Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 17

In the case of fermions, the operator AN acting on a ket where two (or more) numbered particles are in the same individual state yields a zero result: there are no such states in the physical space A(N). Hence we concentrate on the case where all the occupation numbers are either 1 or 0. We denote |ui〉, |uj〉,..,|ul〉,.. all the states having an occupation number equal to 1. The equivalent for fermions of formula (A-7) is written:

(A-10)

Taking into account the 1/N factor appearing in definition (A-3) of AN the right-hand side of this equation is a linear superposition, with coefficients , of N! kets which are all orthogonal to each other (as we have chosen an orthonormal basis for the individual states {|uk〉}); hence its norm is equal to 1. Consequently, Fock states for fermions are defined by (A-10). Contrary to bosons, the main concern is no longer how many particles occupy a state, but whether a state is occupied or not. Another difference with the boson case is that, for fermions, the order of the states matters. If for instance the first two states ui and uj are exchanged, we get the opposite ket:

(A-11)

but it obviously does not change the physical meaning of the ket.