Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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1-b. Grand canonical partition function, grand potential

      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)

       α. Fermions

      For fermions, as nk can only take the values 0 or 1 (two identical fermions never occupy the same individual state), we get:

      (8)

      and:

(9)

The index k must be summed over all the individual states. In case these states are also labeled by orbital and spin subscripts, these must also be included in the summation. Let us consider for example particles having a spin S and contained in a box of volume with periodic boundary conditions. The individual stationary states may be written as |k, ν〉, where k obeys the periodic boundary conditions (Complement C_XIV, § 1-c) and the subscript ν takes (2S + 1) values. Assuming the particles to be free in the box (no spin Hamiltonian), each ν value yields the same contribution to Φ_fermions; in the large volume limit, expression (9) then becomes:

      (10)

       β. Bosons

For bosons, the summation over nk in (7) goes from nk = 0 to infinity, which introduces a geometric series whose sum is readily computed. We therefore get:

(11)

which leads to:

(12)

      For a system of free particles with spin S, confined in a box with periodic boundary conditions, we obtain, in the large volume limit:

      (13)

In a general way, for fermions as well as bosons, the grand potential directly yields the pressure P, as shown in relation (61) of Appendix VI:

(14)

      Using the proper derivatives with respect to the equilibrium parameters (temperature, chemical potential, volume), it also yields the other thermodynamic quantities such as the energy, the specific heats, etc.

2. Average values of symmetric one-particle operators

      Symmetric quantum operators for one, and then for two particles, were introduced in a general way in Chapter XV (§§ B and C). The general expression for a one-particle operator is given by equation (B-12) of that chapter. We can thus write:

(15)

      with, when the state of the system is given by the density operator (1):

(16)

This trace can be computed in the Fock state basis |n₁, ..,ni..,nj,..〉 associated with the eigenstates basis {|uk〉} of . If i ≠ j, operator destroys a particle in the individual state |uj〉 and creates another one in the different state |ui〉; it therefore transforms the Fock state |n₁, ..,ni,.., nj,..〉 into a different, hence orthogonal, Fock state |n₁,.., ni – 1.., nj + 1,..〉. Operator ρeq then acts on this ket, multiplying it by a constant. Consequently, if i ≠ j, all the diagonal elements of the operator whose trace is taken in (16) are zero; the trace is therefore zero. If i = j, this average value may be computed as for the partition function, since the Fock space has the structure of a tensor product of individual state’s spaces. The trace is the product of the i value contribution by all the other k values contributions. We can thus write, in a general way:

(17)

      For

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