Introduction to Quantum Mechanics 1

Ibrahima Sakho is a Doctor of Physical Science, and works at the science and technology training and research unit at the University of Thiès (Senegal). On-site, he teaches quantum mechanics, atomic and nuclear physics, radiation–matter interaction, and environmental chemistry.

• Foreword xiLouis MARCHILDONPreface xiiiChapter 1. Thermal Radiation 11.1. Radiation 21.1.1. Definition 21.1.2. Origin of radiation 21.1.3. Classification of objects 41.2. Radiant flux 41.2.1. Definition of radiant flux, coefficient of absorption 41.2.2. Black body and gray body 51.3. Black body emission spectrum 61.3.1. Isotherms of a black body: experimental facts 61.3.2. Solid angle 71.3.3. Lambert’s law, radiance 91.3.4. Kirchhoff’s laws 101.3.5. Stefan–Boltzmann law, total energy exitance 111.3.6. Wien’s laws, useful spectrum 121.3.7. The Rayleigh–Jeans law, “ultraviolet catastrophe” 151.3.8. Planck’s law, monochromatic radiant exitance 161.4. Exercises 181.4.1. Exercise 1 – Calculation of the Stefan–Boltzmann constant 181.4.2. Exercise 2 – Calculation of the Sun’s surface temperature 181.4.3. Exercise 3 – Average energy of a quantum oscillator, Planck’s formula 191.4.4. Exercise 4 – Deduction of Wien’s first law from Planck’s formula 201.4.5. Exercise 5 – Total electromagnetic energy radiated by the black body 201.5. Solutions 211.5.1. Solution 1 – Calculation of the Stefan–Boltzmann constant 211.5.2. Solution 2 – Calculation of the Sun’s surface temperature 231.5.3. Solution 3 – Average energy of a quantum oscillator, Planck’s formula 241.5.4. Solution 4 – Deduction of Wien’s law from Planck’s law 271.5.5. Solution 5 – Total electromagnetic energy radiated by the black body 29Chapter 2. Wave and Particle Aspects of Light 332.1. Light interferences 342.1.1. Elongation of a light wave 342.1.2. Total elongation of synchronous light sources 352.1.3. Young’s experimental setup 362.1.4. Interference field, fringes of interference 372.1.5. Interpretation, interference as concept 372.1.6. Path difference 392.1.7. Fringe spacing, order of interference 412.2. Photoelectric effect 442.2.1. Experimental setup, definition 442.2.2. Interpretation, photon energy 442.2.3. Einstein relation, energy function 452.2.4. Photoelectric threshold 462.2.5. Stopping potential, saturation current 482.2.6. Quantum efficiency of a photoelectric cell 512.2.7. Sensitivity of a photoelectric cell 512.3. Compton effect 532.3.1. Experimental setup, definition 532.3.2. Energy and linear momentum of a relativistic particle 552.3.3. Interpretation, photon linear momentum, and Compton shift 562.4. Combining the particle- and wave-like aspects of light 592.4.1. Particle- and wave-like properties of the photon 592.4.2. Planck–Einstein relation 602.5. Exercises 612.5.1. Exercise 1 – Single-slit diffraction, interferences 612.5.2. Exercise 2 – Order of interference fringes 622.5.3. Exercise 3 – Experimental measurement of Planck constant and of the work function of an emissive photocathode 632.5.4. Exercise 4 – Experimental study of the behavior of a photoelectric cell, quantum efficiency and sensitivity 642.5.5. Exercise 5 – Compton backscattering 652.5.6. Exercise 6 – Energy and linear momentum of scattered photons and of the electron ejected by Compton effect 652.5.7. Exercise 7 – Inverse Compton effect 662.6. Solutions 662.6.1. Solution 1 – Single-slit diffraction, interferences 662.6.2. Solution 2 – Order of interference fringes 682.6.3. Solution 3 – Experimental measurement of Planck constant and of the work function of an emissive photocathode 702.6.4. Solution 4 – Experimental study of the behaviour of a photoelectric cell, quantum efficiency and sensitivity 742.6.5. Solution 5 – Compton backscattering 762.6.6. Solution 6 – Energy and linear momentum of the scattered photons and of the electron ejected by Compton effect 782.6.7. Solution 7 – Inverse Compton effect 79Chapter 3. Quantum Numbers of the Electron 833.1. Experimental facts 853.1.1. Spectrometer 853.1.2. First lines of the hydrogen atom identified by Ångström 883.1.3. Balmer’s formula 893.1.4. Rydberg constant for hydrogen 903.1.5. Ritz combination principle 923.2. Rutherford’s planetary model of the atom 923.2.1. Rutherford’s scattering, atomic nucleus 923.2.2. Limitations of the planetary model 943.3. Bohr’s quantized model of the atom 953.3.1. Shell model of electron configurations 953.3.2. Bohr’s postulates, principal quantum number 953.3.3. Absorption spectrum, emission spectrum 983.3.4. Principle of angular momentum quantization 993.3.5. Quantized expression of the energy of the hydrogen atom 1003.3.6. Interpretation of spectral series 1043.3.7. Energy diagram of the hydrogen atom, ionization energy 1073.3.8. Advantages and limitations of Bohr’s model 1093.3.9. Reduced Rydberg constant 1103.4. Sommerfeld’s atomic model 1113.4.1. Experimental facts: normal Zeeman effect 1113.4.2. Bohr–Sommerfeld model, angular momentum quantum number 1133.4.3. Atomic orbital, electron configuration 1143.4.4. Interpretation of normal Zeeman effect, angular momentum quantum number 1173.4.5. Advantages and limitations of the Bohr–Sommerfeld model 1203.5. Electron spin 1203.5.1. The Stern–Gerlach experiment 1203.5.2. The Uhlenbeck and Goudsmit hypothesis, electron spin 1213.5.3. Degree of degeneracy of energy levels 1243.5.4. Total quantum number, selection rules 1253.6. Electron magnetic moments 1273.6.1. Orbital and spin magnetic moments 1273.6.2. Magnetic potential energy 1303.6.3. Spin–orbit interaction, spectroscopic notation of states 1313.6.4. Fine structure of the levels of energy of the hydrogen atom 1323.7. Exercises 1353.7.1. Exercise 1 – Spectrum of hydrogen-like ions 1363.7.2. Exercise 2 – Using the energy diagram of the lithium atom 1363.7.3. Exercise 3 – Spectra of the hydrogen atom, application to astrophysics 1373.7.4. Exercise 4 – Atomic resonance 1393.7.5. Exercise 5 – X-ray spectrum 1413.7.6. Exercise 6 – Lifetime of the hydrogen atom according to the planetary model 1433.7.7. Exercise 7 – Correspondence principle, quantization of the angular momentum 1443.7.8. Exercise 8 – Franck–Hertz experiment: experimental confirmation of Bohr’s atomic model 1453.7.9. Exercise 9 – Identification of a hydrogen-like system 1483.7.10. Exercise 10 – Nucleus drag effect: discovery of deuteron 1493.7.11. Exercise 11 – Normal Zeeman effect on the Lyman alpha line of the hydrogen atom 1503.7.12. Exercise 12 – Zeeman–Lorentz triplet, Larmor precession 1503.7.13. Exercise 13 – The Stern–Gerlach experiment, magnetic force 1523.7.14. Exercise 14 – Intensities of the spots in the Stern–Gerlach experiment 1533.7.15. Exercise 15 – Normal Zeeman effect on the 2p level of hydrogen-like systems 1553.7.16. Exercise 16 – Anomalous Zeeman effect on the ground state of hydrogen-like systems 1563.7.17. Exercise 17 – Anomalous Zeeman effect on the 2p level of hydrogen-like systems 1563.7.18. Exercise 18 – Fine structure of the resonance line of the hydrogen atom 1573.7.19. Exercise 19 – Fine structure of n = 2 level of the hydrogen atom 1573.7.20. Exercise 20 – Illustration of complex Zeeman effect on the yellow sodium line, selection rules 1593.7.21. Exercise 21 – Linear oscillator in the phase space, Bohr’s principle for angular momentum quantization 1593.8. Solutions 1613.8.1. Solution 1 – Spectrum of hydrogen-like ions 1613.8.2. Solution 2 – Using the energy diagram of the lithium atom 1643.8.3. Solution 3 – Spectra of the hydrogen atom, application to astrophysics 1663.8.4. Solution 4 – Atomic resonance phenomenon 1683.8.5. Solution 5 – X-ray spectrum 1703.8.6. Solution 6 – Lifetime of the hydrogen atom according to the planetary model 1723.8.7. Solution 7 – Correspondence principle, angular momentum quantization principle 1753.8.8. Solution 8 – Experimental confirmation of Bohr’s model: Franck–Hertz experiment 1793.8.9. Solution 9 – Identification of a hydrogen-like system 1843.8.10. Solution 10 – Nucleus drag effect: discovery of the deuton 1863.8.11. Solution 11 – Normal Zeeman effect on the Lyman alpha line of the hydrogen atom 1883.8.12. Solution 12 – Zeeman–Lorentz triplet, Larmor precession 1893.8.13. Solution 13 – Theoretical interpretation of the Stern–Gerlach experiment, magnetic force 1953.8.14. Solution 14 – Intensities of the spots in the Stern–Gerlach experiment 1983.8.15. Solution 15 – Normal Zeeman effect on the 2p level of hydrogen-like systems 2023.8.16. Solution 16 – Anomalous Zeeman effect on the ground level of hydrogen-like systems 2043.8.17. Solution 17 – Anomalous Zeeman effect on the 2p level of hydrogen-like systems 2053.8.18. Solution 18 – Fine structure of the resonance line of the hydrogen atom 2063.8.19. Solution 19 – Fine structure of n = 2 level of the hydrogen atom 2093.8.20. Solution 20 – Illustration of complex Zeeman effect on the yellow line of sodium, selection rules 2113.8.21. Solution 21 – Linear oscillator in the phase space, Bohr’s angular momentum quantization principle 212Chapter 4. Matter Waves – Uncertainty Relations 2174.1. De Broglie’s matter waves 2184.1.1. From light wave to matter wave 2184.1.2. De Broglie’s relation 2194.1.3. Law of dispersion of matter waves 2214.1.4. Phase velocity and group velocity 2224.1.5. Bohr’s quantization principle and de Broglie hypothesis 2264.1.6. Experimental confirmation, experiment of Davisson and Germer 2284.2. Heisenberg’s uncertainty relations 2364.2.1. Uncertainty principle 2364.2.2. Probabilistic interpretation of the wave function 2374.2.3. Root mean square deviation 2384.2.4. Spatial uncertainty relations, complementary variables 2394.2.5. Time–energy uncertainty relation, width of lines 2414.2.6. Heisenberg’s microscope 2424.3. Exercises 2444.3.1. Group velocity of de Broglie waves in the relativistic case 2444.3.2. Observing an atom with an electron microscope 2454.4. Solutions 2464.4.1. Group velocity of de Broglie waves in the relativistic case 2464.4.2. Observing an atom with an electron microscope 248Appendices 251Appendix 1 253Appendix 2 267Appendix 3 275Appendix 4 287Appendix 5 293References 309Index 313