Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 128

Using relation (B-12) of Chapter XV, we obtain for the average kinetic energy 〈Ĥ₀〉:

(51)

The same argument as that for the evaluation of the matrix elements (49) shows that the average value in the state (47) is only different from zero if r = s; in that case, it is equal to one when r ≤ N, and to zero otherwise. This leads to:

(52)

The subscript 1 was added to the trace to underline the fact that this trace is taken in the one-particle state space and not in the Fock space. The two operators included in the trace only act on that same particle, numbered arbitrarily 1; the subscript 1 could obviously be replaced by the subscript of any other particle, since they all play the same role. The average potential energy coming from the external potential is computed in a similar way and can be written as:

(53)