Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 23

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〉.