Lecture notes Advanced Quantum mechanics

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Fundamental Quantum Mechanics for Engineers Leon van Dommelen 5/5/07 Version 3.1 beta 3.Contents Dedication iii Preface v Why another book on quantum mechanics? . . . . . . . . . . . . . . . . . . . . . . v Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Comments and Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Wish list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Figures xvii List of Tables xxi 1 Mathematical Prerequisites 1 1.1 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Functions as Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Dot, oops, INNER Product . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Additional Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7.1 Dirac notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7.2 Additional independent variables . . . . . . . . . . . . . . . . . . . . . 14 2 Basic Ideas of Quantum Mechanics 15 2.1 The Revised Picture of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The Heisenberg Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 The Operators of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 19 2.4 The Orthodox Statistical Interpretation . . . . . . . . . . . . . . . . . . . . . . 21 2.4.1 Only eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Statistical selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Schr¨odinger’s Cat Background . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 A Particle Confined Inside a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6.1 The physical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.6.2 Mathematical notations . . . . . . . . . . . . . . . . . . . . . . . . . . 26 xi2.6.3 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.4 The Hamiltonian eigenvalue problem . . . . . . . . . . . . . . . . . . . 27 2.6.5 All solutions of the eigenvalue problem . . . . . . . . . . . . . . . . . . 28 2.6.6 Discussion of the energy values . . . . . . . . . . . . . . . . . . . . . . 32 2.6.7 Discussion of the eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 33 2.6.8 Three-dimensional solution . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6.9 Quantum confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 The Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.7.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.7.2 Solution using separation of variables . . . . . . . . . . . . . . . . . . . 41 2.7.3 Discussion of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 44 2.7.4 Discussion of the eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 47 2.7.5 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.7.6 Non-eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 Single-Particle Systems 55 3.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1 Definition of angular momentum . . . . . . . . . . . . . . . . . . . . . 55 3.1.2 Angular momentum in an arbitrary direction . . . . . . . . . . . . . . . 56 3.1.3 Square angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.4 Angular momentum uncertainty . . . . . . . . . . . . . . . . . . . . . . 61 3.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.2.2 Solution using separation of variables . . . . . . . . . . . . . . . . . . . 63 3.2.3 Discussion of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.4 Discussion of the eigenfunctions . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Expectation Value and Standard Deviation . . . . . . . . . . . . . . . . . . . . 74 3.3.1 Statistics of a die . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.2 Statistics of quantum operators . . . . . . . . . . . . . . . . . . . . . . 77 3.3.3 Simplified expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4 The Commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4.1 Commuting operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.4.2 Noncommuting operators and their commutator . . . . . . . . . . . . . 84 3.4.3 The Heisenberg uncertainty relationship . . . . . . . . . . . . . . . . . 84 3.4.4 Commutator reference Reference . . . . . . . . . . . . . . . . . . . . . 86 3.5 The Hydrogen Molecular Ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.5.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.5.2 Energy when fully dissociated . . . . . . . . . . . . . . . . . . . . . . . 89 3.5.3 Energy when closer together . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.4 States that share the electron . . . . . . . . . . . . . . . . . . . . . . . 91 3.5.5 Comparative energies of the states . . . . . . . . . . . . . . . . . . . . 92 3.5.6 Variational approximation of the ground state . . . . . . . . . . . . . . 93 3.5.7 Comparison with the exact ground state . . . . . . . . . . . . . . . . . 95 xii4 Multiple-Particle Systems 97 4.1 Generalization to Multiple Particles . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 The Hydrogen Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.2.2 Initial approximation to the lowest energy state . . . . . . . . . . . . . 99 4.2.3 The probability density . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2.4 States that share the electron . . . . . . . . . . . . . . . . . . . . . . . 100 4.2.5 Variational approximation of the ground state . . . . . . . . . . . . . . 102 4.2.6 Comparison with the exact ground state . . . . . . . . . . . . . . . . . 102 4.3 Two-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5 Instantaneous Interactions Background . . . . . . . . . . . . . . . . . . . . . 106 4.6 Multiple-Particle Systems Including Spin . . . . . . . . . . . . . . . . . . . . . 111 4.6.1 Wave function for a single particle with spin . . . . . . . . . . . . . . . 111 4.6.2 Inner products including spin . . . . . . . . . . . . . . . . . . . . . . . 112 4.6.3 Wave function for multiple particles with spin . . . . . . . . . . . . . . 113 4.6.4 Example: the hydrogen molecule . . . . . . . . . . . . . . . . . . . . . 114 4.6.5 Triplet and singlet states . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.7 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.8 Ways to Symmetrize the Wave Function . . . . . . . . . . . . . . . . . . . . . 116 4.9 Matrix Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.10 Global Symmetrization Background . . . . . . . . . . . . . . . . . . . . . . . 121 5 Examples of Multiple-Particle Systems 123 5.1 Heavier Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1.1 The Hamiltonian eigenvalue problem . . . . . . . . . . . . . . . . . . . 123 5.1.2 Approximate solution using separation of variables . . . . . . . . . . . 124 5.1.3 Hydrogen and helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1.4 Lithium to neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.1.5 Sodium to argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.1.6 Kalium to krypton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.2 Chemical Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.1 Covalent sigma bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.2 Covalent pi bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.2.3 Polar covalent bonds and hydrogen bonds . . . . . . . . . . . . . . . . 133 5.2.4 Promotion and hybridization . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.5 Ionic bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.2.6 Limitations of valence bond theory . . . . . . . . . . . . . . . . . . . . 137 5.3 Confined Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3.1 The Hamiltonian eigenvalue problem . . . . . . . . . . . . . . . . . . . 138 5.3.2 Solution by separation of variables . . . . . . . . . . . . . . . . . . . . 138 5.3.3 Discussion of the solution . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3.4 A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.3.5 The density of states and confinement Advanced . . . . . . . . . . . . 142 5.4 Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 xiii5.4.1 Derivation Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.5 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6 Time Evolution 163 6.1 The Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.1.1 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.1.2 Stationary states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.1.3 Time variations of symmetric two-state systems . . . . . . . . . . . . . 166 6.1.4 Time variation of expectation values . . . . . . . . . . . . . . . . . . . 167 6.1.5 Newtonian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.2 Unsteady perturbations of two-state systems . . . . . . . . . . . . . . . . . . . 169 6.2.1 Schr¨odinger equation for a two-state system . . . . . . . . . . . . . . . 169 6.2.2 Stimulated and spontaneous emission . . . . . . . . . . . . . . . . . . . 171 6.2.3 Absorption of radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.3 Conservation Laws and Symmetries Background . . . . . . . . . . . . . . . . 175 6.4 The Position and Linear Momentum Eigenfunctions . . . . . . . . . . . . . . . 179 6.4.1 The position eigenfunction . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.4.2 The linear momentum eigenfunction . . . . . . . . . . . . . . . . . . . 181 6.5 Wave Packets in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.5.1 Solution of the Schr¨odinger equation. . . . . . . . . . . . . . . . . . . . 183 6.5.2 Component wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.5.3 Wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.5.4 The group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.6 Motion near the Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.6.1 General procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.6.2 Motion through free space . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.6.3 Accelerated motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.6.4 Decelerated motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.6.5 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.7 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.7.1 Partial reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.7.2 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7 Some Additional Topics 197 7.1 All About Angular Momentum Advanced . . . . . . . . . . . . . . . . . . . . 197 7.1.1 The fundamental commutation relations . . . . . . . . . . . . . . . . . 198 7.1.2 Ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.1.3 Possible values of angular momentum . . . . . . . . . . . . . . . . . . . 202 7.1.4 A warning about angular momentum . . . . . . . . . . . . . . . . . . . 203 7.1.5 Triplet and singlet states . . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.1.6 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . 207 7.1.7 Pauli spin matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2 The Relativistic Dirac Equation Advanced . . . . . . . . . . . . . . . . . . . 213 7.2.1 The Dirac idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.2.2 Emergence of spin from relativity . . . . . . . . . . . . . . . . . . . . . 215 xiv7.3 The Electromagnetic Field Advanced . . . . . . . . . . . . . . . . . . . . . . 218 7.3.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.3.2 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.3.3 Electrons in magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . 227 7.4 Nuclear Magnetic Resonance Advanced . . . . . . . . . . . . . . . . . . . . . 229 7.4.1 Description of the method . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.4.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.4.3 The unperturbed system . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.4.4 Effect of the perturbation . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.5 Some Topics Not Covered Advanced . . . . . . . . . . . . . . . . . . . . . . . 236 7.6 The Meaning of Quantum Mechanics Background . . . . . . . . . . . . . . . 239 7.6.1 Failure of the Schr¨odinger Equation? . . . . . . . . . . . . . . . . . . . 240 7.6.2 The Many-Worlds Interpretation . . . . . . . . . . . . . . . . . . . . . 242 Notes 249 Bibliography 273 Web Pages 275 Notations 277 Index 295 xvList of Figures 1.1 The classical picture of a vector. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Spike diagram of a vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 More dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Infinite dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 The classical picture of a function. . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Forming the dot product of two vectors. . . . . . . . . . . . . . . . . . . . . . 6 1.7 Forming the inner product of two functions. . . . . . . . . . . . . . . . . . . . 7 2.1 A visualization of an arbitrary wave function. . . . . . . . . . . . . . . . . . . 16 2.2 Combined plot of position and momentum components. . . . . . . . . . . . . . 18 2.3 The uncertainty principle illustrated. . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 Classical picture of a particle in a closed pipe. . . . . . . . . . . . . . . . . . . 25 2.5 Quantum mechanics picture of a particle in a closed pipe. . . . . . . . . . . . . 25 2.6 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 One-dimensional energy spectrum for a particle in a pipe. . . . . . . . . . . . . 32 2.8 One-dimensional ground state of a particle in a pipe. . . . . . . . . . . . . . . 34 2.9 Second and third lowest one-dimensional energy states. . . . . . . . . . . . . . 34 2.10 Definition of all variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.11 True ground state of a particle in a pipe. . . . . . . . . . . . . . . . . . . . . . 37 2.12 True second and third lowest energy states. . . . . . . . . . . . . . . . . . . . . 38 2.13 The harmonic oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.14 The energy spectrum of the harmonic oscillator. . . . . . . . . . . . . . . . . . 45 2.15 Ground state ψ of the harmonic oscillator . . . . . . . . . . . . . . . . . . . 47 000 2.16 Wave functions ψ and ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 100 010 2.17 Energy eigenfunction ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 213 2.18 Arbitrary wave function (not an energy eigenfunction). . . . . . . . . . . . . . 52 3.1 Spherical coordinates of an arbitrary point P. . . . . . . . . . . . . . . . . . . 56 3.2 Spectrum of the hydrogen atom. . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 Ground state wave function ψ of the hydrogen atom. . . . . . . . . . . . . . 70 100 3.4 Eigenfunction ψ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 200 3.5 Eigenfunction ψ , or 2p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 210 z 3.6 Eigenfunction ψ (and ψ ). . . . . . . . . . . . . . . . . . . . . . . . . . . 72 211 21−1 3.7 Eigenfunctions 2p , left, and 2p , right. . . . . . . . . . . . . . . . . . . . . . . 73 x y 3.8 Hydrogen atom plus free proton far apart. . . . . . . . . . . . . . . . . . . . . 89 3.9 Hydrogen atom plus free proton closer together. . . . . . . . . . . . . . . . . . 90 xvii3.10 The electron being anti-symmetrically shared. . . . . . . . . . . . . . . . . . . 91 3.11 The electron being symmetrically shared. . . . . . . . . . . . . . . . . . . . . . 92 4.1 State with two neutral atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.2 Symmetric state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3 Antisymmetric state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4 Separating the hydrogen ion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.5 The Bohm experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.6 The Bohm experiment, after the Venus measurement. . . . . . . . . . . . . . . 108 4.7 Spin measurement directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.8 Earth’s view of events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.9 A moving observer’s view of events. . . . . . . . . . . . . . . . . . . . . . . . . 110 5.1 Approximate solutions for hydrogen (left) and helium (right). . . . . . . . . . 126 5.2 Approximate solutions for lithium (left) and beryllium (right). . . . . . . . . 128 5.3 Example approximate solution for boron. . . . . . . . . . . . . . . . . . . . . . 129 5.4 Covalent sigma bond consisting of two 2p states. . . . . . . . . . . . . . . . . 132 z 5.5 Covalent pi bond consisting of two 2p states. . . . . . . . . . . . . . . . . . . 132 x 5.6 Covalent sigma bond consisting of a 2p and a 1s state. . . . . . . . . . . . . . 133 z 3 5.7 Shape of an sp hybrid state.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2 5.8 Shapes of the sp (left) and sp (right) hybrids. . . . . . . . . . . . . . . . . . . 136 5.9 Allowed wave number vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.10 Schematic energy spectrum of the free electron gas. . . . . . . . . . . . . . . . 141 5.11 Occupied wave number states and Fermi surface in the ground state . . . . . . 141 5.12 Density of states for the free electron gas.. . . . . . . . . . . . . . . . . . . . . 144 5.13 Energy states, top, and density of states, bottom, when there is confinement in the y-direction, as in a quantum well. . . . . . . . . . . . . . . . . . . . . . . . 144 5.14 Energy states, top, and density of states, bottom, when there is confinement in both the y- and z-directions, as in a quantum wire. . . . . . . . . . . . . . . . 146 5.15 Energy states, top, and density of states, bottom, when there is confinement in all three directions, as in a quantum dot or artificial atom. . . . . . . . . . . . 147 5.16 Sketch of free electron and banded energy spectra. . . . . . . . . . . . . . . . . 148 5.17 Cross section of the full wave number space. . . . . . . . . . . . . . . . . . . . 150 5.18 The k-grid and k-sphere in wave number space. . . . . . . . . . . . . . . . . . 155 5.19 Tearing apart of the wave number space energies. . . . . . . . . . . . . . . . . 158 5.20 Energy, as radial distance from the origin, for varying wave number vector directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.21 Occupied levels in the ground state for two valence electrons per lattice cell. . 159 6.1 Emission and absorption of radiation by an atom. . . . . . . . . . . . . . . . . 171 6.2 ApproximateDiracdeltafunctionδ (x−ξ)isshownleft. Thetruedeltafunction ε δ(x−ξ) is the limit when ε becomes zero, and is an infinitely high, infinitely thin spike, shown right. It is the eigenfunction corresponding to a position ξ. . 180 6.3 The real part (red) and envelope (black) of an example wave. . . . . . . . . . . 184 6.4 The wave moves with the phase speed. . . . . . . . . . . . . . . . . . . . . . . 185 xviii6.5 The real part (red) and magnitude or envelope (black) of a typical wave packet 185 6.6 The velocities of wave and envelope are not equal. . . . . . . . . . . . . . . . . 186 6.7 A particle in free space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.8 An accelerating particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.9 An decelerating particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.10 Unsteady solution for the harmonic oscillator. The third picture shows the maximum distance from the nominal position that the wave packet reaches. . . 192 6.11 A partial reflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.12 An tunneling particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.13 Penetration of an infinitely high potential energy barrier. . . . . . . . . . . . . 194 7.1 Example bosonic ladders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.2 Example fermionic ladders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 7.3 Triplet and singlet states in terms of ladders . . . . . . . . . . . . . . . . . . . 207 7.4 Clebsch-Gordan coefficients of two spin 1/2 particles. . . . . . . . . . . . . . . 208 7.5 Clebsch-Gordan coefficients for l =1/2. . . . . . . . . . . . . . . . . . . . . . 209 b 7.6 Clebsch-Gordan coefficients for l =1.. . . . . . . . . . . . . . . . . . . . . . . 210 b 7.7 Relationship of Maxwell’s first equation to Coulomb’s law. . . . . . . . . . . . 221 7.8 Maxwell’s first equation for a more arbitrary region. The figure to the right includes the field lines through the selected points.. . . . . . . . . . . . . . . . 222 7.9 The net number of field lines leaving a region is a measure for the net charge inside that region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 7.10 Since magnetic monopoles do not exist, the net number of magnetic field lines leaving a region is always zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.11 Electric power generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.12 Twowaystogenerateamagneticfield: usingacurrent(left)orusingavarying electric field (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.13 Larmorprecessionoftheexpectationspin(ormagneticmoment)vectoraround the magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.14 Probability of being able to find the nuclei at elevated energy versus time for a given perturbation frequency ω. . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.15 Maximum probability of finding the nuclei at elevated energy. . . . . . . . . . 235 7.16 A perturbing magnetic field, rotating at precisely the Larmor frequency, causes the expectation spin vector to come cascading down out of the ground state. . 236 7.17 Bohm’s version of the Einstein, Podolski, Rosen Paradox . . . . . . . . . . . . 242 7.18 Non entangled positron and electron spins; up and down. . . . . . . . . . . . . 243 7.19 Non entangled positron and electron spins; down and up. . . . . . . . . . . . . 243 7.20 The wave functions of two universes combined . . . . . . . . . . . . . . . . . . 243 7.21 The Bohm experiment repeated. . . . . . . . . . . . . . . . . . . . . . . . . . . 245 7.22 Repeated experiments on the same electron. . . . . . . . . . . . . . . . . . . . 246 xixList of Tables 2.1 One-dimensional eigenfunctions of the harmonic oscillator, 3, p. 56. . . . . . . 43 3.1 The first few spherical harmonics, from 3, p. 139. . . . . . . . . . . . . . . . . 59 3.2 The first few radial wave functions for hydrogen, from 3, p. 154. . . . . . . . 66 5.1 Abbreviated periodic table of the elements, showing element symbol, atomic number, ionization energy, and electronegativity. . . . . . . . . . . . . . . . . . 127 xxiChapter 1 Mathematical Prerequisites Quantum mechanics is based on a number of advanced mathematical ideas that are described in this section. 1.1 Complex Numbers Quantum mechanics is full of complex numbers, numbers involving √ i= −1. √ Notethat −1isnotanordinary, “real”, number, sincethereisnoreal numberwhosesquare is −1; the square of a real number is always positive. This section summarizes the most important properties of complex numbers. First, any complex number, call it c, can by definition always be written in the form c= c +ic (1.1) r i √ where both c and c are ordinary real numbers, not involving −1. The number c is called r i r the real part of c and c the imaginary part. i We can think of the real and imaginary parts of a complex number as the components of a two-dimensional vector: c i © © © c © © © © c r 12 CHAPTER 1. MATHEMATICAL PREREQUISITES The length of that vector is called the “magnitude,” or “absolute value” c of the complex number. It equals q 2 2 c= c +c . r i Complexnumberscanbemanipulatedprettymuchinthesamewayasordinarynumberscan. A relation to remember is: 1 =−i (1.2) i which can be verified by multiplying top and bottom of the fraction by i and noting that by 2 definition i =−1 in the bottom. ∗ The complex conjugate of a complex number c, denoted by c , is found by replacing i every- where by −i. In particular, if c = c +ic , where c and c are real numbers, the complex r i r i conjugate is ∗ c = c −ic (1.3) r i The following picture shows that graphically, you get the complex conjugate of a complex number by flipping it over around the horizontal axis: c i © © © c © © © © H c r H H ∗ H c H H jH −c i Youcangetthemagnitudeofacomplexnumbercbymultiplyingcwithitscomplexconjugate ∗ c and taking a square root: √ ∗ c= c c (1.4) ∗ If c = c +ic , where c and c are real numbers, multiplying out c c shows the magnitude of r r r i c to be q 2 2 c= c +c r i which is indeed the same as before. From the above graph of the vector representing a complex number c, the real part is c = r ccosαwhereαistheanglethatthevectormakeswiththehorizontalaxis,andtheimaginary part is c =csinα. So we can write any complex number in the form i c=c(cosα+isinα) The critically important Euler identity says that: iα cosα+isinα = e (1.5)1.1. COMPLEX NUMBERS 3 So, any complex number can be written in “polar form” as iα c=ce where both the magnitudec and the angle α are real numbers. iα Any complex number of magnitude one can therefor be written as e . Note that the only two real numbers of magnitude one, 1 and −1, are included for α = 0, respectively α = π. The number i is obtained for α = π/2 and−i for α =−π/2. (See note1 if you want to know where the Euler identity comes from.) Key Points ¦ Complex numbers include the square root of minus one, i, as a valid number. ¦ All complex numbers can be written as a real part plus i times an imaginary part, where both parts are normal real numbers. ¦ The complex conjugate of a complex number is obtained by replacing i everywhere by −i. ¦ The magnitude of a complex number is obtained by multiplying the number by its complex conjugate and then taking a square root. ¦ The Euler identity relates exponentials to sines and cosines. 1.1 Review Questions 2 1 Multiply out (2+3i) and then find its real and imaginary part. 2 Show more directly that 1/i =−i. 3 Multiply out (2+3i)(2−3i) and then find its real and imaginary part. 4 Find the magnitude or absolute value of 2+3i. 2 2 5 Verify that (2−3i) is still the complex conjugate of (2+3i) if both are multiplied out. −2i 2i 6 Verify that e is still the complex conjugate of e after both are rewritten using the Euler identity. ¡ ¢ iα −iα 7 Verify that e +e /2 = cosα. ¡ ¢ iα −iα 8 Verify that e −e /2i = sinα.4 CHAPTER 1. MATHEMATICAL PREREQUISITES 1.2 Functions as Vectors The second mathematical idea that is critical for quantum mechanics is that functions can be treated in a way that is fundamentally not that much different from 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, for that matter, in thirty dimensions, can be represented by a spike diagram: Figure 1.3: More dimensions.1.2. FUNCTIONS AS VECTORS 5 Foralargenumberofdimensions, andinparticularinthelimitofinfinitelymanydimensions, the large values of i can be rescaled into a continuous coordinate, call it x. For example, x might be defined as i divided by the number of dimensions. In any case, the spike diagram becomes a function f(x): Figure 1.4: Infinite 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 infinitely many dimensions. Key Points ¦ Functions can be thought of as vectors with infinitely many components. ¦ This allows quantum mechanics do the same things with functions as we can do with vectors. 1.2 Review Questions 1 Graphically compare the spike diagram of the 10-dimensional vector v with compo- nents (0.5,1,1.5,2,2.5,3,3.5,4,4.5,5) with the plot of the function f(x) = 0.5x. 2 Graphically compare the spike diagram of the 10-dimensional unit vector ˆı , with 3 components (0,0,1,0,0,0,0,0,0,0), with the plot of the function f(x) = 1. (No, they do not look alike.)6 CHAPTER 1. MATHEMATICAL PREREQUISITES 1.3 The Dot, oops, INNER Product The dot product of vectors is an important tool. It makes it possible to find the length of a vector, by multiplying the vector by itself and taking the square root. It is also used to check if two vectors are orthogonal: if their dot product is zero, they are. In this subsection, the dot product is defined for complex vectors and functions. The usual dot product of two vectors f and g can be found by multiplying components with the same index i together and summing that: f·g≡ f g +f g +f g 1 1 2 2 3 3 (The emphatic equal, ≡, is commonly used to indicate “is by definition equal” or “is always equal.”) Figure 1.6 shows multiplied components using equal colors. Figure 1.6: Forming the dot product of two vectors. Notetheuseofnumericsubscripts,f ,f ,andf ratherthanf ,f ,andf ; itmeansthesame 1 2 3 x y z thing. Numeric subscripts allow the three term sum above to be written more compactly as: X f·g≡ f g i i all i The Σ is called the “summation symbol.” The length of a vector f, indicated byf or simply by f, is normally computed as q s X 2 f= f·f = f i all i However, this does not work correctly for complex vectors. The difficulty is that terms of the 2 2 form f are no longer necessarily positive numbers. For example, i =−1. i Therefore, itisnecessarytouseageneralized“innerproduct”forcomplexvectors, whichputs a complex conjugate on the first vector: X ∗ hfgi≡ f g (1.6) i i all i1.3. THE DOT, OOPS, INNER PRODUCT 7 Ifvectorf isreal,thecomplexconjugatedoesnothing,andtheinnerproducthfgiisthesame asthedotproductf·g. Otherwise,intheinnerproductf andg arenolongerinterchangeable; theconjugatesareonlyonthe firstfactor, f. Interchangingf andg changestheinnerproduct value into its complex conjugate. The length of a nonzero vector is now always a positive number: s q X 2 f= hffi= f (1.7) i all i Physicists take the inner product “bracket” verbally apart as hf gi bra /c ket and refer to vectors as bras and kets. Theinnerproductoffunctionsisdefinedinexactlythesamewayasforvectors,bymultiplying values at the same x position together and summing. But since there are infinitely many x- values, the sum becomes an integral: Z ∗ hfgi= f (x)g(x)dx (1.8) all x as illustrated in figure 1.7. Figure 1.7: Forming the inner product of two functions. The equivalent of the length of a vector is in case of a function called its “norm:” s Z q 2 f≡ hffi= f(x) dx (1.9) all x The double bars are used to avoid confusion with the absolute value of the function. A vector or function is called “normalized” if its length or norm is one: hffi=1 iff f is normalized. (1.10)

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