Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

else⁶, taking into account (A-55):

      (A-61)

      We have thus shown that the operators .. act on the vacuum state in the same way as the operators defined by (A-51), raised to the powers ps, pt, ..

      When the occupation numbers ps, pt, .. can take on any values, the kets (A-56) span the entire Fock space. Writing the previous equality for ps and ps + 1, we see that the action on all the basis kets of and of yields the same result, establishing the equality between these two operators. Relation (A-52) can be readily obtained by Hermitian conjugation.

      (ii) Fermions

      The demonstration is identical, with the constraint that the occupation numbers are 0 or 1 . As this requires no changes in the operator or state order, it involves no sign changes.

B. One-particle symmetric operators

      Using creation and annihilation operators makes it much easier to deal, in the Fock space, with physical operators that are thus symmetric (§ C-4-a-β of Chapter XIV). We first study the simplest of such operators, those which act on a single particle and are called “one-particle operators”.

B-1. Definition

      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.

B-2. Expression in terms of the operators a and a^†

      We choose a basis {|ui〉} for the individual states. The matrix elements fkl of the one-particle operator are given by:

      (B-3)

      They can be used to expand the operator itself as follows:

(B-4)

      B-2-a. Action of F^(N) on a ket with N particles

Using in (B-1) the expression (B-4) for leads to:

(B-5)

The action of on a symmetrized ket written as (A-9) therefore includes a sum over k and l of terms:

      (B-6)

      with coefficients fkl. Let us use (A-7) or (A-10) to compute this ket for given values of k and l. As the operator contained in the bracket is symmetric with respect to the exchange of particles, it commutes with the two operators SN and AN (§ C-4-a-β of Chapter XIV)), and the ket can be written as:

      (B-7)

In the summation over q, the only non-zero terms are those for which the individual state |ul〉 coincides with the individual state |um〉 occupied in the ket on the right by the particle labeled q; there are nl different values of q that obey this condition (i.e. none or one for fermions). For these nl terms, the operator |q : |uk〉 〈q : ul| transforms the state |um〉 into |ui〉, then SN (or AN) reconstructs a symmetrized (but not normalized) ket:

(B-8)