Quantum corrections to the Hamiltonian function

In Section 2.5.3 we derived the semiclassic expression for the canonical partition function [see Eq. (2.110)] based on the assumption that at sufficiently high temperatures we may replace the Hamiltonian operator by its classic analog, the Hamiltonian function [see Eq. (2.100)]. In this section we will sketch a more refined treatment of the seniiclassic theory developed in Section 2.5 originally due to Hill and presented in detail in his classical work on statistical mechanic s [326]. Because of Hill s clear and detailed exposition and because we need the final result mainly as a justiheation to treat conBued fluids l means of classic statistical thermodynamics, we will just briefly outline the key ideas of Hill s treatment for reasons of completeness of the current work. 

Vi = dldvi. In deriving Eq. (B.106) we employed the space representation of the Hamiltonian operator [see Eq. (2.95)], and the fact that the classic Hamiltonian firnction ran be. split into kinetic- and potential-energy contributions according to Eq. (2.100). Terms proportional to in Eq. (B.106) arise from the kinetic part of II applied to the product of terms on the right side of Eq. (B.104) (using, of course, the product rule of conventional calculus). 

Equating in this expression terms of equal power in h, one immediately finds that 

Inserting this result back into Eq. (B.109), the first-order function is obtained as 

In writing Eq. (B.113) we have already neglected the term proportional to yP 0)- Because this coefficient is linear in p, it will vanish upon integration over momentum subspace [see, for example, Eq. (2.111)]. From Eq. (B.113) one realizes that the correction 

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