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On the Classical Limits of Quantum Statistical Distributions
Author(s) -
Fujita S.,
Schubert R.,
Ho C. T.,
Lee E. K.
Publication year - 1990
Publication title -
fortschritte der physik/progress of physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/prop.2190381102
Subject(s) - boson , physics , classical limit , fermion , quantum mechanics , statistical mechanics , hamiltonian (control theory) , quantum statistical mechanics , partition function (quantum field theory) , quantum , identical particles , boltzmann distribution , distribution function , thermodynamic limit , boltzmann constant , mathematical physics , mathematics , mathematical optimization
In the classical mechanical limit in which quantum cells of volume (2πħ) 2 are reduced to points in phase space, the Bose and Fermi distributions are shown to approach the Boltzmann distribution. This limit is a continuous limit and therefore does not alter the indistinguishability of bosons (fermions) which is defined in terms of a discontinuous permutation symmetry of the density operator and Hamiltonian for the system. This means that the classical particles which correspond to the identical boson (fermions) remain indistinguishable. In the same limit, the partition function for N identical bosons approaches (2πħ) −3 N ( N !) −1 f . f d 3 r 1 d 3 p 1 … d 3 r N d 3 pN exp (–β H ) Z c . The two factors, (2πħ) −3 N and ( N !) −1 which are often added in an ad hoc manner in many books on statistical mechanics, are thus derived from first principles. In the classical statistical limit in which the number density n and the reciprocal temperature β become small, the quantum statistical distributions also approach the Boltzmann distribution. This is shown by examining the chemical potential μ as a function of n and β. The classical approximation is valid if the thermal de Broglie wavelength is much shorter than the interparticle distance irrespective of any interparticle interaction .