Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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rel="nofollow" href="#ulink_6feccab3-a261-55ee-997b-eb502b191fa5">(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 taking twice in a row the Hermitian conjugate of an operator brings you back to the initial operator.

      A-3-a. Bosons

      For bosons, we deduce from (A-16) that the only non-zero matrix elements of

in the Fock states orthonormal basis are:

(A-20)

      They link two vectors having equal occupation numbers except for ni, which increases by one going from the ket to the bra.

The matrix elements of the Hermitian conjugate of

are obtained from relation (A-20), using the general definition (B-49) of Chapter II. The only non-zero matrix elements of aui are thus:

      (A-21)

      Since the basis we use is complete, we can deduce the action of the operators ai on kets having given occupation numbers:

(A-22)

      (note that we have replaced ni by ni — 1). As opposed to

, which adds a particle in the state |ui〉, the operator aui takes one away; it yields zero when applied on a ket where the state |ui〉 is empty to begin with, such as the vacuum state:

(A-23)

      We call aui “the annihilation operator” for the state |ui〉.

      A-3-b. Fermions

      For fermions, relation (A-18) allows writing the matrix elements:

      (A-24)

      The only non-zero elements are those where all the individual occupied states are left unchanged in the bra and the ket, except for the state ui only present in the bra, but not in the ket. As for the occupation numbers, none change, except for ni which goes from 0 (in the ket) to 1 (in the bra).

      The Hermitian conjugation operation then yields the action of the corresponding annihilation operator:

(A-25)

      or, if initially the state |ui〉 is not occupied:

(A-26)

Relations (A-22) and (A-23) are also valid for fermions, with the usual condition that all occupation numbers should be equal to 0 or 1 ; otherwise, the relations amount to 0 = 0.

       Comment:

To use relation (A-25) when the state |ui〉 is already occupied but not listed in the first position, we first have to bring it there; if it requires an odd permutation, a change of sign will occur. For example:

      (A-27)

      For fermions, the operators a and a^† therefore act on the individual state that is listed in the first position in the N-particle ket; a destroys the first state in the list, and at creates a new state placed at the beginning of the list. Forgetting this could lead to errors in sign.

A-4. Occupation number operators (bosons and fermions)

      Consider the operator defined by:

(A-28)

and its action on a Fock state. For bosons, if we apply successively formulas (A-22) and (A-16), we see that this operator yields the same Fock state, but multiplied by its occupation number ni. For fermions, if |ui〉 is empty in the Fock state, relation (A-26) shows that the action of the operator yields

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