Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3

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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 zero. If the state |ui〉 is already occupied, we must first permute the states to bring |ui〉 to the first position, which may eventually change the sign in front of the Fock space ket. The successive application on this ket of (A-25) and (A-19) shows that the action of the operator leaves this ket unchanged; we then move the state |ui〉 back to its initial position, which may introduce a second change in sign, canceling the first one. We finally obtain for fermions the same result as for bosons, except that the ni can only take the values 1 and 0. In both cases the Fock states are the eigenvectors of the operator with the occupation numbers as eigenvalues; consequently, this operator is named the “occupation number operator of the state |ui〉”. The operator associated with the total number of particles is simply the sum:

(A-29)

A-5. Commutation and anticommutation relations

Creation and annihilation operators have very simple commutation (for the bosons) and anticommutation (for the fermions) properties, which make them easy tools for taking into account the symmetrization or antisymmetrization of the state vectors.

To simplify the notation, each time the equations refer to a single basis of individual states |ui〉, we shall write ai instead of aui. If, however,

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