Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

      (the sums go to infinity). In both cases, we have included on the right-hand side a first term associated with a total number of particles equal to zero. The corresponding space, S,A(0), is defined as a one-dimensional space, containing a single state called “vacuum” and denoted |0〉 or |vac〉. For bosons as well as fermions, an orthonormal basis for the Fock space can be built with the Fock states |n₁, n₂, …, ni, nl..〉, relaxing the constraint (A-4): the occupation numbers may then take on any (integer) values, including zeros for all, which corresponds to the vacuum ket |0〉. Linear combinations of all these basis vectors yield all the vectors of the Fock space, including linear superpositions of kets containing different particle numbers. It is not essential to attribute a physical interpretation to such superpositions since they can be considered as intermediate states of the calculation. Obviously, the Fock space contains many kets with well defined particle numbers: all those belonging to a single sub-space S(N) for bosons, or A(N) for fermions. Two kets having different particle numbers N are necessarily orthogonal; for example, all the kets having a non-zero total population are orthogonal to the vacuum state.

       Comments:

(i) Contrary to the distinguishable particle case, the Fock space is not the tensor product of the spaces of states associated with particles numbered 1, 2,…, q, etc. First of all, for a fixed N, it only includes the totally symmetric (or antisymmetric) subspace of this tensor product; furthermore, the Fock space is the direct sum of such subspaces associated with each value of the particle number N.

      The Fock space is, however, the tensor product of Fock spaces

associated with the individual orthogonal states |ui〉, each

being spanned by the kets |ni〉 where ni takes on all integer values (from zero to infinity for bosons, from zero to one for fermions):

      (A-14)

      This is because the Fock states, which are a basis for

, may be written as the tensor product:

      (A-15)

      It is often said that each individual state defines a “mode” of the system of identical particles. Decomposing the Fock state into a tensor product allows considering the modes as describing different and distinguishable variables. This will be useful on numerous occasions (see for example Complements B_XV, D_XV and E_XV).

      (ii) One should not confuse a Fock state with an arbitrary state of the Fock space. The occupation numbers of individual states are all well defined in a Fock state (also called “number state”), whereas an arbitrary state of the Fock space is a linear superposition of these eigenstates, with several non-zero coefficients.

A-2. Creation operators a^†

      Choosing a basis of individual states {|ui〉}, we now define the action in the Fock space of the creation operator⁵

on a particle in the state |ui〉.

      A-2-a. Bosons

      For bosons, we introduce the linear operator

defined by:

(A-16)

      As all the states of the Fock space may be obtained by a linear superposition of |n₁, n₂, .., ni, ..〉, the action of

is defined in the entire space. It adds a particle to the system, which goes from a state of S(N) to a state of S(N + 1), and in particular from the vacuum to a state having one single occupied state.

Creation operators acting on the vacuum allow building occupied states. Recurrent application of (A-16), leads to:

(A-17)

       Comment:

      Why was the factor

introduced in (A-16)? We shall see later (§ B) that, together with the factors of (A-7), it simplifies the computations.

      A-2-b. Fermions

      For fermions, we define the operator

by:

(A-18)

      where the newly created state |ui〉 appears first in the list of states in the ket on the right-hand side. If we start from a ket where the individual state |ui〉 is already occupied (ni = 1), the action of

leads to zero, as in this case (A-10) gives:

(A-19)

Formulas (A-16) and (A-17) are also valid for fermions, with all the occupation numbers equal to 0 or 1 (or else both members are zero).

       Comment:

Definition (A-18) must not depend on the specific order of the individual states uj, .., uk, .., ul, .. in the ket on which the operator

acts. It can be easily verified that any permutation of the states simply multiply by its parity both members of the equality. It therefore remains valid independently of the order chosen for the individual states in the initial ket.

A-3. Annihilation operators a

      We now study the Hermitian conjugate operator of

, that we shall simply call aui since