Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 52

Using relations (42) and (43) of Appendix VI, we can write the grand canonical density operator ρeq (whose trace has been normalized to 1) as:

(1)

where Z is the grand canonical partition function:

(2)

In these relations, β = 1/(kBT) is the inverse of the absolute temperature T multiplied by the Boltzmann constant kB, and μ, the chemical potential (which may be fixed by a large reservoir of particles). Operators Ĥ and are, respectively, the system Hamiltonian and the particle number operator defined by (B-15) in Chapter XV.

Assuming the particles do not interact, equation (B-1) of Chapter XV allows writing the system Hamiltonian Ĥ as a sum of one-particle operators, in each subspace having a total number of particles equal to N:

(3)

Let us call {|uk〉} the basis of the individual states that are the eigenstates of the operator . Noting and ak the creation and annihilation operators of a particle in these states, Ĥ may be written as in (B-14):

(4)

where the ek, are the eigenvalues of . Operator (1) can also be written as:

(5)

We shall now compute the average values of all the one- or two-particle operators for a system described by the density operator (1).