As part of a study of Planck’s blackbody radiation theory, H. Poincaré (in 1911–12) advanced a theory which analyzed the partition of energy between “resonators” and the kinetic motion of atoms. Resonators (the objects of Poincaré’s theory) facilitate the exchange of energy between radiation and matter, but otherwise their identity has remained unresolved. Poincaré considered resonators characterized by a particular mean energy ε/[exp(ε/kT)−1], which he showed to necessarily imply quantized energies nε (n=0,1,2,…). We additionally consider resonators characterized by a mean energy ε/[exp(ε/kT)+1], which (using Poincaré’s methodology) we show to necessarily imply quantized energies nε (n=0 and 1). Resonators are here identified with transitions between internal quantum states of atoms. This includes normal electronic atoms characterized by possible energies nε (n=0 and 1), as well as atoms populated by subatomic bosons (such as pions) and characterized by multiple occupancy of quantum states and possible energies nε (n=0,1,2,…). We distinguish between Poincaré’s theory and the closely related analysis by P. Ehrenfest of quantization amongst cavity modes.
As part of a study of Planck’s blackbody radiation theory, H. Poincaré (in 1911–12) advanced a theory which analyzed the partition of energy between “resonators” and the kinetic motion of atoms. Resonators (the objects of Poincaré’s theory) facilitate the exchange of energy between radiation and matter, but otherwise their identity has remained unresolved. Poincaré considered resonators characterized by a particular mean energy ε/[exp(ε/kT)−1], which he showed to necessarily imply quantized energies nε (n=0,1,2,…). We additionally consider resonators characterized by a mean energy ε/[exp(ε/kT)+1], which (using Poincaré’s methodology) we show to necessarily imply quantized energies nε (n=0 and 1). Resonators are here identified with transitions between internal quantum states of atoms. This includes normal electronic atoms characterized by possible energies nε (n=0 and 1), as well as atoms populated by subatomic bosons (such as pions) and characterized by multiple occupancy of quantum states and possible energies nε (n=0,1,2,…). We distinguish between Poincaré’s theory and the closely related analysis by P. Ehrenfest of quantization amongst cavity modes.
In his last memoir on mathematical physics, Henri Poincaré presented one of the most profound and compelling proofs of the hypothesis of quanta. This highly original proof, which is actually three separate proofs, is based on first principles and is full of physical insight, mathematical rigor, and elegant simplicity. The memoir is refreshingly uncluttered by some of the conventional, and more abstract concepts, such as temperature and entropy, that Planck and others relied on in their work. Poincaré’s analysis is based on an ingenious physical model consisting of long-period resonators interacting with short-period resonators. A unique formulation of statistical mechanics, based on the calculus of probabilities, Fourier’s integral, and complex analysis, logically unfolds throughout the memoir. Poincaré invents an ‘‘inverse statistical mechanics’’ that allows him to prove a crucial result that no one had proved before: The hypothesis of quanta is both a sufficient and a necessary condition to account for Planck’s law of radiation. In a separate, more universal proof, Poincaré proves that the existence of a discontinuity in the motion of a resonator is necessary to explain any observed law of radiation. Given the significant impact of Poincaré’s memoir on quantum theory and statistical physics, it is surprising that most physicists are not aware of its valuable mathematical and physical ideas. Poincaré’s tour de force proofs are presented here in a form suitable for use in a standard course in quantum mechanics, statistical mechanics, or mathematical physics.
The origin of the relativity theories was closely bound up with the development of electromagnetic concepts, a development that approached a coherent fieldtheoretical formulation, according to which all actions may vary in a continuous manner. In contrast, quantum theory arose out of the development of atomic concepts, a development that was characterized by the acknowledgment of a fundamental limitation to classical physical ideas when applied to atomic phenomena. This restriction was expressed in the so-called quantum postulate, which attributed to any atomic process an essential discontinuity that was symbolized by Planck's quantum of action. Quantum field theory (QFT) is a later phase of the conceptual developments of quantum theory, and has the old quantum theory and non-relativistic quantum mechanics, essentially the preliminary analyses of the interactions between atoms and radiation, as its predecessors. This chapter will review some features of quantum physics that are relevant to the rise of QFT. The quantization of motion In solving the problem of the equilibrium between matter and radiation, Max Planck (1900) showed that the laws of heat radiation demanded an element of discontinuity in the description of atomic processes. In the statistical behavior of atoms represented, in Planck's description, by linear resonators in their interactions with radiation, only states of vibration should be taken into account whose energy was an integral multiple of a quantum, hv , where h is Planck's constant and v is the frequency of the resonator. Planck himself believed that the discontinuity of energy was only a property of atoms, and was not ready to apply the idea of energy quantization to radiation itself.
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).
In deriving his radiation law in 1900, Max Planck employed a simple harmonic oscillator to model the exchange of energy between radiation and matter. Traditionally the harmonic oscillator has been viewed as modelling an entity which is itself oscillating, although a suitable oscillating entity has not been forthcoming. (Opinion is divided between a material oscillator, an imaginary oscillator and a need to revise Planck"s derivation to apply to cavity modes of oscillation). We offer a novel, atomistic interpretation of Planck"s derivation wherein the harmonic oscillator models a transition between the internal quantum states of an atom|not a normal electronic atom characterised by possible energies 0 and hv , but an atom populated by subatomic bosons (such as pions) and characterised by multiple occupancy of quantum states and possible energies nhv ( n = 0;1;2; :::). We show how Planck"s derivation can be varied to accommodate electronic atoms. A corollary to the atomistic interpretation is that Planck"s derivation can no longer be construed as support for the postulate that material oscillating entities can have only those energies that are multiples of hv .
An analysis of applying the property of speed instantaneity to electronic transitions (i.e., transitions of the atomic electron that are not instantaneous), instead of quantum jumps (i.e., instantaneous transitions), during a complete oscillation for the process of Bohr’s model of the atom absorbing and emitting electromagnetic radiation (i.e., photon) reveals a discontinuity that is conserved as an emitted electromagnetic radiation (i.e., photon) discontinuity. The frequency of the oscillations of the atomic electron (i.e., electric charge) and induced magnetic field for the transitions of the atomic electron (i.e., electric charge) during the complete oscillation for the absorption and emission process (of Bohr’s atomic model) is conserved as the frequency of the oscillating electric and magnetic fields of the emitted electromagnetic radiation (i.e., photon). The conserved frequency and discontinuity (based on speed instantaneity and electronic transitions that are not instantaneous) of the atomic absorption and emission process (of Bohr’s atomic model) lead to predicting the need for Planck’s constant (i.e., quantum of action) without the use of probability. Planck claimed the interpretation of reality must change if his method of deriving the radiation law stood on a physical concept. Einstein, whom Planck credited as the first to make an advancement in the quantum field, unknowingly introduced a quantized two-dimensional discontinuum geometry that is one of the foundations on which a table of five realities used in physics was developed for making predictions. Using the table of realities, locality is redefined, and entanglement is demystified.
‘At the present state of science it does not seem possible to avoid the formal character of the quantum theory which is shown by the fact that the interpretation of atomic phenomena does not involve a description of the mechanism of the discontinuous processes […]. On the correspondence principle it seems nevertheless possible to […] arrive at a consistent description of optical phenomena by connecting the discontinuous effects occurring in atoms with the continuous radiation field […]’. This announced a turning point in atomic physics research; for the first time there were reasons to assert that discontinuity was not in itself incompatible with the theory's descriptive content, and rational solutions were provided that related a whole set of phenomena dependent on the radiation properties of matter to the continuous picture of the radiation field. This was the conclusion arrived at in a new paper by Bohr which took up and developed the hypothesis put forward by the American physicist John Slater that ‘the atom, even before a process of transition between two stationary states takes place, is capable of communication with distant atoms through a virtual radiation field’ was taken up and developed. The article, ‘The Quantum Theory of Radiation’, signed by Bohr, Kramers and Slater, was published in the Philosophical Magazine in January 1924. Slater had arrived in Copenhagen at the end of the preceding year with the idea that his hypothesis might lead to very different theoretical results.