Abstract Einstein’s statistical-thermodynamic calculations of the random variations of the radiation density constitute the ground upon which the light quantum hypothesis was originally based. According to these calculations such a variation of the radiation energy prevails—superposed above the fluctuations caused by the interferences calculated according to the classical theory—as if the radiation consisted of mutually independently mobile quanta hv of energy. According to Einstein, Maxwell’s theory correctly renders mean time values, which alone have been directly observable, as proved by the complete agreement between theory and experiment in optics; but Maxwell’s theory leads to laws respecting the thermic properties of radiation which are incompatible with the entropy-probability relation. After the wave-mechanical theory of de Broglie and Schrödinger had been generally accepted, the idea concerning the statistical character of the wave-field—already presented by Einstein in 1905—has, of course, been taken up again by Born in a more general way. As is well known, Bohr and Heisenberg have tried to conquer the difficulties to which the radiation theory has led by a radical change in our conception of energy.
Abstract Einstein’s statistical-thermodynamic calculations of the random variations of the radiation density constitute the ground upon which the light quantum hypothesis was originally based. According to these calculations such a variation of the radiation energy prevails—superposed above the fluctuations caused by the interferences calculated according to the classical theory—as if the radiation consisted of mutually independently mobile quanta hv of energy. According to Einstein, Maxwell’s theory correctly renders mean time values, which alone have been directly observable, as proved by the complete agreement between theory and experiment in optics; but Maxwell’s theory leads to laws respecting the thermic properties of radiation which are incompatible with the entropy-probability relation. After the wave-mechanical theory of de Broglie and Schrödinger had been generally accepted, the idea concerning the statistical character of the wave-field—already presented by Einstein in 1905—has, of course, been taken up again by Born in a more general way. As is well known, Bohr and Heisenberg have tried to conquer the difficulties to which the radiation theory has led by a radical change in our conception of energy.
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.
Abstract The question of fluctuations in electromagnetic radiation played an important part during the first period of the development of quantum theory. After having introduced (Einstein 1905) the conception of light quanta or photons in order to explain the observed phenomena of the photoelectric effect, Einstein (1909) considered the consequences of this idea for other properties of the radiation. Planck’s formula for the energy density of radiation implies, by arguments of general thermodynamics and statistics, the following expression for the mean square fluctuation of the energy contained in a volume v in terms of the mean energy Ēv.
Einstein never deviated from his belief in the reality of light quanta. In 1909, he published a remarkable paper in which he showed that fluctuations in the intensity of black-body radiation consisted of the statistical sum of the wave and particle properties of light. Following the 1911 Solvay conference, most of the participants were converted to the existence of light quanta. Bohr first applied the concepts of quanta to the structure of atoms. The experiment which finally convinced everyone of the light quantum hypothesis was Compton's demonstration of the energy change of X-rays in their scattering by electrons. This resulted in the concept of the wave-particle duality of light, one of the major foundation stones of quantum mechanics.
Light quanta The history of the corpuscular and wave controversy in optical theory is well known. At the end of the last century physicists had no doubt as to the wave character of light, and the acceptance of the wave theory meant the rejection of the corpuscular theory. The possibility of both being correct and helpful to one another had not been much thought about and still less explored. At about this time Planck's quantum theory appeared and was soon followed in 1905 by Einstein's theory of ‘light quanta’ to explain the photo-electric effect, for which the wave theory alone had utterly failed to account. Einstein assumed that radiation exists in discrete ‘quanta’ of energy hν , where ν is the frequency; thus a quantum is a separate thing, like a corpuscle, and has a frequency associated with it, like a wave. There was a blend of the two older theories involved. This idea has been generalised by the speculations of Louis de Broglie, who pointed out the important clue given by the theory of relativity, namely, that of the identity of mass and energy, so that the conservation of mass is therefore also the conservation of energy. This suggested to him that all forms of energy (including radiation) should have an atomic structure, like matter; and that the atoms of energy are grouped round certain singular points, forming electrons, light quanta, and the like.
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
Abstract This paper is based on the idea that wave mechanics is primarily suitable for giving an indirect description of nature. The world to which the equations of the, theory here proposed are directly applicable is an unnatural one, containing sources and sinks of matter which give to its radiational laws a symmetry lacking in nature. The theory may be regarded as consisting of two postulates. The first (which may, if desired, be regarded as a definition) specifies the character and properties of an unnatural assembly of systems, each consisting of an electron in a static potential field U. It states that, of the energy radiated in an elementary solid angle of standard magnitude, the part which will pass through an analyzer which transmits the component of electric field in a given direction (perpendicular to the line of observation) is determined by applying the usual Schrödinger interpretations (of density, current and material energy) to a Schrödinger equation perturbed by a term which is a certain constant multiplied by the electric field, in that direction, acting on the electron, due to the potential field U. (For comparison with classical radiation theory, it will be noted that the perturbation term vanishes in any region in which, from the classical standpoint, the electron is not accelerated, in that direction, by the field U). In the process of specifying the radiation from the assembly, the Schrödinger interpretations will have specified also the nature and magnitude of the sources and sinks of electrons (each in a field U). Their presence is due, mathematically speaking, to the multiplying constant mentioned above being a complex quantity. From these results the properties of a natural assembly are specified by means of the second postulate, which states that when the sources and sinks are removed, any process which in their absence is meaningless or physically inconceivable will cease, and all other processes will remain unaffected. The calculation is readily extended to include the effect of an external field of radiation. The theory requires no representation of radiation other than the Maxwellian wave, and will be free from the difficulties encountered in other theories.