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De Broglie’s Matter Waves and Einstein’s Quantum Theory of the Ideal Gas

Anthony Duncan A, Michel Janssen B

Published Aug 17, 2023• 0 citations
Publisher

Abstract

Abstract This chapter covers De Broglie’s extension to matter of wave–particle duality, which Einstein had introduced for radiation. It fills in the background in the tradition of extremal principles in classical physics that De Broglie drew on to formulate his theory. It then covers Bose’s introduction of a new statistics for light quanta and Einstein’s application of those statistics to matter in his quantum theory of the ideal gas. Providing support for De Broglie’s ideas, Einstein derived a formula for energy fluctuations in such a gas that has the exact same form as the formula for energy fluctuations in black-body radiations: it is the sum of two terms, one suggesting waves, the other one suggesting particles.

DOI
10.1093/oso/9780198883906.003.0006

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Abstract This chapter covers De Broglie’s extension to matter of wave–particle duality, which Einstein had introduced for radiation. It fills in the background in the tradition of extremal principles in classical physics that De Broglie drew on to formulate his theory. It then covers Bose’s introduction of a new statistics for light quanta and Einstein’s application of those statistics to matter in his quantum theory of the ideal gas. Providing support for De Broglie’s ideas, Einstein derived a formula for energy fluctuations in such a gas that has the exact same form as the formula for energy fluctuations in black-body radiations: it is the sum of two terms, one suggesting waves, the other one suggesting particles.
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.
This chapter examines the wave properties of matter. Early attempts to extend the quantum ideas to the theory of matter had some limited success, but did not progress very far. Another radical idea was required before a satisfactory theory could be developed. As light has both wave and particle properties, perhaps the same is true of 'ordinary' particles such as electrons. This bold suggestion was made by Louis de Broglie in 1923, and within a few years it had been confirmed experimentally. De Broglie's wave hypothesis formed the basis for Schrödinger's equation, which was found capable of predicting the allowed energy levels in atoms and molecules, and must be regarded as one of the most important mathematical equations of modern science. Before discussing these ideas, the chapter begins by considering the classical mechanics of particles, reviewing some simple concepts that form an essential background to the quantum theory of matter.
Abstract Planck’s original introduction of the quantum idea was based on a statistical mechanical treatment of black body radiation. S. Bose, a physicist working in India, found a new way of deriving Planck’s law. Einstein applied Bose’s method to a gas of massive particles. This led to the prediction of a new physical effect (Bose-Einstein condensation). With the introduction of the exclusion principle by Pauli, Fermi, and Dirac independently introduced a new kind of statistics. Considerations by Einstein and Schrödinger introduced a preliminary form of second quantization.
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.
Wave-particle duality is widely regarded as one of the conceptual pillars of modern quantum theory, providing the framework through which the behavior of microscopic matter can be consistently interpreted. According to this principle, microscopic entities such as electrons, photons, atoms and even complex molecules exhibit both wave-like and particle-like properties depending on the experimental configuration used to observe them. This duality is quantitatively formalized through the De Broglie relation, which associates a wavelength with any particle possessing momentum. The present work provides a comprehensive theoretical analysis of the physical and mathematical foundations of wave-particle duality, focusing particularly on the De Broglie wavelength and the kinetic energy of the electron in both non-relativistic and relativistic regimes. The connection between the De Broglie hypothesis and Schrödinger’s equation is discussed in detail, highlighting the role of the kinetic energy operator and the interpretation of matter waves in terms of quantum wave packets. Experimental evidence supporting the wave nature of matter, including electron diffraction experiments and interferometry techniques, is also reviewed. Beyond their conceptual importance, these principles underpin several modern technologies, including electron microscopy, nanoscale semiconductor devices and quantum metrology. The analysis demonstrates that the De Broglie relation provides a unifying framework connecting momentum, wavelength and kinetic energy, thereby forming a fundamental bridge between classical mechanics, quantum theory and modern technological applications.
Abstract This chapter examines some crucial experiments and basic principles of quantum mechanics that have strongly influenced the development of quantum theory in its modern form: starting from Planck’s law of black-body radiation and its derivation using the quantization of energy; Einstein’s photon hypothesis for the puzzle presented by the photoelectric effect and its test via the Millikan experiment; the Compton effect; Bohr’s theses for the structure of the hydrogen atom and the complementarity of waves and particles i.e., the wave-particle duality, culminating in de Broglie’s ideas for matter waves and their confirmation in the Davisson-Germer experiment for electron diffraction from a crystal; Heisenberg’s uncertainty principle for position and momentum as well as for energy an time; the famous double-slit experiment and Born’s probability interpretation, and finally the provocative thought experiment of Schrödinger’s cat