Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

leads to:

(C-8)

We can then use general relations (A-49) to transform the operator product:

      (C-9)

Including this form in the first term on the right-hand side of (C-8) yields, for the δjk contribution:

      (C-10)

      which exactly cancels the second term of (C-8). Consequently, we are left with:

(C-11)

      As the right-hand side of this expression has the same form in all spaces having a fixed N, it is also valid for the operator acting in the entire Fock space.

C-3. General case

      Any two-particle operator may be decomposed as a sum of products of single particle operators:

(C-12)

where the coefficients cα, β are numbers⁷. Hence expression (C-1) can be written as:

      (C-13)

In this linear combination with coefficients cα, β, each term (corresponding to a given α and β) is of the form (C-5) and can therefore be replaced by expression (C-11). This leads to:

      (C-14)

The right-hand side of this equation has the same form in all the spaces of fixed N; hence it is valid in the entire Fock space. Furthermore, we recognize in the summation over α and β the matrix element of as defined by (C-12):

(C-15)

      The final result is then:

(C-16)

      which is the general expression for a two-particle symmetric operator.

As for the one-particle operators, each term of expression (C-16) for the two-particle operators contains equal numbers of creation and annihilation operators. Consequently, these symmetric operators do not change the total number of particles, as was obvious from their initial definition.

C-4. Two-particle reduced density operator

      Relation (C-16) implies that the average value of any two-particle operator may be written as:

(C-17)

Figure 1: Physical interaction between two identical particles: initially in the states |ukα〉 and |ukβ〉 (schematized by the letters α and β), the particles are transferred to the states |ukγ〉 and |uks〉 (schematized by the letters γ and δ)

      This expression is similar to the average value of an operator for a two-particle system having a density operator :

      (C-18)

      which leads us to define a two-particle reduced density operator :

      (C-19)

In this definition we have left out the factor 1/2 of (C-17) since this will lead to a normalization of often more handy: the matrix element of in the position representation yields directly the double density (as well as the field correlation function that we shall study in § B-3-b of Chapter XVI). The trace of is then written:

      (C-20)

      It is obviously possible to divide the right-hand side of the definition of either by the factor 2, or else by the factor if we wish its trace to be equal to 1.

C-5. Physical discussion; consequences of the exchange

      As mentioned in the introduction of this chapter, the equations no longer contain labeled particles, permutations, symmetrizers and antisymmetrizers; the total number of particles N has also disappeared. We may now continue the discussion begun in § D-2 of Chapter XIV concerning the exchange terms, but in a more