above graph of the vector representing a complex number c, thereal part is cr = [c[ cos where is the angle that the vector makes with thehorizontal axis, and the imaginary part is ci = [c[ sin . So you can write anycomplex number in the formc = [c[ (cos + i sin )The critically important Euler formula says that:cos + i sin = ei(1.5)1.1. COMPLEX NUMBERS 5So, any complex number can be written in polar form asc = [c[ei(1.6)where both the magnitude [c[ and the phase angle (or argument) are realnumbers.Any complex number of magnitude one can therefor be written as ei. Notethat the only two real numbers of magnitude one, 1 and 1, are included for = 0, respectively = . The number i is obtained for = /2 and i for = /2.(See note A.6 if you want to know where the Euler formula comes from.)Key Points Complex numbers include the square root of minus one, i, as a validnumber. All complex numbers can be written as a real part plus i times animaginary part, where both parts are normal real numbers. The complex conjugate of a complex number is obtained by replacingi everywhere by i. The magnitude of a complex number is obtained by multiplying thenumber by its complex conjugate and then taking a square root. The Euler formula relates exponentials to sines and cosines.1.1 Review Questions1 Multiply out (2 + 3i)2and then nd its real and imaginary part.2 Show more directly that 1/i = i.3 Multiply out (2+3i)(23i) and then nd its real and imaginary part.4 Find the magnitude or absolute value of 2 + 3i.5 Verify that (2 3i)2is still the complex conjugate of (2 +3i)2if bothare multiplied out.6 Verify that e2iis still the complex conjugate of e2iafter both arerewritten using the Euler formula.7 Verify that

ei+ei

/2 = cos .8 Verify that

eiei

/2i = sin .6 CHAPTER 1. MATHEMATICAL PREREQUISITES1.2 Functions as VectorsThe second mathematical idea that is critical for quantum mechanics is thatfunctions can be treated in a way that is fundamentally not that much dierentfrom vectors.A vector f (which might be velocity v, linear momentum p = mv, force F,or whatever) is usually shown in physics in the form of an arrow:Figure 1.1: The classical picture of a vector.However, the same vector may instead be represented as a spike diagram,by plotting the value of the components versus the component index:Figure 1.2: Spike diagram of a vector.(The symbol i for the component index is not to be confused with i =1.)In the same way as in two dimensions, a vector in three dimensions, or, forthat matter, in thirty dimensions, can be represented by a spike diagram:Figure 1.3: More dimensions.1.2. FUNCTIONS AS VECTORS 7For a large number of dimensions, and in particular in the limit of innitelymany dimensions, the large values of i can be rescaled into a continuous coor-dinate, call it x. For example, x might be dened as i divided by the numberof dimensions. In any case, the spike diagram becomes a function f(x):Figure 1.4: Innite dimensions.The spikes are usually not shown:Figure 1.5: The classical picture of a function.In this way, a function is just a vector in innitely many dimensions.Key Points Functions can be thought of as vectors with innitely many compo-nents. This allows quantum mechanics do the same things with functions asyou can do with vectors.1.2 Review Questions1 Graphically compare the spike diagram of the 10-dimensional vectorv with components (0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) with the plot of thefunction f(x) = 0.5x.2 Graphically compare the spike diagram of the 10-dimensional unitvector 3, with components (0,0,1,0,0,0,0,0,0,0), with the plot of thefunction f(x) = 1. (No, they do not look alike.)8 CHAPTER 1. MATHEMATICAL PREREQUISITES1.3 The Dot, oops, INNER ProductThe dot product of vectors is an important tool. It makes it possible to ndthe length of a vector, by multiplying the vector by itself and taking the squareroot. It is also used to check if two vectors are orthogonal: if their dot productis zero, they are. In this subsection, the dot product is dened for complexvectors and functions.The usual dot product of two vectors f and g can be found by multiplyingcomponents with the same index i together and summing that:

f g f1g1 + f2g2 + f3g3(The emphatic equal, , is commonly used to indicate is by denition equal oris always equal.) Figure 1.6 shows multiplied components using equal colors.Figure 1.6: Forming the dot product of two vectors.Note the use of numeric subscripts, f1, f2, and f3 rather than fx, fy, and fz;it means the same thing. Numeric subscripts allow the three term sum aboveto be written more compactly as:

f g all ifigiThe is called the summation symbol.The lengt