Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 40

Let us first assume the operator can be factored as:

(C-4)

The operator written in (C-1) then becomes:

(C-5)

The right-hand side of this expression starts with a product of one-particle operators, each of which can be replaced, following (B-11), by its expression as a function of the creation and annihilation operators:

(C-6)

As for the last term on the right-hand side of (C-5), it is already a single particle operator:

(C-7)

This 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.