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  Quantum MechanicsMathematical StructureandPhysical StructurePart 2 John R. BoccioEmeritus Professor of PhysicsSwarthmore CollegeCopyright c  2014 by Professor John BoccioDo not reproduce or distribute copies of this book without permission.June 25, 2014  Contents 11  Time-Dependent Perturbation Theory 921 11.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92111.1.1 What is the physical meaning of this result? . . . . . . . . 92711.2 Atomic Radiation and Selection Rules . . . . . . . . . . . . . . . 94011.2.1 The Electric Dipole Approximation . . . . . . . . . . . . . 94111.2.2 Induced Emission and Absorption . . . . . . . . . . . . . 94311.3 A Real Physical Process - Ionization . . . . . . . . . . . . . . . . 94611.3.1 Evaluation of the Matrix Element . . . . . . . . . . . . . 94811.4 Adiabatic and Sudden Approximations . . . . . . . . . . . . . . . 95011.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95811.5.1 Square Well Perturbed by an Electric Field . . . . . . . . 95811.5.2 3-Dimensional Oscillator in an electric field . . . . . . . . 95911.5.3 Hydrogen in decaying potential . . . . . . . . . . . . . . . 95911.5.4 2 spins in a time-dependent potential . . . . . . . . . . . 95911.5.5 A Variational Calculation of the Deuteron Ground StateEnergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96011.5.6 Sudden Change - Don’t Sneeze . . . . . . . . . . . . . . . 96011.5.7 Another Sudden Change - Cutting the spring . . . . . . . 96011.5.8 Another perturbed oscillator . . . . . . . . . . . . . . . . 96111.5.9 Nuclear Decay . . . . . . . . . . . . . . . . . . . . . . . . 96111.5.10Time Evolution Operator . . . . . . . . . . . . . . . . . . 96111.5.11Two-Level System . . . . . . . . . . . . . . . . . . . . . . 96211.5.12Instantaneous Force . . . . . . . . . . . . . . . . . . . . . 96211.5.13Hydrogen beam between parallel plates . . . . . . . . . . 96211.5.14Particle in a Delta Function and an Electric Field . . . . 96211.5.15Nasty time-dependent potential [complex integration needed]96311.5.16Natural Lifetime of Hydrogen . . . . . . . . . . . . . . . . 96411.5.17Oscillator in electric field . . . . . . . . . . . . . . . . . . 96411.5.18Spin Dependent Transitions . . . . . . . . . . . . . . . . . 96511.5.19The Driven Harmonic Oscillator . . . . . . . . . . . . . . 96611.5.20A Novel One-Dimensional Well . . . . . . . . . . . . . . . 96611.5.21The Sudden Approximation . . . . . . . . . . . . . . . . . 96711.5.22The Rabi Formula . . . . . . . . . . . . . . . . . . . . . . 967i  ii  CONTENTS  11.5.23Rabi Frequencies in Cavity QED . . . . . . . . . . . . . . 968 12  Identical Particles 971 12.1 Theoretical ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . 97112.2 Bosons with Spin = 0 . . . . . . . . . . . . . . . . . . . . . . . . 97612.3 Spin = 1/2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . 97812.4 The N-Electron Atom . . . . . . . . . . . . . . . . . . . . . . . . 98412.5 The Helium Atom . . . . . . . . . . . . . . . . . . . . . . . . . . 99012.6 Multielectron Atoms . . . . . . . . . . . . . . . . . . . . . . . . . 99812.6.1 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . 100012.6.2 Shell Structure . . . . . . . . . . . . . . . . . . . . . . . . 100112.7 Angular Momentum Coupling . . . . . . . . . . . . . . . . . . . . 100512.7.1 LS Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 100612.7.2 Hund’s Rules . . . . . . . . . . . . . . . . . . . . . . . . . 101312.7.3 JJ-Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 101412.8 Spherical Harmonics Addition Theorem . . . . . . . . . . . . . . 101812.8.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . 101812.8.2 The Addition Theorem . . . . . . . . . . . . . . . . . . . 102112.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102312.9.1 Two Bosons in a Well . . . . . . . . . . . . . . . . . . . . 102312.9.2 Two Fermions in a Well . . . . . . . . . . . . . . . . . . . 102412.9.3 Two spin − 1 / 2 particles . . . . . . . . . . . . . . . . . . . 102412.9.4 Hydrogen Atom Calculations . . . . . . . . . . . . . . . . 102512.9.5 Hund’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . 102612.9.6 Russell-Saunders Coupling in Multielectron Atoms . . . . 102612.9.7 Magnetic moments of proton and neutron . . . . . . . . . 102712.9.8 Particles in a 3-D harmonic potential . . . . . . . . . . . . 102812.9.9 2 interacting particles . . . . . . . . . . . . . . . . . . . . 102812.9.10LS versus JJ coupling . . . . . . . . . . . . . . . . . . . . 102912.9.11In a harmonic potential . . . . . . . . . . . . . . . . . . . 102912.9.122 particles interacting via delta function . . . . . . . . . . 102912.9.132 particles in a square well . . . . . . . . . . . . . . . . . 103012.9.142 particles interacting via a harmonic potential . . . . . . 103012.9.15The Structure of helium . . . . . . . . . . . . . . . . . . . 1030 13  Scattering Theory and Molecular Physics 1033 13.1 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 103313.1.1 Greens Functions . . . . . . . . . . . . . . . . . . . . . . . 103513.1.2 Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . 104113.1.3 Partial Waves . . . . . . . . . . . . . . . . . . . . . . . . . 104213.1.4 The Optical Theorem . . . . . . . . . . . . . . . . . . . . 104713.1.5 Born Approximation . . . . . . . . . . . . . . . . . . . . . 104713.1.6 A General Property of Phase Shifts . . . . . . . . . . . . . 105013.1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 105113.1.8 Born Approximation Examples . . . . . . . . . . . . . . . 1055
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