Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 32

Consider an operator defined in the space of individual states; acts in the state space of particle q. It could be for example the momentum of the q-th particle, or its angular momentum with respect to the origin. We now build the operator associated with the total momentum of the N-particle system, or its total angular momentum, which is the sum over q of all the associated with the individual particles.

A one-particle symmetric operator acting in the space S(N) for bosons - or A(N) for fermions - is therefore defined by:

(B-1)

(contrary to states, which are symmetric for bosons and antisymmetric for fermions, the physical operators are always symmetric). The operator acting in the Fock space is defined as the operator acting either in S(N) or in A(N), depending on the specific case. Since the basis for the entire Fock space is the union of the bases of these spaces for all values of N, the operator is thus well defined in the direct sum of all these subspaces. To summarize:

(B-2)

Using (B-1) directly to compute the matrix elements of often leads to tedious manipulations. Starting with an operator involving numbered particles, we place it between states with numbered particles; we then symmetrize the bra, the ket, and take into account the symmetry of the operator (cf. footnote 1). This introduces several summations (on the particles and on the permutations) that have to be properly regrouped to be simplified. We will now show that expressing in terms of creation and annihilation operators avoids all these intermediate calculations, taking nevertheless into account all the symmetry properties.