Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 41

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.