Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 53

In statistical mechanics, the “grand potential” Φ associated with the grand canonical equilibrium is defined as the (natural) logarithm of the partition function, multiplied by -kBT (cf. Appendix VI, § 1-cβ):

(6)

where Z is given by (2). The trace appearing in this equation is easily computed in the basis of the Fock states built from the individual states {|uk〉}, as we now show. The trace of a tensor product of operators (Chapter II, § F-2-b) is simply the product of the traces of each operator. The Fock space has the structure of a tensor product of the spaces associated with each of the |uk〉 (each being spanned by kets having a population nk ranging from zero to infinity - see comment (i) of § A-1-c in Chapter XV); we must thus compute a product of traces in each of these spaces. For a fixed k, we sum all the diagonal elements over all the values of nk, then take the product over all k’s, which leads to:

(7)