Mathematical Physics of BlackBody Radiation

Contents 

I Old Picture 3
1 Blackbody Radiation 5
1.1 Birth of Modern Physics . . . . . . . . . . . . . . . . . . . . . 5
1.2 Planck, Einstein and Schr¨odinger . . . . . . . . . . . . . . . . 6
1.3 Finite Precision Computation . . . . . . . . . . . . . . . . . . 7
2 Blackbody as Blackpiano 9
3 Interaction Light-Matter 13
4 Planck-Stefan-Boltzmann Laws 17
4.1 Planck’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Stefan-Boltzmann’s Law . . . . . . . . . . . . . . . . . . . . . 18
4.3 The Enigma of the Photoelectric Effect . . . . . . . . . . . . . 23
4.4 The Enigma of Blackbody Radiation . . . . . . . . . . . . . . 24
4.5 Confusion in Media . . . . . . . . . . . . . . . . . . . . . . . . 24
4.6 Confessions by Confused Scientists . . . . . . . . . . . . . . . 25
4.7 Towards Enigma Resolution . . . . . . . . . . . . . . . . . . . 27
5 Planck/Einstein Tragedy 29
5.1 James Jeans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Max Planck . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 Planck and Einstein . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Classical Derivation of Rayleigh-Jeans Law 35
6.1 Counting Cavity Degrees of Freedom . . . . . . . . . . . . . . 35
6.2 Dependence of Space Dimension . . . . . . . . . . . . . . . . . 36
7 Statistics vs Computation 37
7.1 Cut-Off by Statistics . . . . . . . . . . . . . . . . . . . . . . . 37
7.2 Cut-Off by Finite Precision Computation . . . . . . . . . . . . 37

II New Analysis 39
8 Wave Equation with Radiation 41
8.1 A Basic Radiation Model . . . . . . . . . . . . . . . . . . . . . 41
9 Spectral Analysis of Radiation 45
9.1 Basic Energy Balance R = F . . . . . . . . . . . . . . . . . . . 45
9.2 Rayleigh-Jeans Law . . . . . . . . . . . . . . . . . . . . . . . . 48
9.3 Radiation from Near-Resonance . . . . . . . . . . . . . . . . . 49
9.4 Thermal Equilibrium from Near-Resonance . . . . . . . . . . . 49
9.5 The Poynting Vector vs ∥f∥ 2 . . . . . . . . . . . . . . . . . . . 50
10 Acoustic Near-Resonance 53
10.1 Radiation vs Acoustic Resonance . . . . . . . . . . . . . . . . 53
10.2 Resonance in String Instrument . . . . . . . . . . . . . . . . . 53
10.3 Fourier Analysis of Near-Resonance . . . . . . . . . . . . . . . 55
10.4 Application to Acoustical Resonance . . . . . . . . . . . . . . 56
10.5 Computational Resonance . . . . . . . . . . . . . . . . . . . . 57
11 Model of Blackbody Radiation 63
11.1 Finite Precision Computation . . . . . . . . . . . . . . . . . . 63
11.2 Radiation and Heating . . . . . . . . . . . . . . . . . . . . . . 64
11.3 Planck as Rayleigh-Jeans with Cut-off . . . . . . . . . . . . . 65
11.4 Planck’s Law: R + H = F . . . . . . . . . . . . . . . . . . . . 65
11.5 Connection to Uncertainty Principle . . . . . . . . . . . . . . . 66
11.6 Stefan-Boltzmann’s Law . . . . . . . . . . . . . . . . . . . . . 66
11.7 Radiative Interaction . . . . . . . . . . . . . . . . . . . . . . . 67
11.8 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
11.9 Radiative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 69
11.10Interaction by Shared Force . . . . . . . . . . . . . . . . . . . 69
11.11Generic Nature of Blackbody . . . . . . . . . . . . . . . . . . 70
11.12Cut-Off by Residual Stabilization . . . . . . . . . . . . . . . . 71
11.13Cordination Length . . . . . . . . . . . . . . . . . . . . . . . . 71
12 Universal Blackbody 73
12.1 Kirchhoff and Universality . . . . . . . . . . . . . . . . . . . . 73
12.2 Blackbody as Cavity with Graphite Walls . . . . . . . . . . . 75
13 Model of Universal Blackbody 77
14 Radiative Heat Transfer 79
14.1 Stefan-Boltzmann for Two Blackbodies . . . . . . . . . . . . . 79
14.2 Non-Physical Two-Way Heat Transfer . . . . . . . . . . . . . . 80
15 Greybody vs Blackbody 83
16 2nd Law of Radiation 85
16.1 Irreversible Heating . . . . . . . . . . . . . . . . . . . . . . . . 85
16.2 Mystery of 2nd Law . . . . . . . . . . . . . . . . . . . . . . . 86
16.3 Stefan-Boltzmann Law as 2nd Law . . . . . . . . . . . . . . . 86
17 Reflection vs Blackbody Absorption/Emission 87
18 Blackbody as Transformer of Radiation 89
19 Hot Sun and Cool Earth 91 19.1 Emission Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 91
20 Blackbody Dynamics 93
20.1 Recollection of Model . . . . . . . . . . . . . . . . . . . . . . . 93
20.2 Radiative Interaction of Two Blackbodies . . . . . . . . . . . . 95
21 The Photoelectric Effect 97
21.1 Nobel Prize to Einstein . . . . . . . . . . . . . . . . . . . . . . 97
21.2 The photoelectric effect I . . . . . . . . . . . . . . . . . . . . . 97
21.3 Remark on Viscosity Models . . . . . . . . . . . . . . . . . . . 101
21.4 The Photolelectric Effect II . . . . . . . . . . . . . . . . . . . 101
22 The Compton Effect 103
22.1 The Compton Effect I . . . . . . . . . . . . . . . . . . . . . . 103
22.2 The Compton Effect II . . . . . . . . . . . . . . . . . . . . . . 103


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