
Contents
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7.1 The photon and the Compton effect 7.1 The photon and the Compton effect
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7.2 Towards Bose–Einstein statistics 7.2 Towards Bose–Einstein statistics
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7.2.1 Boltzmann statistics 7.2.1 Boltzmann statistics
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7.2.2 Bose’s counting of photon states 7.2.2 Bose’s counting of photon states
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Density of photon states Density of photon states
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Bose’s derivation of the Planck distribution Bose’s derivation of the Planck distribution
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Implicit assumptions in Bose’s derivation Implicit assumptions in Bose’s derivation
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7.2.3 Einstein’s elaboration of Bose’s counting statistics 7.2.3 Einstein’s elaboration of Bose’s counting statistics
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Density of nonrelativistic particle states Density of nonrelativistic particle states
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Distribution of identical bosons Distribution of identical bosons
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The discovery of Bose–Einstein condensation The discovery of Bose–Einstein condensation
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7.3 Quantum mechanics and identical particles 7.3 Quantum mechanics and identical particles
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7.3.1 Wave mechanics: de Broglie–Einstein–Schrödinger 7.3.1 Wave mechanics: de Broglie–Einstein–Schrödinger
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7.3.2 Identical particles are truly identical in quantum mechanics 7.3.2 Identical particles are truly identical in quantum mechanics
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7.3.3 Spin and statistics 7.3.3 Spin and statistics
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7.3.4 The physical implications of symmetrization 7.3.4 The physical implications of symmetrization
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The final resolution of the counting schemes of Bose (1924), Einstein (1905), and Planck (1900) The final resolution of the counting schemes of Bose (1924), Einstein (1905), and Planck (1900)
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7.4 Bose–Einstein condensation 7.4 Bose–Einstein condensation
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7.4.1 Condensate occupancy calculated 7.4.1 Condensate occupancy calculated
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7.4.2 The condensation temperature 7.4.2 The condensation temperature
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for noninteracting helium for noninteracting helium
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Behavior of the chemical potential when Behavior of the chemical potential when
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Overlap of wavefunctions when Overlap of wavefunctions when
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7.4.3 Laboratory observation of Bose–Einstein condensation 7.4.3 Laboratory observation of Bose–Einstein condensation
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7.5 SuppMat: Radiation pressure due to a gas of photons 7.5 SuppMat: Radiation pressure due to a gas of photons
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7.6 SuppMat: Planck’s original analysis in view of Bose–Einstein statistics 7.6 SuppMat: Planck’s original analysis in view of Bose–Einstein statistics
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7.7 SuppMat: The role of particle indistinguishability in Bose–Einstein condensation 7.7 SuppMat: The role of particle indistinguishability in Bose–Einstein condensation
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7 Bose–Einstein statistics and condensation
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Published:January 2013
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Abstract
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.
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