Contents PART I: PROBLEMS 4. Thermodynamics and Statistical Physics Introductory Thermodynamics 4.1. Why Bother? (Moscow 4.2. Space Station Pressure (MIT) 4.3. Baron von Münchausen and Intergalactic Travel (Moscow 4.4. Railway Tanker (Moscow 4.5. Magic Carpet (Moscow 4.6. Teacup Engine (Princeton, Moscow 4.7. Grand Lunar Canals (Moscow 4.8. Frozen Solid (Moscow 4.9. Tea in Thermos (Moscow 4.10. Heat Loss (Moscow 4.11. Liquid Solid Liquid (Moscow 4.12. Hydrogen Rocket (Moscow 4.13. Maxwell Boltzmann Averages (MIT) 4.14. Slowly Leaking Box (Moscow Phys-Tech, Stony Brook (a,b)) 4.15. Surface Contamination (Wisconsin-Madison) 4.16. Bell Jar (Moscow 4.17. Hole in Wall (Princeton) 4.18. Ballast Volume Pressure (Moscow 4.19. Rocket in Drag (Princeton) 4.20. Adiabatic Atmosphere (Boston, Maryland) 3 3 3 4 4 5 5 6 7 7 8 8 9 9 9 9 10 10 11 11 12 13 xiii
xiv Contents 4.21. Atmospheric Energy (Rutgers) 4.22. Puncture (Moscow Heat and Work 4.23. 4.24. 4.25. 4.26. 4.27. 4.28. 4.29. 4.30. 4.31. 4.32. 4.33. 4.34. Cylinder with Massive Piston (Rutgers, Moscow Spring Cylinder (Moscow Isothermal Compression and Adiabatic Expansion of Ideal Gas (Michigan) Isochoric Cooling and Isobaric Expansion (Moscow Venting (Moscow Cylinder and Heat Bath (Stony Brook) Heat Extraction (MIT, Wisconsin-Madison) Heat Capacity Ratio (Moscow Otto Cycle (Stony Brook) Joule Cycle (Stony Brook) Diesel Cycle (Stony Brook) Modified Joule Thomson (Boston) Ideal Gas and Classical Statistics 4.35. Poisson Distribution in Ideal Gas (Colorado) 4.36. Polarization of Ideal Gas (Moscow 4.37. Two-Dipole Interaction (Princeton) 4.38. Entropy of Ideal Gas (Princeton) 4.39. Chemical Potential of Ideal Gas (Stony Brook) 4.40. Gas in Harmonic Well (Boston) 4.41. Ideal Gas in One-Dimensional Potential (Rutgers) 4.42. Equipartition Theorem (Columbia, Boston) 4.43. Diatomic Molecules in Two Dimensions (Columbia) 4.44. Diatomic Molecules in Three Dimensions (Stony Brook, Michigan State) 4.45. Two-Level System (Princeton) 4.46. Zipper (Boston) 4.47. Hanging Chain (Boston) 4.48. Molecular Chain (MIT, Princeton, Colorado) Nonideal Gas 4.49. Heat Capacities (Princeton) 4.50. Return of Heat Capacities (Michigan) 4.51. Nonideal Gas Expansion (Michigan State) 4.52. van der Waals (MIT) 13 14 14 14 15 15 16 16 16 16 17 17 18 18 19 19 19 20 20 20 21 21 21 21 22 23 24 24 24 25 26 26 26 27 27
Contents xv 4.53. Critical Parameters (Stony Brook) Mixtures and Phase Separation 4.54. Entropy of Mixing (Michigan, MIT) 4.55. Leaky Balloon (Moscow 4.56. Osmotic Pressure (MIT) 4.57. Clausius Clapeyron (Stony Brook) 4.58. Phase Transition (MIT) 4.59. Hydrogen Sublimation in Intergalactic Space (Princeton) 4.60. Gas Mixture Condensation (Moscow 4.61. Air Bubble Coalescence (Moscow 4.62. Soap Bubble Coalescence (Moscow 4.63. Soap Bubbles in Equilibrium (Moscow Quantum Statistics 4.64. 4.65. 4.66. 4.67. 4.68. 4.69. 4.70. 4.71. 4.72. 4.73. 4.74. 4.75. 4.76. 4.77. 4.78. 4.79. 4.80. 4.81. 4.82. 4.83. 4.84. 4.85. 4.86. 4.87. 4.88. Fermi Energy of a 1D Electron Gas (Wisconsin-Madison) Two-Dimensional Fermi Gas (MIT, Wisconson-Madison) Nonrelativistic Electron Gas (Stony Brook, Wisconsin-Madison, Michigan State) Ultrarelativistic Electron Gas (Stony Brook) Quantum Corrections to Equation of State (MIT, Princeton, Stony Brook) Speed of Sound in Quantum Gases (MIT) Bose Condensation Critical Parameters (MIT) Bose Condensation (Princeton, Stony Brook) How Hot the Sun? (Stony Brook) Radiation Force (Princeton, Moscow Phys-Tech, MIT) Hot Box and Particle Creation (Boston, MIT) D-Dimensional Blackbody Cavity (MIT) Fermi and Bose Gas Pressure (Boston) Blackbody Radiation and Early Universe (Stony Brook) Photon Gas (Stony Brook) Dark Matter (Rutgers) Einstein Coefficients (Stony Brook) Atomic Paramagnetism (Rutgers, Boston) Paramagnetism at High Temperature (Boston) One-Dimensional Ising Model (Tennessee) Three Ising Spins (Tennessee) N Independent Spins (Tennessee) N Independent Spins, Revisited (Tennessee) Ferromagnetism (Maryland, MIT) Spin Waves in Ferromagnets (Princeton, Colorado) 28 28 28 28 28 29 30 30 30 31 31 31 32 32 32 32 33 33 33 34 34 34 35 35 36 36 37 37 38 39 39 40 40 40 41 41 41 42
xvi Contents Fluctuations 4.89. 4.90. 4.91. 4.92. 4.93. 4.94. 4.95. 4.96. Magnetization Fluctuation (Stony Brook) Gas Fluctuations (Moscow Quivering Mirror (MIT, Rutgers, Stony Brook) Isothermal Compressibility and Mean Square Fluctuation (Stony Brook) Energy Fluctuation in Canonical Ensemble (Colorado, Stony Brook) Number Fluctuations (Colorado (a,b), Moscow Phys-Tech (c)) Wiggling Wire (Princeton) LC Voltage Noise (MIT, Chicago) Applications to Solid State 4.97. Thermal Expansion and Heat Capacity (Princeton) 4.98. Schottky Defects (Michigan State, MIT) 4.99. Frenkel Defects (Colorado, MIT) 4.100. Two-Dimensional Debye Solid (Columbia, Boston) 4.101. Einstein Specific Heat (Maryland, Boston) 4.102. Gas Adsorption (Princeton, MIT, Stanford) 4.103. Thermionic Emission (Boston) 4.104. Electrons and Holes (Boston, Moscow 4.105. Adiabatic Demagnetization (Maryland) 4.106. Critical Field in Superconductor (Stony Brook, Chicago) 42 42 43 43 43 44 44 44 44 45 45 45 45 46 46 47 47 47 48 49 5. Quantum Mechanics One-Dimensional Potentials 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11. 5.12. Shallow Square Well I (Columbia) Shallow Square Well II (Stony Brook) Attractive Delta Function Potential I (Stony Brook) Attractive Delta Function Potential II (Stony Brook) Two Delta Function Potentials (Rutgers) Transmission Through a Delta Function Potential (Michigan State, MIT, Princeton) Delta Function in a Box (MIT) Particle in Expanding Box (Michigan State, MIT, Stony Brook) One-Dimensional Coulomb Potential (Princeton) Two Electrons in a Box (MIT) Square Well (MIT) Given the Eigenfunction (Boston, MIT) 51 51 51 52 52 53 54 54 54 55 55 55 56 56
Contents xvii 5.13. Combined Potential (Tennessee) 56 Harmonic Oscillator 5.14. 5.15. 5.16. 5.17. 5.18. 5.19. 5.20. 5.21. Angular Momentum and Spin 5.22. 5.23. 5.24. 5.25. Dipolar Interactions (Stony Brook) 5.26. Spin-Dependent Potential (MIT) 5.27. 5.28. 5.29. 5.30. Variational Calculations 5.31. Anharmonic Oscillator (Tennessee) 5.32. Linear Potential I (Tennessee) 5.33. Linear Potential II (MIT, Tennessee) 5.34. Return of Combined Potential (Tennessee) 5.35. Quartic in Three Dimensions (Tennessee) 5.36. Halved Harmonic Oscillator (Stony Brook, Chicago (b), Princeton (b)) 5.37. Helium Atom (Tennessee) Perturbation Theory 5.38. Momentum Perturbation (Princeton) 5.39. Ramp in Square Well (Colorado) 5.40. Circle with Field (Colorado, Michigan State) 5.41. Rotator in Field (Stony Brook) 5.42. Finite Size of Nucleus (Maryland, Michigan State, 5.43. 5.44. Given a Gaussian (MIT) Harmonic Oscillator ABCs (Stony Brook) Number States (Stony Brook) Coupled Oscillators (MIT) Time-Dependent Harmonic Oscillator I (Wisconsin-Madison) Time-Dependent Harmonic Oscillator II (Michigan State) Switched-on Field (MIT) Cut the Spring! (MIT) Given Another Eigenfunction (Stony Brook) Algebra of Angular Momentum (Stony Brook) Triplet Square Well (Stony Brook) Three Spins (Stony Brook) Constant Matrix Perturbation (Stony Brook) Rotating Spin (Maryland, MIT) Nuclear Magnetic Resonance (Princeton, Stony Brook) Princeton, Stony Brook) U and Perturbation (Princeton) Relativistic Oscillator (MIT, Moscow Phys-Tech, Stony Brook (a)) 56 56 57 57 58 58 59 59 60 60 60 60 61 61 61 62 62 63 63 63 63 63 64 64 64 64 65 66 66 66 66 67 67 67 68
xviii Contents 5.45. Spin Interaction (Princeton) 5.46. Spin Orbit Interaction (Princeton) 5.47. Interacting Electrons (MIT) 5.48. Stark Effect in Hydrogen (Tennessee) 5.49. Hydrogen with Electric and Magnetic Fields (MIT) 5.50. Hydrogen in Capacitor (Maryland, Michigan State) 5.51. Harmonic Oscillator in Field (Maryland, Michigan State) 5.52. of Tritium (Michigan State) WKB 5.53. Bouncing Ball (Moscow Phys-Tech, Chicago) 5.54. Truncated Harmonic Oscillator (Tennessee) 5.55. Stretched Harmonic Oscillator (Tennessee) 5.56. Ramp Potential (Tennessee) 5.57. Charge and Plane (Stony Brook) 5.58. Ramp Phase Shift (Tennessee) 5.59. Parabolic Phase Shift (Tennessee) 5.60. Phase Shift for Inverse Quadratic (Tennessee) Scattering Theory 5.61. Step-Down Potential (Michigan State, MIT) 5.62. Step-Up Potential (Wisconsin-Madison) 5.63. Repulsive Square Well (Colorado) 5.64. 3D Delta Function (Princeton) 5.65. Two-Delta-Function Scattering (Princeton) 5.66. Scattering of Two Electrons (Princeton) 5.67. Spin-Dependent Potentials (Princeton) 5.68. Rayleigh Scattering (Tennessee) 5.69. Scattering from Neutral Charge Distribution (Princeton) General 5.70. 5.71. 5.72. 5.73. 5.74. 5.75. 5.76. Spherical Box with Hole (Stony Brook) Attractive Delta Function in 3D (Princeton) Ionizing Deuterium (Wisconsin-Madison) Collapsed Star (Stanford) Electron in Magnetic Field (Stony Brook, Moscow Electric and Magnetic Fields (Princeton) Josephson Junction (Boston) 68 68 69 69 69 70 70 70 71 71 71 71 72 72 73 73 73 73 73 74 75 75 76 76 76 77 77 77 77 78 78 78 79 79 79
Contents xix PART II: SOLUTIONS 4. Thermodynamics and Statistical Physics Introductory Thermodynamics 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10. 4.11. 4.12. 4.13. 4.14. 4.15. 4.16. 4.17. 4.18. 4.19. 4.20. 4.21. 4.22. Heat and Work 4.23. 4.24. 4.25. 4.26. 4.27. 4.28. 4.29. 4.30. Why Bother? (Moscow Space Station Pressure (MIT) Baron von Münchausen and Intergalactic Travel (Moscow Railway Tanker (Moscow Phys-Tech ) Magic Carpet (Moscow Phys-Tech ) Teacup Engine (Princeton, Moscow Grand Lunar Canals (Moscow Frozen Solid (Moscow Tea in Thermos (Moscow Heat Loss (Moscow Liquid Solid Liquid (Moscow Hydrogen Rocket (Moscow Maxwell Boltzmann Averages (MIT) Slowly Leaking Box (Moscow Phys-Tech, Stony Brook (a,b)) Surface Contamination (Wisconsin-Madison) Bell Jar (Moscow Hole in Wall (Princeton) Ballast Volume Pressure (Moscow Rocket in Drag (Princeton) Adiabatic Atmosphere (Boston, Maryland) Atmospheric Energy (Rutgers) Puncture (Moscow Cylinder with Massive Piston (Rutgers, Moscow Spring Cylinder (Moscow Isothermal Compression and Adiabatic Expansion of Ideal Gas (Michigan) Isochoric Cooling and Isobaric Expansion (Moscow Venting (Moscow Cylinder and Heat Bath (Stony Brook) Heat Extraction (MIT, Wisconsin-Madison) Heat Capacity Ratio (Moscow 83 83 83 84 84 85 87 89 90 92 92 94 95 96 97 99 101 102 103 104 106 107 108 110 112 112 113 115 117 118 119 120 122
xx Contents 4.31. Otto Cycle (Stony Brook) 4.32. Joule Cycle (Stony Brook) 4.33. Diesel Cycle (Stony Brook) 4.34. Modified Joule Thomson (Boston) Ideal Gas and Classical Statistics 4.35. 4.36. 4.37. 4.38. 4.39. 4.40. 4.41. 4.42. 4.43. 4.44. 4.45. 4.46. 4.47. 4.48. Poisson Distribution in Ideal Gas (Colorado) Polarization of Ideal Gas (Moscow Two-Dipole Interaction (Princeton) Entropy of Ideal Gas (Princeton) Chemical Potential of Ideal Gas (Stony Brook) Gas in Harmonic Well (Boston) Ideal Gas in One-Dimensional Potential (Rutgers) Equipartition Theorem (Columbia, Boston) Diatomic Molecules in Two Dimensions (Columbia) Diatomic Molecules in Three Dimensions (Stony Brook, Michigan State) Two-Level System (Princeton) Zipper (Boston) Hanging Chain (Boston) Molecular Chain (MIT, Princeton, Colorado) Nonideal Gas 4.49. Heat Capacities (Princeton) 4.50. Return of Heat Capacities (Michigan) 4.51. Nonideal Gas Expansion (Michigan State) 4.52. van der Waals (MIT) 4.53. Critical Parameters (Stony Brook) Mixtures and Phase Separation 4.54. Entropy of Mixing (Michigan, MIT) 4.55. Leaky Balloon (Moscow 4.56. Osmotic Pressure (MIT) 4.57. Clausius Clapeyron (Stony Brook) 4.58. Phase Transition (MIT) 4.59. Hydrogen Sublimation in Intergalactic Space (Princeton) 4.60. Gas Mixture Condensation (Moscow 4.61. Air Bubble Coalescence (Moscow 4.62. Soap Bubble Coalescence (Moscow 4.63. Soap Bubbles in Equilibrium (Moscow Quantum Statistics 4.64. Fermi Energy of a 1D Electron Gas (Wisconsin-Madison) 4.65. Two-Dimensional Fermi Gas (MIT, Wisconson-Madison) 123 125 126 127 128 128 130 131 133 135 136 137 138 141 142 146 147 148 149 151 151 152 154 155 156 158 158 159 160 162 163 164 165 166 167 168 170 170 171
Contents xxi 4.66. 4.67. 4.68. 4.69. 4.70. 4.71. 4.72. 4.73. 4.74. 4.75. 4.76. 4.77. 4.78. 4.79. 4.80. 4.81. 4.82. 4.83. 4.84. 4.85. 4.86. 4.87. 4.88. Fluctuations 4.89. 4.90. 4.91. 4.92. 4.93. 4.94. 4.95. 4.96. Nonrelativistic Electron Gas (Stony Brook, Wisconsin-Madison, Michigan State) Ultrarelativistic Electron Gas (Stony Brook) Quantum Corrections to Equation of State (MIT, Princeton, Stony Brook) Speed of Sound in Quantum Gases (MIT) Bose Condensation Critical Parameters (MIT) Bose Condensation (Princeton, Stony Brook) How Hot the Sun? (Stony Brook) Radiation Force (Princeton, Moscow Phys-Tech, MIT) Hot Box and Particle Creation (Boston, MIT) D-Dimensional Blackbody Cavity (MIT) Fermi and Bose Gas Pressure (Boston) Blackbody Radiation and Early Universe (Stony Brook) Photon Gas (Stony Brook) Dark Matter (Rutgers) Einstein Coefficients (Stony Brook) Atomic Paramagnetism (Rutgers, Boston) Paramagnetism at High Temperature (Boston) One-Dimensional Ising Model (Tennessee) Three Ising Spins (Tennessee) N Independent Spins (Tennessee) N Independent Spins, Revisited (Tennessee) Ferromagnetism (Maryland, MIT) Spin Waves in Ferromagnets (Princeton, Colorado) Magnetization Fluctuation (Stony Brook) Gas Fluctuations (Moscow Quivering Mirror (MIT, Rutgers, Stony Brook) Isothermal Compressibility and Mean Square Fluctuation (Stony Brook) Energy Fluctuation in Canonical Ensemble (Colorado, Stony Brook) Number Fluctuations (Colorado (a,b), Moscow Phys-Tech (c)) Wiggling Wire (Princeton) LC Voltage Noise (MIT, Chicago) Applications to Solid State 4.97. Thermal Expansion and Heat Capacity (Princeton) 4.98. Schottky Defects (Michigan State, MIT) 4.99. Frenkel Defects (Colorado, MIT) 172 173 174 177 180 181 182 183 185 189 189 191 192 194 196 197 200 203 204 204 205 205 206 207 207 209 210 210 212 216 219 221 223 223 226 226
xxii Contents 5. 4.100. 4.101. 4.102. 4.103. 4.104. 4.105. 4.106. Two-Dimensional Debye Solid (Columbia, Boston) Einstein Specific Heat (Maryland, Boston) Gas Adsorption (Princeton, MIT, Stanford) Thermionic Emission (Boston) Electrons and Holes (Boston, Moscow Adiabatic Demagnetization (Maryland) Critical Field in Superconductor (Stony Brook, Chicago) Quantum Mechanics One-Dimensional Potentials 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11. 5.12. 5.13. Shallow Square Well I (Columbia) Shallow Square Well II (Stony Brook) Attractive Delta Function Potential I (Stony Brook) Attractive Delta Function Potential II (Stony Brook) Two Delta Function Potentials (Rutgers) Transmission Through a Delta Function Potential (Michigan State, MIT, Princeton) Delta Function in a Box (MIT) Particle in Expanding Box (Michigan State, MIT, Stony Br0ook) One-Dimensional Coulomb Potential (Princeton) Two Electrons in a Box (MIT) Square Well (MIT) Given the Eigenfunction (Boston, MIT) Combined Potential (Tennessee) Harmonic Oscillator 5.14. 5.15. 5.16. 5.17. 5.18. 5.19. 5.20. 5.21. Given a Gaussian (MIT) Harmonic Oscillator ABCs (Stony Brook) Number States (Stony Brook) Coupled Oscillators (MIT) Time-Dependent Harmonic Oscillator I (Wisconsin-Madison) Time-Dependent Harmonic Oscillator II (Michigan State) Switched-on Field (MIT) Cut the Spring! (MIT) Angular Momentum and Spin 5.22. 5.23. 5.24. 5.25. 5.26. Given Another Eigenfunction (Stony Brook) Algebra of Angular Momentum (Stony Brook) Triplet Square Well (Stony Brook) Dipolar Interactions (Stony Brook) Spin-Dependent Potential (MIT) 228 230 232 234 236 238 241 243 243 243 244 245 247 248 250 250 251 253 253 255 255 256 257 257 258 260 262 263 263 264 265 266 266 267 269 271 272
Contents xxiii 5.27. Three Spins (Stony Brook) 5.28. Constant Matrix Perturbation (Stony Brook) 5.29. Rotating Spin (Maryland, MIT) 5.30. Nuclear Magnetic Resonance (Princeton, Stony Brook) Variational Calculations 5.31. Anharmonic Oscillator (Tennessee) 5.32. Linear Potential I (Tennessee) 5.33. Linear Potential II (MIT, Tennessee) 5.34. Return of Combined Potential (Tennessee) 5.35. Quartic in Three Dimensions (Tennessee) 5.36. Halved Harmonic Oscillator (Stony Brook, Chicago (b), Princeton (b)) 5.37. Helium Atom (Tennessee) Perturbation Theory 5.38. Momentum Perturbation (Princeton) 5.39. Ramp in Square Well (Colorado) 5.40. Circle with Field (Colorado, Michigan State) 5.41. Rotator in Field (Stony Brook) 5.42. Finite Size of Nucleus (Maryland, Michigan State, Princeton, Stony Brook) 5.43. 5.44. 5.45. 5.46. 5.47. 5.48. 5.49. 5.50. 5.51. 5.52. U and Perturbation (Princeton) Relativistic Oscillator (MIT, Moscow Phys-Tech, Stony Brook (a)) Spin Interaction (Princeton) Spin Orbit Interaction (Princeton) Interacting Electrons (MIT) Stark Effect in Hydrogen (Tennessee) Hydrogen with Electric and Magnetic Fields (MIT) Hydrogen in Capacitor (Maryland, Michigan State) Harmonic Oscillator in Field (Maryland, Michigan State) of Tritium (Michigan State) WKB 5.53. Bouncing Ball (Moscow Phys-Tech, Chicago) 5.54. Truncated Harmonic Oscillator (Tennessee) 5.55. Stretched Harmonic Oscillator (Tennessee) 5.56. Ramp Potential (Tennessee) 5.57. Charge and Plane (Stony Brook) 5.58. Ramp Phase Shift (Tennessee) 5.59. Parabolic Phase Shift (Tennessee) 5.60. Phase Shift for Inverse Quadratic (Tennessee) 272 274 275 276 278 278 279 280 281 282 283 286 287 287 288 289 290 290 292 293 297 297 298 299 300 302 303 305 305 305 306 307 308 309 310 311 311
xxiv Contents Scattering Theory 312 5.61. Step-Down Potential (Michigan State, MIT) 312 5.62. Step-Up Potential (Wisconsin-Madison) 312 5.63. Repulsive Square Well (Colorado) 313 5.64. 3D Delta Function (Princeton) 315 5.65. Two-Delta-Function Scattering (Princeton) 316 5.66. Scattering of Two Electrons (Princeton) 317 5.67. Spin-Dependent Potentials (Princeton) 318 5.68. Rayleigh Scattering (Tennessee) 320 5.69. Scattering from Neutral Charge Distribution (Princeton) 321 General 5.70. 5.71. 5.72. 5.73. 5.74. 5.75. 5.76. Spherical Box with Hole (Stony Brook) Attractive Delta Function in 3D (Princeton) Ionizing Deuterium (Wisconsin-Madison) Collapsed Star (Stanford) Electron in Magnetic Field (Stony Brook, Moscow Electric and Magnetic Fields (Princeton) Josephson Junction (Boston) 322 322 323 324 324 328 329 330 PART III: APPENDIXES Approximate Values of Physical Constants Some Astronomical Data Other Commonly Used Units Conversion Table from Rationalized MKSA to Gaussian Units Vector Identities Vector Formulas in Spherical and Cylindrical Coordinates Legendre Polynomials Rodrigues Formula Spherical Harmonics Harmonic Oscillator Angular Momentum and Spin Variational Calculations Normalized Eigenstates of Hydrogen Atom Conversion Table for Pressure Units Useful Constants Bibliography 335 336 336 337 337 338 341 342 342 342 343 344 344 345 345 347