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Quantum statistics and the origin of wave mechanics

Robert Golub, S. K. Lamoreaux

Published Feb 28, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780198822189.003.0009

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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.
Satyendra Nath (S.N.) Bose is one of the great Indian scientists. His remarkable work on the black body radiation or derivation of Planck’s law led to quantum statistics, in particular, the statistics of photon. Albert Einstein applied Bose’s idea to a gas made of atoms and predicted a new state of matter now called Bose-Einstein condensate. It took 70 years to observe the predicted condensation phenomenon in the laboratory. With a brief introduction to the formative period of Professor Bose, this research survey begins with the founding works on quantum statistics and, subsequently, provides a brief account of the series of events terminating in the experimental realization of Bose-Einstein condensation. We also provide two simple examples to visualize the role of synthetic spin-orbit coupling in a quasi-one-dimensional condensate with attractive atom-atom interaction.
Abstract With the failure of classical physics to explain the experimentally observed black body radiation spectrum, the photoelectric effect and the existence of narrow spectral lines in the light emitted by excited atoms, physicists were forced to reach out for new ideas. Following Planck’s initial “drastic hypothesis” of light “quanta”, Einstein led the way in exploring some of the myriad implications of this idea. In particular, the study of fluctuations led to a new understanding of the interaction of electromagnetic radiation with atoms. Einstein invented the photon by showing, through fluctuation studies, that light emitted by atoms must be in the form of plane waves. These ideas led to the first notions regarding the quantization of matter and its dynamics.
In 1924, Satyendra Nath Bose's pioneering work laid the foundation for Bose-Einstein statistics, which describes particles with integral spins. His derivation of Planck's law for blackbody radiation bypassed classical assumptions, relying instead on the statistical mechanics of light quanta. Bose's methodology addressed limitations in existing theories, such as Einstein's dependence on classical concepts like Wien's displacement law and Bohr's correspondence principle. Further, his work underscored the incompatibility between classical electrodynamics and quantum theory, proposing innovative statistical approaches to thermodynamic equilibrium. The insights from Bose's work extend beyond theoretical physics. As was immediately noticed by Einstein, for temperatures below a critical threshold, Bose-Einstein statistics predicts the formation of a Bose-Einstein condensate (BEC), where particles condense en-masse into the ground state. This quantum phenomenon on a macroscopic scale opened avenues to explore new technologies in recent times, apart from throwing light on new phases of matter. This article revisits Bose's groundbreaking contributions, highlighting their enduring impact on quantum mechanics, statistical physics, and field theory.
This paper gives a historical perspective to the gestation and birth of Bose-Einstein (BE).As is known today, particles whose spin is an even multiple of hslash/2\\hslash/2hslash/2 obey BE statistics, those whose spin is an odd multiple of hslash/2\\hslash/2hslash/2 obey FD statistics. (These particles may be elementary or composite.) Both statistics are derived from the fact that particles in the quantum realm are indistinguishable, the difference being that in the FD case at most one particle can occupy any given quantum state (Pauli principle) while in the BE case any number of particles can be in a particular state. Little of this was known when the statistics were first developed by Bose. The paper gives the background of attempts to explain the black-body spectrum, which led eventually to Bose's combinatorical derivation of the Planck law using the concept of indistinguishability. (Bose was professor of physics at Dacca, in what is now Bangladesh, then India.) Einstein was the first to appreciate Bose's work, translating the paper into German for publication in Zeitschrift für Physik, and applying the methods to a quantum gas (hence, "Bose-Einstein condensation", which explains the bizarre behavior of liquid rmHe4{\\rm He}^4rmHe4). \n\n
The principal mathematical idea behind the statistical properties of black-body radiation (photons) was introduced already by L. Boltzmann (1877/2015) and used by M. Planck (1900; 1906) to derive the frequency distribution of radiation (Planck’s law) when its discrete (quantum) structure was additionally added to the reasoning. The fundamental physical idea – the principle of indistinguishability of the quanta (photons) – had been somewhat hidden behind the formalism and evolved slowly. Here the role of P. Debye (1910), H. Kamerlingh Onnes and P. Ehrenfest (1914) is briefly elaborated and the crucial role of W. Natanson (1911a; 1911b; 1913) is emphasized. The reintroduction of this Natanson’s statistics by S. N. Bose (1924/2009) for light quanta (called photons since the late 1920s), and its subsequent generalization to material particles by A. Einstein (1924; 1925) is regarded as the most direct and transparent, but involves the concept of grand canonical ensemble of J. W. Gibbs (1902/1981), which in a way obscures the indistinguishability of the particles involved. It was ingeniously reintroduced by P. A. M. Dirac (1926) via postulating (imposing) the transposition symmetry onto the many-particle wave function. The above statements are discussed in this paper, including the recent idea of the author (Spałek 2020) of transformation (transmutation) – under specific conditions – of the indistinguishable particles into the corresponding to them distinguishable quantum particles. The last remark may serve as a form of the author’s post scriptum to the indistinguishability principle.
Abstract This chapter presents the history of quantum statistics, from its beginnings in Planck’s work, through the development of Bose-Einstein and Fermi-Dirac statistics, to the more recent work on forms of parastatistics, focusing on the role played by considerations of particle identity and individuality. The new physics was taken to imply that quantum particles are non-individuals in some sense. This provides the framework for a novel account of Bohr’s view of particle individuality, which further illuminates his complementarity interpretation as well as Born and Schrödinger’s broadly structuralist approaches. Schrödinger’s view that identity does not apply to quantum particles is emphasized, as is Weyl’s incorporation of this view into his treatment of aggregates of particles.