Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 39

Consider a physical quantity involving two particles, labeled q and q′. It is associated with an operator acting in the state space of these two particles (the tensor product of the two individual state’s spaces). Starting from this binary operator, the easiest way to obtain a symmetric N-particle operator is to sum all the over all the particles q and q′, where the two subscripts q and q′ range from 1 to N. Note, however, that in this sum all the terms where q = q′ add up to form a one-particle operator of exactly the same type as those studied in § B-1. Consequently, to obtain a real two-particle operator we shall exclude the terms where q = q′ and define:

(C-1)

The factor 1/2 present in this expression is arbitrary but often handy. If for example the operator describes an interaction energy that is the sum of the contributions of all the distinct pairs of particles, and corresponding to the same pair are equal and appear twice in the sum over q and q′: the factor 1/2 avoids counting them twice. Whenever , it is equivalent to write in the form:

(C-2)

As with the one-particle operators, expression (C-1) defines symmetric operators separately in each physical state’s space having a given particle number N. This definition may be extended to the entire Fock space, which is their direct sum over all N. This results in a more general operator , following the same scheme as for (B-2):

(C-3)