Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 24

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.