Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 56

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 i = j, this expression yields the average particle number in the individual state |ui〉.