Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 25

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)