Copyright Chris H. Greene 2009

Graduate Quantum Mechanics I

Lecture Notes

by Chris H. Greene

Department of Physics and JILA

University of Colorado at Boulder

Fall 2009

Copyright Chris H. Greene 2009

Table of Contents Chris Greene’s Quantum Mechanics I Notes

Fall, 2009 Two Slit Interference Experiment.......................................................................1 Review of Classical Mechanics ..........................................................................8 Time Dependence and Poisson Brackets ..........................................................11 Formal Quantum Mechanics.............................................................................15 Coordinate Representation................................................................................16 Momentum Representation...............................................................................17 Eigenvalues and Eigenstates .............................................................................19 Operators...........................................................................................................23 Postulates of Quantum Mechanics....................................................................25 Some Properties of Operators ...........................................................................29 Representations of States and Operators...........................................................30 Changing Representations ................................................................................32 Eigenvalue Equations........................................................................................32 Properties of Hermitian Operators ....................................................................34 Compatibility of Observables ...........................................................................36 Why is Compatibility Important? .....................................................................38 Complete Sets of Commuting Observables ......................................................44 Time Evolution Operator ..................................................................................46 QM Description of Light Polarization ..............................................................49 Circular Polarization .........................................................................................55 Time-Dependent Schrodinger Wave Mechanics ..............................................58 Ehrenfest’s Theorem.........................................................................................63 Reality of Expectation Values ..........................................................................64 Separation of the Time-Dependent Schrodinger Equation ...............................65 Localized Wave Packets ...................................................................................68 Energy Normalization, Continuum States ........................................................70 Heisenberg Versus Schrodinger Representations/Pictures ...............................71 Density Operator and Density Matrix...............................................................73 Baker-Campbell-Hausdorff Identity .................................................................76 States of Thermal Equilibrium..........................................................................78 Two-Level Systems ..........................................................................................80 One-Dimensional QM Problems.......................................................................81 Using Symmetry to Solve Scattering Problems in 1D......................................87 Time Delay in Scattering ..................................................................................89 Simple 1D Harmonic Oscillator........................................................................92 Operator Solution of the Harmonic Oscillator..................................................98 Coherent States of the Harmonic Oscillator ...................................................106 Path Integral Method.......................................................................................114 Equivalence to the Time-Dependent Schrodinger Equation...........................121 Semiclassical Approximations........................................................................122 Systems with N Degrees of Freedom..............................................................123 Direct Product Space.......................................................................................124

Copyright Chris H. Greene 2009

Conservation of Total Momentum of N Particles...........................................130 Symmetry ==> Conservation Law..................................................................132 Two Non-Interacting Distinguishable Particles..............................................134 Identical Particles............................................................................................135 Exchange as a Super-Operator........................................................................138 Pauli Exclusion Principle................................................................................141 Exchange Interaction ......................................................................................141 Three Scattering Experiments (Feynman Lectures) .......................................148 Schrodinger Equation in 3 Dimensions ..........................................................152 Rotational Invariance and Angular Momentum .............................................153 Commutation Relations as Definition of Angular Momentum.......................158 Angular Momentum Matrices.........................................................................166 Orbital Angular Momentum Eigenfunctions (Spherical Harmonics).............167 Spherically-Symmetric Problems in 3D .........................................................172 Free Particle in Spherically Coordinates.........................................................175 Relating Cartesian and Spherical Coordinates for a Free Particle ..................177 The Hydrogen Atom (Spherical Treatment) ...................................................179 Pauli’s Operator Solution of the Hydrogen Atom ..........................................183 Spin .................................................................................................................189 Paramagnetic Resonance ................................................................................196