Quantum Theory

By Marcel Pope

Chapter 1 Introduction ......................................................................................... 1 Black Body Radiation ................................................................................ 5 Rayleigh-Jeans-Law .................................................................................. 8 Wien’s Law................................................................................................ 8 Planck’s Law ............................................................................................. 8 Thermal Photon Statistics ........................................................................ 10 Mode Counting ......................................................................................... 11 Photo-Electric Effect ............................................................................... 13 Spontaneous and Induced Emission ......................................................... 14 Planck’s Constant .................................................................................... 17 Reduced Planck’s Constant ..................................................................... 19 Planck’s Route to the Black Body Radiation Formula and Quantization ... 21 Wien’s Radiation Law.............................................................................. 21 Planck’s Thermodynamic Approach: Oscillators in the Oven Wall ......... 21 A Classical Result Relating Radiation Intensity to Oscillator Energy...... 22 Oscillator Thermodynamics: Planck Focuses On Entropy ...................... 23 New Experiments, New Theory .............................................................. 23 The Great Breakthrough: Birth of the Quantum...................................... 24 Meanwhile in England.............................................................................. 26 Photoelectric Effect ................................................................................. 27 Einstein’s Concept of Light Quanta ......................................................... 28 Quantum Theory and Consciousness ...................................................... 30 The Attractions ........................................................................................ 30 The Difficulties ........................................................................................ 31 vi Chapter 2 Origins of Quantum Theory and Wave Aspects of Matter ............ 32 Origin of the Quantum Theory ................................................................ 32 Planck’s Quantum Theory ....................................................................... 33 Experiment the Photoelectric Effect ....................................................... 35 Apparatus ................................................................................................. 35 Experimental Design ................................................................................ 37 Reactions to Bohr’s Model ...................................................................... 39 The Bohr Atom........................................................................................ 45 Bohr’s Semiclassical Argument to Fix the Quantum of Angular Momentum .... 46 The Bohr Atom is Still Important ............................................................. 48 An Accident at the Phone Company Makes Everything Crystal Clear .. 48 De Broglie Hypothesis in Quantum Mechanics ...................................... 49 Qualitative Definition of Mechanical Waves ........................................... 60 Energy and Power of a Traveling Wave on a String ............................... 60 Superposition of Mechanical Waves ........................................................ 61 The Difference between Relativistic and NR QM.................................. 62 Antiparticles ............................................................................................. 64 QED: Photon Couples to E+E- .......................................................................................65 Fluctuations of the Vacuum ..................................................................... 66 Feynman Diagrams .................................................................................. 66 Infinities and Renormalisation .................................................................. 68 The Wave Theory of Matter ................................................................... 68 Wave Mechanics in a Ring Cavity........................................................... 70 Plane Bound ............................................................................................. 71 The Uncertainty Principle ........................................................................ 78 Trying to Beat the Uncertainty Principle ................................................. 79 Electrons in the Double Slit Experiment .................................................. 80 Uncertainty Principle Determines the Size of Everything ....................... 81 Chapter 3 Semi-Classical Theory of Radiation ................................................. 83 Classical Theory ...................................................................................... 83 Experimental Evidence ............................................................................ 87 Classical Geometry and Mechanics ......................................................... 92 Maxwell’s Wave Equation ..................................................................... 104 What Does the Wave Equation Tell Us about the Photon ..................... 104 Constructing a Wave Equation for a Particle With Mass ...................... 105 A Non-relativistic Wave Equation.......................................................... 105 How Does a Varying Potential Affect a De Broglie Wave ................... 106 Schrodinger Equation for a Particle in a Potential ................................. 107 Current Conservation ............................................................................. 108 Photons and Electrons ........................................................................... 109 Wavepackets and the Principle of Superposition .................................... 111 The First Step in Building a Wavepacket: Adding Two Sine Waves ...... 112 vii Phase Velocity and Group Velocity ......................................................... 112 The Gaussian Wavepacket ..................................................................... 114 Expectation values and the Uncertainty Principle .................................. 115 Einstein Suggests an Explanation ............................................................ 115 Millikan’s Attempts to Disprove Einstein’s Theory................................. 116 The Nature of Light ................................................................................ 116 Ionizing Versus Non-Ionizing Radiation .................................................. 118 Biological Effects of Ionizing Radiation.................................................. 119 Einstein’s Photoelectric Equation........................................................... 122 Stopping Potential .................................................................................. 122 The Planck Radiation Law .................................................................... 124 Chapter 4 Quantum Theory of Scattering ....................................................... 128 Scattering ............................................................................................... 128 Scattering Alphas ................................................................................... 132 Scattering Theory .................................................................................. 136 In Theoretical Physics ........................................................................... 137 Direct Scattering Problems .................................................................... 137 Concept of Group Velocity and Phase Velocity ..................................... 140 Phase Velocity ....................................................................................... 141 Relative Velocity .................................................................................... 143 Scattering States and Barriers ............................................................... 144 Streams of Particles............................................................................... 144 Barriers .................................................................................................. 145 Alpha Decay .......................................................................................... 146 Theory for New Class of Quantum Phase Transition ........................... 147 Pigeonholing Quantum Phase Transitions .............................................. 148 Classification.......................................................................................... 148 Quantum Phase Transitions in Correlated Electron Systems ................ 150 Quantum State ....................................................................................... 151 Bra–Ket Notation .................................................................................. 151 Basis States ............................................................................................ 151 Superposition of States .......................................................................... 152 Pure and Mixed States .......................................................................... 152 Chapter 5 Particle in a Central Potential ........................................................ 154 Motion of a Particle ............................................................................... 154 Particle Physics ..................................................................................... 155 Probing Particles .................................................................................... 156 Subatomic Particles ............................................................................... 157 Extraterrestrial Particle Physics ............................................................ 159 Balloon Experiments .............................................................................. 159 Rectilinear Motion.................................................................................. 160 viii Dynamics of Circular Motion ................................................................ 162 Centripetal Force ................................................................................... 162 Direction of Centripetal Force and Circular Trajectory ......................... 163 Force Analysis of Uniform Circular Motion .......................................... 164 Uniform Circular Motion in Horizontal Plane ........................................ 164 Motion of a Space Shuttle ...................................................................... 165 Horizontal Circular Motion in a Rotor ................................................... 165 Angular Momentum Quantization .......................................................... 169 Momentum and Angular Momentum with Spherical Coordinates ......... 170 Total Angular Momentum and Waves on a Balloon .............................. 170 Angular Momentum and the Uncertainly Principle ............................... 171 The Schrodinger Equation in (R, ?, ?) Coordinates .............................. 172 Separating the Variables: The Messy Details ........................................ 172 Separating Out and Solving the ?(?) Equation.................................... 172 Separating Out the ?(?) Equation ........................................................ 173 The R(R) Equation ................................................................................ 174 The Hydrogen Atom .............................................................................. 175 Bohr’s Model of Hydrogen Atom.......................................................... 177 Line Spectra ........................................................................................... 177 Bohr’s Relationship................................................................................ 178 Bohr Model of Atom.............................................................................. 181 Emission Spectrum of Hydrogen ........................................................... 185 Schröedinger Equation ........................................................................... 190 Introduction ............................................................................................ 190 Curvature of Wave Functions ................................................................ 190 The Wave Function, ?.......................................................................... 195 Resonances of Wave Function .............................................................. 197 Uncertainty and Non-Commutation ....................................................... 198 A Quantitative Measure of “Uncertainty” ............................................. 198 The Generalized Uncertainty Principle .................................................. 199 Proof of the Theorem: ........................................................................... 199 Minimizing the Uncertainty .................................................................... 200 Schrödinger Equation for One Particle .................................................. 201 Schrödinger’s Equation Numerically...................................................... 205 Spreadsheet for the Finite Square Well ................................................. 206 Finding the Ground State Energy ........................................................... 207 Automating the Process ......................................................................... 207 Finding Excited States............................................................................ 208 Curvature of Wave Functions ................................................................ 208 Stationary States of Schrödinger Equation ............................................ 209 The Infinite Well .................................................................................... 209 Periodic Boundary Conditions................................................................ 219 The Schrödinger Equation for the Hydrogen Atom ............................... 228 ix Wave functions as Vector Spaces ......................................................... 229 Vibrations and Waves ............................................................................ 230 Concepts of Probability .......................................................................... 237 Physical Quantum States and Probability .............................................. 238 Physical Meanings: From Measurement to Relationship ....................... 241 Probability for Individuals. Changes in the Meaning ............................. 242 Theoretical and Empirical Probabilities ................................................. 246 Probability Axioms ................................................................................. 249 Methods of Assigning Probabilities ........................................................ 249 Probability in Quantum Mechanics ........................................................ 254 Quantum Mechanical Imponderables with Nonlinear Parallels ............ 255 Chapter 6 General Formalism of Quantum Mechanics ................................. 261 Some Basic Ideas About Quantum Mechanics ..................................... 261 Waves and Particles .............................................................................. 261 Need of Quantum Mechanics ................................................................ 265 Linear Vector Spaces in Quantum Mechanics ...................................... 266 Algebra for Quantum Mechanics .......................................................... 270 Eigenvalues and Eigenstates of Hermitian Matrices ............................. 270 Diagonalizing a Hermtian Matrix ........................................................... 271 Functions of Matrices ............................................................................ 273 Vector Space .......................................................................................... 275 The Schwartz Inequality ........................................................................ 279 Black Body Radiation ............................................................................ 282 Electronic Wavefunction ........................................................................ 284 Electrical Properties............................................................................... 284 Chapter 7 Relativistic Quantum Mechanics ................................................... 287 Relativity and Quantum Mechanics ....................................................... 287 Philosophical Implications ...................................................................... 289 Applications............................................................................................ 290 Examples ................................................................................................ 291 Relativity of Light .................................................................................. 292 The Lorentz Transformations ................................................................ 301 Problems with the Galilean Transformations ......................................... 301 Deriving the Lorentz Transformations ................................................... 302 Spheres of Light .................................................................................... 303 Lorentz Invariants .................................................................................. 304 The Light Cone ...................................................................................... 306 “Lorentz Contraction” and “Time Dilation” Effects of Relativity ......... 306 Universal or General Relativity Theory ................................................. 307 Newtonian Gravitational Theory ............................................................ 308 Completely Generic Coordinate System................................................ 310 x Importance of Relativistic Effects in Atoms .......................................... 313 Direct, Indirect and Spin-Orbit Effects ................................................. 314 The Hydrogen Atom .............................................................................. 317 The Photoelectric Effect ....................................................................... 319 Hertz Finds Maxwell’s Waves: and Something Else ............................. 319 Hallwachs’ Simpler Approach ............................................................... 320 J.J. Thomson Identifies the Particles ..................................................... 321 Lenard Finds Some Surprises ................................................................ 321 Millikan’s Attempts to Disprove Einstein’s Theory................................ 322 Sparks in Gases ..................................................................................... 323 Cathode Rays ......................................................................................... 323 Thomson Discovers the Electron ........................................................... 323 X-Rays ................................................................................................... 325 What are X-Rays? ................................................................................. 326 X-Rays Come in Quanta ....................................................................... 327 More Rays ............................................................................................. 327 Marie Curie Investigates Becquerel’s Rays .......................................... 328 Ernest Rutherford Investigates Becquerel’s Rays ................................ 329 Identifying the Beta Rays ...................................................................... 329 Identifying the Alpha Rays .................................................................... 330