The Planck’s law of blackbody radiation is derived in several methods in this paper. The original Planck’s derivation in his 1900 paper, the derivation by Einstein, and another derivation based on the Bose-Einstein statistics are reviewed with comments on each derivation. Statistical properties of radiation are briefly reviewed in the second part of the paper. Different concepts of photon statistics are applied on the thermal (chaotic) light and laser light. Nonclassical properties of light that manifest its quantum nature are briefly highlighted with a short review of the experimental evidence embodied in ……….. 1-Introduction and historical perspective It was Robert Kirchhoff who first argued in 1859 that the thermal radiation of the blackbody was of a fundamental nature.1 Since that time, several attempts were made to derive a precise relation that matches the experimental results. Wien, suggested that the radiation density per
The Planck’s law of blackbody radiation is derived in several methods in this paper. The original Planck’s derivation in his 1900 paper, the derivation by Einstein, and another derivation based on the Bose-Einstein statistics are reviewed with comments on each derivation. Statistical properties of radiation are briefly reviewed in the second part of the paper. Different concepts of photon statistics are applied on the thermal (chaotic) light and laser light. Nonclassical properties of light that manifest its quantum nature are briefly highlighted with a short review of the experimental evidence embodied in ……….. 1-Introduction and historical perspective It was Robert Kirchhoff who first argued in 1859 that the thermal radiation of the blackbody was of a fundamental nature.1 Since that time, several attempts were made to derive a precise relation that matches the experimental results. Wien, suggested that the radiation density per
Abstract There are two rather different phenomena that furthered the development of quantum theory: blackbody radiation and atomic spectra. In fact quantum theory began in 1900 with Planck’s radiation law and his introduction of the quantum of action h (Episode 7). His derivation of the law was by no means stringent. He assumed the existence of many resonators (atoms or molecules) of different but fixed frequencies ν, which could emit or absorb radiation only in quanta of energy E = hν. Einstein, in his light-quantum hypothesis of 1905, postulated that the energy quantization is a property of the radiation field itself and not due to the material resonators. He found a similarity between a cavity filled with radiation consisting of quanta and a vessel filled with gas consisting of atoms. However, he could not obtain Planck’s law but only Wien’s (Episode 10).
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.
The experimental foundations of quantum theory are presented in some detail in this chapter: on the one hand, the investigation of black-body radiation, which helps in developing an interdisciplinary view of physics, besides having historical interest; on the other hand, the energy and linear momentum of photons, atomic spectra, discrete energy levels, wave-like properties of electrons, interference phenomena and uncertainty relations. Black-body radiation Black-body radiation is not just a topic of historical interest. From a pedagogical point of view, it helps in developing an interdisciplinary view of physics, since it involves, among the other, branches of physics such as electrodynamics and thermodynamics, as well as a new constant of nature, the Planck constant, which is peculiar to quantum theory and quantum statistics. Moreover, looking at modern developments, the radiation that pervades the whole universe (Gamow 1946, Penzias and Wilson 1965, Smoot et al . 1992, Spergel et al . 2003) is a black-body radiation, and the expected emission of particles from black holes (space–time regions where gravity is so strong that no light ray can escape to infinity, and all nearby matter gets eaten up) is also (approximately) a black-body radiation (Hawking 1974, 1975). In this section, relying in part on Born (1969), we are aiming to derive the law of heat radiation, following Planck's method. We think of a box for which the walls are heated to a definite temperature T .
Planck’s 1900 introduction of the “energy elements”, i.e., electromagnetic radiation quanta of energy hν each, for founding his radiation law marks the origin of quantum theory and is discussed in all courses on this subject. Surprisingly, the question whether Planck really implied quantisation is debated among historians of physics. We present a simplified account of that debate which also sheds light on the issue of indistinguishability and Einstein’s light quantum hypothesis. Here, the relevance of the notion ‘(in)distinguishability’ is often overemphasized. It has to be treated together with ‘equality’, ‘identity’, and ‘interchangeability’, where the latter one is the decisive notion in this context. As a result, the difference between Planck’s and Einstein’s concepts of ‘light particles’ becomes more easily to grasp. For that and many other reasons, we believe that the teaching of quantum physics benefits from including the material presented here, and that beyond the demand of historical completeness. Nevertheless, for the sake of our readers’ convenience, we almost completely reprint Planck’s former students Hettner’s remembering of Planck’s way to his radiation law, together with its translation into English. It seems that Planck has not really done the calculations described in his December 1900 lecture. For this, following Becker, we show how to overcome the difficulties over there, and we sketch his simplifying approach in his subsequent 1901 article. Strangely enough, in his 1906/07 lectures on the theory of heat radiation, Planck deviates from those calculations and obtains a non-extensive entropy. This, too, is most easily analyzed using Bach’s three-level scheme ‘configuration. occupation. occupancy’ for the counting combinatorics involved. Boltzmann’s 1877, Planck’s 1900, Einstein’s 1905, Bose’s 1924, and other treatments are systematized using that scheme
Einstein's papers on quantum theory over the period 1905-1925 are reviewed. These papers played a fundamental role in the development of quantum concepts. This review traces the relationship between these papers, on the one hand, and Wien's and Planck's theoretical work on thermal radiation and Einstein's own work on statistical thermodynamics, on the other. Einstein's 1905 development of the concept of light quanta from Wien's radiation law, the application of this concept to elementary interactions of radiation with matter, Einstein's 1906–1907 analysis of the Planck radiation law, and Einstein's development of the foundations of a quantum theory of specific heat are all discussed. Particular attention is paid to Einstein's 1909 development of the concept of a particle-wave duality for radiation from a study of the energy and momentum fluctuations in a radiation field. Einstein's 1916 papers on the quantum theory of radiation are analyzed, and it is emphasized that Einstein was the first to suggest a probabilistic interpretation of the particle-wave duality, in addition to introducing coefficients to characterize the probabilities for optical transitions and offering his well-known derivation of the Planck radiation law. There is a special discussion of Einstein's expression, in his 1925 paper on the quantum theory of monatomic ideal gases, of high regard for de Broglie's concept of a particle-wave duality for material particles.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.