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Satyendra Nath Bose
Quantum Statistics for Photons (1924)
In 1924, Albert Einstein received an amazing very short paper from India by Satyendra Nath Bose. Einstein must have been pleased to read the title, "Planck's Law and the Hypothesis of Light Quanta." It was more attention to Einstein's 1905 work than anyone had paid in nearly twenty years. The paper began by claiming that the "phase space" (a combination of 3-dimensional coordinate space and 3-dimensional momentum space) should be divided into small volumes of h^{3}, the cube of Planck's constant. By simply counting the number of possible distributions of light quanta over these cells, Bose claimed he could calculate the entropy and all other thermodynamic properties of the radiation, including the famous Planck law.
ρ_{ν}dν = (8πhν^{2}/c^{3}) / (e^{ - hν / kT } -1)
Maxwell and Boltzmann had derived their distribution law for material particles by analogy with the Gaussian exponential tail of probability in the theory of errors. The number of gas particles with velocity between v and v + dv is
N ( v ) dv = (4 / α^{2} √π) v^{2} e^{ - v2 / α2 } dv.
Max Planck had simply guessed his expression from Wien's law for high frequency radiation
ρ_{ν}dν = aν^{3} e^{ - bν/T}.
and from Lord Rayleigh's expression for low-frequency (long-wavelength) radiation.
ρ_{ν}dν = ν^{2}T.
Planck simply added the term - 1 in the denominator of Wien's expression (1 / e^{ - bν / T}). Planck later acknowledged that his breakthrough was luckily finding a mathematical expression that interpolates between Wien's Law at high frequencies and the Rayleigh-Jeans Law at low frequencies. He called it a glückliche Interpolationformel. Einstein said Planck's work was "monstrous," but obviously in perfect agreement with experiment. How had he done it? All Einstein's derivations of the Planck law, including that of 1916-17 (which Bose called "remarkably elegant"), used classical electromagnetic theory to derive the density of radiation as the number of "modes" or "degrees of freedom" of the radiation field,
ρ_{ν}dν = (8πν^{2}dν / c^{3}) E
In 1906, Einstein had criticized Planck's use of this classical expression in deriving his "quantum" radiation law. He called it a contradiction... "this assumption...contradicts the theoretical basis from which [this expression] was developed. |