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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Did Planck, Einstein, and Bose count indivisible photons or discrete emission/absorption processes in a black-body cavity?

Michael Ambroselli, Chandrasekhar Roychoudhuri

Published Sep 10, 2015• 0 citations
Publisher

Abstract

Planck, Einstein and Bose all had to introduce statistics, and thus counting, in order to successfully derive an equation for the energy distribution within the black-body radiation spectrum, and what we now call Bose-Einstein statistics. Some of the details involved in the counting procedure vary while still giving the same result. However, the interpretation of what we count may differ dramatically from one another (as, for example, between Planck and Bose), without impacting the final, mathematical result. We demonstrate here a further alternative, which varies both, in the details of the counting, as well as in the interpretation, while still producing the same well known statistics. This approach puts the "quantumness" back into the radiation emission/absorption process, possibly dispensing with the requirement of quantized light, at least in the context of black-body radiation.

DOI
10.1117/12.2188291

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Planck, Einstein and Bose all had to introduce statistics, and thus counting, in order to successfully derive an equation for the energy distribution within the black-body radiation spectrum, and what we now call Bose-Einstein statistics. Some of the details involved in the counting procedure vary while still giving the same result. However, the interpretation of what we count may differ dramatically from one another (as, for example, between Planck and Bose), without impacting the final, mathematical result. We demonstrate here a further alternative, which varies both, in the details of the counting, as well as in the interpretation, while still producing the same well known statistics. This approach puts the "quantumness" back into the radiation emission/absorption process, possibly dispensing with the requirement of quantized light, at least in the context of black-body radiation.
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The interplay between optical and statistical physics is a rich and exciting field of study. Black body radiation was the first application of photon statistics, although it was initially treated as a problem of the cavity oscillators in equilibrium with the photon field. However Planck surprisingly resisted the idea that anything physical would be quantized for a long time after he had solved the problem. We trace this development. Then, after the invention of the laser itself, it proved difficult to develop a theory of laser action that could account for photon statistics, i.e. fluctuations near threshold. This was accomplished in 1965. After Bose-Einstein condensation was successfully achieved, the same problem arose in this case. The fluctuation problem had not been treated adequately even for the ideal Bose gas. However this problem has now been solved using the same techniques as in the theory of laser action.
A historical overview is given on the basic results which appeared by the year 1926 concerning Einstein's fluctuation formula of black-body radiation, in the context of light-quanta and wave-particle duality. On the basis of the original publications (from Planck's derivation of the black-body spectrum and Einstein's introduction of the photons up to the results of Born, Heisenberg and Jordan on the quantization of a continuum) a comparative study is presented on the first line of thoughts that led to the concept of quanta. The nature of the particle-like fluctuations and the wave-like fluctuations are analysed by using several approaches. With the help of the classical probability theory, it is shown that the infinite divisibility of the Bose distribution leads to the new concept of classical poissonian photo-multiplets or to the binary photo-multiplets of fermionic character. As an application, Einstein's fluctuation formula is derived as a sum of fermion type fluctuations of the binary photo-multiplets.
A historical overview is given on the basic results which appeared by the year 1926 concerning Einstein's fluctuation formula of black-body radiation, in the context of light-quanta and wave-particle duality. On the basis of the original publications – from Planck's derivation of the black-body spectrum and Einstein's introduction of the photons up to the results of Born, Heisenberg and Jordan on the quantization of a continuum – a comparative study is presented on the first lines of thoughts that led to the concept of quanta. The nature of the particle-like fluctuations and the wave-like fluctuations are analysed by using several approaches. With the help of classical probability theory, it is shown that the infinite divisibility of the Bose distribution leads to the new concept of classical "poissonian photo-multiplets" or to the "binary photo-multiplets" of fermionic character. As an application, Einstein's fluctuation formula is derived as a sum of fermion type fluctuations of the binary photo-multiplets.
This work examines the physical and conceptual evolution of the light quantum from Planck's blackbody theory to the theoretical and experimental developments that led to the quantization of the electromagnetic field. The present study shows that the decisive transition occurred in Einstein's quantum theory of radiation (1916-1917). Absorption, stimulated emission, and spontaneous emission were formulated as elementary probabilistic mechanisms whose statistical balance alone reproduces the blackbody spectrum. In particular, spontaneous emission requires the emission of a light quantum, thereby implicitly proving its physical necessity before its theoretical status was clarified. At the same time, the already existing term photon began to acquire a stable usage following Lewis's 1926 proposal and became increasingly associated with Einstein's light quantum. By the mid-1920s, the central problem had shifted from whether light quanta were physically required to how radiation could be incorporated into the emerging quantum-mechanical formalism. This transition marks a key stage, illustrating how initial debates about the existence of light quanta gave way to their integration into a comprehensive theoretical structure. The resulting asymmetry between the novel quantum description of matter and the still-classical description of radiation, called into question by the phenomenon of spontaneous emission, identifies the physical problem that led to the quantization of the electromagnetic field.
Classical electromagnetics in free space leaves no room for accommodation of a light particle. To underline this, it is sufficient to remember that Einstein in order to reach his lightquantum (energy-quantum) hypothesis had to mix classical theoretical physics with a piece of experimental information that defies description in classical terms. In his 1905-paper Einstein concluded [60]:Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of mutually independent energy quanta of magnitude hν. The introduction of light quanta in the Wien regime (the high-frequency part of the Planck blackbody radiation law) is the first step toward the concept of electromagnetic radiation as a Bose gas of photons [180, 39, 68, 69]. Although a photon gas obeys Bose-Einstein statistics for all frequencies the statistical independence assumed by Einstein in 1905 is not true in general. Einstein used Boltzmann statistics when he conjectured the existence of light quanta. In the Wien regime Boltzmann and Bose-Einstein statistics lead to the same conclusion since the mean number of particles (photons) with a given energy is much less than one. The circumstance that the light quanta are not statistically independent in Bose-Einstein counting finds its explanation in quantum mechanics. Hence, the statistical correlation is induced by the requirement of totally symmetric wave functions.