A partial differential equation for the number density of a D‐dimensional grand canonical ensemble is used to derive an explicit expression for the semiclassical number density. This is accomplished by first using the Bloch equation and its solutions to derive the classical number density and then combining the classical result with a Planck’s constant expansion of the partial differential equation to obtain the semiclassical formula. Similar results are also derived for the entropy density, energy density, and grand potential density of a D‐dimensional grand canonical ensemble.
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B. Kahn, On the Theory of the Equation of State (North-Holland, Amsterdam, 1938), reprinted in Studies in Statistical Mechanics, edited by J. DeBoer and G. E. Uhlenbeck (North-Holland, Amsterdam, 1965), Vol. III, pp. 277ff.
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L. B. W. Jolley, Summation of Series (Dover, New York, 1961).
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© 1995 American Institute of Physics.
1995
American Institute of Physics
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