Table Of Content

LECTURE NOTES ON QUANTUM MECHANICS Dr. Shun-Qing Shen Department of Physics The University of Hong Kong September 2004 Contents 0.1 General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Fundamental Concepts 1 1.1 Relation between experimental interpretations and theoretical inferences . . . 2 1.2 Materials from Britannica Online . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Photoelectric e(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Frank-Hertz experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Compton e(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 The Stern-Gerlach experiment . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 Sequential Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . 9 1.3.3 Analogy with Polarization of Light . . . . . . . . . . . . . . . . . . . . 11 1.4 Dirac Notation and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Base kets and Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.1 Eigenkets of an Observable . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.2 Eigenkets as Base kets: . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.3 Matrix Representation: . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Measurements, Observables & The Uncertainty Relation . . . . . . . . . . . . 18 ii CONTENTS – MANUSCRIPT 1.6.1 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.6.2 Spin1/2 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6.3 Probability Postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6.4 S and S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 (cid:123) (cid:124) 1.6.5 The Algebra of Spin Operators . . . . . . . . . . . . . . . . . . . . . . 23 1.6.6 Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.7.1 Transformation Operator . . . . . . . . . . . . . . . . . . . . . . . 27 1.7.2 Transformation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.7.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.8 Position, Momentum, and Translation . . . . . . . . . . . . . . . . . . . . . . 31 1.8.1 Continuous Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8.2 Some properties of the (cid:17)(cid:3)function. . . . . . . . . . . . . . . . . . . . . 32 1.8.3 Position Eigenkets and Position Measurements . . . . . . . . . . . . . 33 1.8.4 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.9 The Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Quantum Dynamics 46 2.1 Time Evolution and the Schrödinger Equation . . . . . . . . . . . . . . . . . . 46 2.1.1 Time Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1.2 The Schrodinger Equation. . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.3 Time Dependence of Expectation Value: Spin Precession. . . . . . . . 52 2.2 The Schrodinger versus the Heisenberg Picture . . . . . . . . . . . . . . . . . 54 2.2.1 Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2.2 Two Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iii CONTENTS – MANUSCRIPT 2.2.3 The Heisenberg Equation of Motion. . . . . . . . . . . . . . . . . . . . 56 2.2.4 How to construct a Hamiltonian . . . . . . . . . . . . . . . . . . . . . 57 2.3 Simple Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3.1 Eigenvalue and eigenstates . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3.2 Time Development of the Oscillator . . . . . . . . . . . . . . . . . . . 66 2.3.3 The Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4 Schrodinger Wave Equation: Simple Harmonic Oscillator . . . . . . . . . . . 68 2.5 Propagators and Feynman Path Integrals . . . . . . . . . . . . . . . . . . . . 70 2.5.1 Propagators in Wave Mechanics. . . . . . . . . . . . . . . . . . . . . . 70 2.5.2 Propagator as a Transition Amplitude. . . . . . . . . . . . . . . . . . . 75 2.6 The Gauge Transformation and Phase of Wave Function . . . . . . . . . . . . 79 2.6.1 Constant Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.6.2 Gauge Transformation in Electromagnetism . . . . . . . . . . . . . . . 81 2.6.3 The Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . 84 2.6.4 The Aharonov-Bohm E(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . 86 2.7 Interpretation of Wave Function. . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.7.1 What’s (cid:12) ((cid:123))? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 (cid:14) 2.7.2 The Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.8.1 One dimensional square well potential . . . . . . . . . . . . . . . . . . 91 2.8.2 A charged particle in a uniform magnetic field . . . . . . . . . . . . . 95 3 Theory of Angular Momentum 98 3.1 Rotation and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . 98 3.1.1 Finite versus infinitesimal rotation . . . . . . . . . . . . . . . . . . . . 99 iv CONTENTS – MANUSCRIPT 3.1.2 Orbital angular momentum . . . . . . . . . . . . . . . . . . . . . . . . 104 3.1.3 Rotation operator for spin 1/2 . . . . . . . . . . . . . . . . . . . . . . 106 3.1.4 Spin precession revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.2 Rotation Group and the Euler Angles . . . . . . . . . . . . . . . . . . . . . . 108 3.2.1 The Group Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.2 Orthogonal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2.3 “Special”? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.2.4 Unitary Unimodular Group . . . . . . . . . . . . . . . . . . . . . . . . 110 3.2.5 Euler Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.3 Eigenvalues and Eigenkets of Angular Momentum . . . . . . . . . . . . . . . 114 3.3.1 Representation of Rotation Operator . . . . . . . . . . . . . . . . . . . 120 3.4 Schwinger Oscillator Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.4.1 Spin 1/2 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.4.2 Two-spin—1/2 system . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.3 Explicit Formula for Rotation Matrices. . . . . . . . . . . . . . . . . . 128 3.5 Combination of Angular Momentum and Clebsh-Gordan Coe(cid:33)cients . . . . . 130 3.5.1 Clebsch-Gordan coe(cid:33)cients . . . . . . . . . . . . . . . . . . . . . . . . 133 3.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.6.1 Two spin-1/2 systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.6.2 Spin-orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4 Symmetries in Physics 140 4.1 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . 141 4.1.1 Symmetry in Classical Physics . . . . . . . . . . . . . . . . . . . . . . 141 4.1.2 Symmetry in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 142 v CONTENTS – MANUSCRIPT 4.1.3 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.1.4 Symmetry and symmetry breaking . . . . . . . . . . . . . . . . . . . . 144 4.1.5 Summary: symmetries in physics . . . . . . . . . . . . . . . . . . . . . 147 4.2 Discrete Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.2.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.2.2 The Momentum Operator . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2.3 The Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.2.4 Lattice Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.3 Permutation Symmetry and Identical Particles . . . . . . . . . . . . . . . . . 157 4.3.1 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.4 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4.1 Classical cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4.2 Antilinear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4.3 Antiunitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4.4 T for a zero spin particle . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.4.5 T for a nonzero spin particle . . . . . . . . . . . . . . . . . . . . . . . 162 5 Approximation Methods for Bound States 163 5.1 The Variation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1.1 Expectation value of the energy . . . . . . . . . . . . . . . . . . . . . . 164 5.1.2 Particle in a one-dimensional infinite square well . . . . . . . . . . . . 165 5.1.3 Ground State of Helium Atom . . . . . . . . . . . . . . . . . . . . . . 166 5.2 Stationary Perturbation Theory: Nondegenerate Case . . . . . . . . . . . . . 168 5.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.2.2 The Two-State Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 169 vi CONTENTS – MANUSCRIPT 5.2.3 Formal Development of Perturbation··· . . . . . . . . . . . . . . . . . 170 5.3 Application of the Perturbation Expansion . . . . . . . . . . . . . . . . . . . 174 5.3.1 Simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.3.2 Atomic hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.4 Stationary Perturbation Theory: Degenerate Case . . . . . . . . . . . . . . . 179 5.4.1 Revisited two-state problem . . . . . . . . . . . . . . . . . . . . . . . . 180 5.4.2 The basic procedure of degenerate perturbation theory . . . . . . . . . 181 5.4.3 Example: Zeeman E(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.4.4 Example: First Order Stark E(cid:30)ect in Hydrogen . . . . . . . . . . . . . 184 5.5 The Wentzel-Kramers-Brillouin (WKB) approximation . . . . . . . . . . . . . 186 5.6 Time-dependent Problem: Interacting Picture and Two-State Problem . . . . 186 5.6.1 Time-dependent Potential and Interacting Picture . . . . . . . . . . . 187 5.6.2 Time-dependent Two-State Problem . . . . . . . . . . . . . . . . . . . 188 5.7 Time-dependent Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . 191 5.7.1 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.7.2 Time-independent perturbation . . . . . . . . . . . . . . . . . . . . . . 193 5.7.3 Harmonic perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.7.4 The Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6 Collision Theory 196 6.1 Collisions in one- and three-dimensions . . . . . . . . . . . . . . . . . . . . . . 197 6.1.1 One-dimensional square potential barriers . . . . . . . . . . . . . . . . 197 6.2 Collision in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.3 Scattering by Spherically Symmetric Potentials . . . . . . . . . . . . . . . . . 204 6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 vii CONTENTS – MANUSCRIPT 6.4.1 Scattering by a square well . . . . . . . . . . . . . . . . . . . . . . . . 209 6.4.2 Scattering by a hard-sphere potential . . . . . . . . . . . . . . . . . . 211 6.5 Approximate Collision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.5.1 The Lippman-Schwinger Equation . . . . . . . . . . . . . . . . . . . . 212 6.5.2 The Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.5.3 Application: from Yukawa potential to Coloumb potential . . . . . . . 217 6.5.4 Identical Particles and Scattering . . . . . . . . . . . . . . . . . . . . . 218 6.6 Landau-Zener Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7 Selected Topics 220 7.1 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.1.1 Density Operator and Ensembles . . . . . . . . . . . . . . . . . . . . . 220 7.1.2 Quantum Statistical Mechanism . . . . . . . . . . . . . . . . . . . . . 223 7.1.3 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.1.4 Systems of non-interaction particles . . . . . . . . . . . . . . . . . . . 227 7.1.5 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . 230 7.1.6 Free fermion gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.2 Quantum Hall E(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.2.1 Hall E(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.2.2 Quantum Hall E(cid:30)ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.2.3 Laughlin’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.2.4 Charged particle in the presence of a magnetic field . . . . . . . . . . 240 7.2.5 Landau Level and Quantum Hall E(cid:30)ect . . . . . . . . . . . . . . . . . 243 7.3 Quantum Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7.3.1 Spin Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 viii CONTENTS – MANUSCRIPT 7.3.2 Two-Site Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.3.3 Ferromagnetic Exchange ((cid:77) (cid:63) 0) . . . . . . . . . . . . . . . . . . . . . 249 7.3.4 Antiferromagnetic Exchange . . . . . . . . . . . . . . . . . . . . . . . . 251 0.1 General Information • Aim/Following-up: The course provides an introduction to advanced techniques in quantum mechanics and their application to several selected topics in condensed matter physics. • Contents: Dirac notation and formalism, time evolution of quantum systems, angu- lar momentum theory, creation and annihilation operators (the second quantization representation), symmetries and conservation laws, permutation symmetry and identi- cal particles, quantum statistics, non-degenerate and degenerate perturbation theory, time-dependent perturbation theory, the variational method • Prerequisites: PHYS2321, PHYS2322, PHYS2323, and PHYS2325 • Co-requisite: Nil • Teaching: 36 hours of lectures and tutorial classes • Duration: One semester (1st semester) • Assessment: One-three hour examination (70%) and course assessment (30%) • Textbook: J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994) • Web page: http://bohr.physics.hku.hk/~phys3332/ All lectures notes in pdf files can be download from the site. ix CONTENTS – MANUSCRIPT • References: L. Schi(cid:30), Quantum Mechanics (McGraw-Hill, 1968, 3nd ed.); Richard Feynman, Robert B. Leighton, and Matthew L. Sands, Feynman Lectures on Physics Vol. III, (Addison-Wesley Publishing Co., 1965); L. D. Landau and E. M. Lifshitz, Quantum Mechanics Time and Venue: Year 2004 (Weeks 2 — 14) 11:40 -12:30: Monday/S802 11:40 -12:30: Wednesday/S802 11:40 -12:30: Friday/S802 x

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Lecture notes on quantum mechanics PDF

265 Pages·2004·2.706 MB·English
by  Shen S.-Q.
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