Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bose statistics : before and after

Narasimhaiengar Mukunda

Published Jun 1, 1994• 3 citations
OpenAlex

Abstract

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 $\\hslash/2$ obey BE statistics, those whose spin is an odd multiple of $\\hslash/2$ 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 ${\\rm He}^4$). \n\n

Related papers

Quantum statistics and the origin of wave mechanics

Robert Golub, S. K. Lamoreaux

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.

•Feb 28, 2023•0 citations

Quantization of the electromagnetic field, entropy of an ideal monoatomic gas, and the birth of the Bose-Einstein statistics

Masud Mansuripur

In 1924, Einstein received a short manuscript in the mail from the Indian physicist S.N. Bose. He quickly translated Bose’s manuscript to German and submitted it to Zeitschrift für Physik. Within a few weeks, Einstein presented his own findings (using a generalization of Bose’s counting method) to a session of the Prussian Academy of Sciences. Whereas Bose had suggested a new counting method for the quanta of the electromagnetic field—one that yielded Planck’s blackbody radiation formula—Einstein applied Bose’s method to an ideal monoatomic gas. Shortly afterward, Einstein presented to the Academy a follow-up paper in which he described the Bose-Einstein condensation for the first time. The present paper describes some of the fascinating issues that Einstein struggled with as he attempted to unify the quantum-statistical properties of matter with those of the electromagnetic field.

arXiv:2504.03689•Quantum Sensing and Nano Electronics and Photonics•Mar 21, 2025•0 citations

Satyendra Nath Bose: quantum statistics to Bose-Einstein condensation

Golam Ali Sekh, B. Talukdar

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.

arXiv:2210.13967•The Moldavian Journal of the Physical Sciences•Dec 1, 2023•0 citations

Bose–Einstein statistics and condensation

Ta-Pei Cheng

We discuss another of Einstein’s statistical contributions to quantum theory: his work on quantum statistics. It was prompted by a paper that S. Bose sent to him in 1924 when the physics community had finally accepted the photon idea upon the discovery and analysis of Compton scattering. We present in detail Bose’s derivation of Planck’s distribution, using a particle approach. Einstein extended Bose’s analysis of radiation to systems of matter particles. Here he made the discovery of the astounding phenomenon of Bose–Einstein condensation. The ultimate understanding of Planck’s spectral distribution came about in modern quantum mechanics with its notion of indistinguishable particles, and the connection between spin and statistics. We also provide a modern discussion of Bose–Einstein condensation.

Oxford University Press eBooks•Jan 31, 2013•0 citations

The Story of Bose, Photon Spin and Indistinguishability

Partha Ghose

As we approach the centenary of the discovery of quantum statistics in 1924, it is important to revisit Bose's original derivation of Planck's law usually ignored in most standard presentations of Bose-Einstein statistics. It introduced not only the novel concept of the indistinguishability of photons but also of their intrinsic spin, a fact unknown to most physicists.

arXiv:2308.01909•Jun 14, 2023•3 citations

The Bose-Einstein statistics: Remarks on Debye, Natanson, and Ehrenfest contributions and the emergence of indistinguishability principle for quantum particles

Józef Spałek

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.

Studia Historiae Scientiarum•Sep 30, 2020•6 citations
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
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.
In 1924, Einstein received a short manuscript in the mail from the Indian physicist S.N. Bose. He quickly translated Bose’s manuscript to German and submitted it to Zeitschrift für Physik. Within a few weeks, Einstein presented his own findings (using a generalization of Bose’s counting method) to a session of the Prussian Academy of Sciences. Whereas Bose had suggested a new counting method for the quanta of the electromagnetic field—one that yielded Planck’s blackbody radiation formula—Einstein applied Bose’s method to an ideal monoatomic gas. Shortly afterward, Einstein presented to the Academy a follow-up paper in which he described the Bose-Einstein condensation for the first time. The present paper describes some of the fascinating issues that Einstein struggled with as he attempted to unify the quantum-statistical properties of matter with those of the electromagnetic field.
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.
We discuss another of Einstein’s statistical contributions to quantum theory: his work on quantum statistics. It was prompted by a paper that S. Bose sent to him in 1924 when the physics community had finally accepted the photon idea upon the discovery and analysis of Compton scattering. We present in detail Bose’s derivation of Planck’s distribution, using a particle approach. Einstein extended Bose’s analysis of radiation to systems of matter particles. Here he made the discovery of the astounding phenomenon of Bose–Einstein condensation. The ultimate understanding of Planck’s spectral distribution came about in modern quantum mechanics with its notion of indistinguishable particles, and the connection between spin and statistics. We also provide a modern discussion of Bose–Einstein condensation.
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.