Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 154

We assume the Hamiltonian is of the form:

(1)

which is the sum of the particles’ kinetic energy Ĥ₀, their coupling energy with an external potential:

(2)

and their mutual interaction , which can be expressed as:

(3)

We are going to use the “grand canonical” ensemble (Appendix VI, § 1-c), where the particle number is not fixed, but takes on an average value determined by the chemical potential μ. In this case, the density operator ρ is an operator acting in the entire Fock space εF (where N can take on all the possible values), and not only in the state space εN for N particles (which is is more restricted since it corresponds to a fixed value of N). We set, as usual:

(4)

where kB is the Boltzmann constant and T the absolute temperature. At the grand canonical equilibrium, the system density operator depends on two parameters, β and the chemical potential μ, and can be written as:

(5)

with the relation that comes from normalizing to 1 the trace of ρ_eq:

(6)

The function Z is called the “grand canonical partition function” (see Appendix VI, § 1-c). The operator associated with the total particle number is defined in (B-17) of Chapter XV. The temperature T and the chemical potential μ are two intensive quantities, respectively conjugate to the energy and the particle number.

Because of the particle interactions, these formulas generally lead to calculations too complex to be carried to completion. We therefore look, in this complement, for approximate expressions of ρ_eq and Z that are easier to use and are based on the mean field approximation.