By Gerard G. Emch
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Extra resources for Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Physics & Astronomical Monograph)
Signifies that there is no state for < 0. Hence, the density of states depends on the square root of the energy, 1=2 . In general, one can show that the density of states for free particles in d dimensions behaves as D. 1). This difference in energy dependence manifests itself in distinct physical properties among systems of different dimensions. 17) for the density of states, expressing it more generally as D. / D A Á 1 Â. /; 48 4 Statistical Mechanics of Ideal Gases where A and Á > 0 are constants.
K˛. 16) is k D 2 =L, we can convert the sum into an integral in the limit L ! 1 to obtain D. / D X k˛ ı. k/3 ı. Ã3 Z d3 k ı. 2s C 1/L3 4 dkk 2 ı. 2s C 1/V 2m 3=2 1 d"k D " ı. 2s C 1/V 4 2 Â sum ! integral "k / k 2m „2 adopt polar coordinates Â use k D 2m"k „2 Ã1=2 "k / k Ã3=2 1=2 Â. 17) In the last equality, the step function Â. / signifies that there is no state for < 0. Hence, the density of states depends on the square root of the energy, 1=2 . In general, one can show that the density of states for free particles in d dimensions behaves as D.
14) A particle with an integer (a half-integer) spin is called boson (fermion) after Bose (Fermi), who introduced the rule upon studying statistical mechanics of photons with s D 1 (electrons with s D 12 ) without referring to the connection with spin. The remarkable relationship was formulated by Fiertz  and proved by Pauli  in the context of relativistic quantum field theory. Although the statement itself has been confirmed experimentally without doubt, a direct proof in the non-relativistic framework seems yet to be performed.