Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 62

As two fermions cannot occupy the same quantum state, the product is zero if i = j; we therefore assume i ≠ j which allows, using for ρeq expression (5) (which is a product), to perform independent calculations for the different modes. The case i = l and j = k yields, using the anticommutation relations:

(30)

and the case i = k and j = l yields:

(31)

We begin with term (30). As i and j are different, operators and act on different modes, which belong to different factors in the density operator (5). The average value of the product is thus simply the product of the average values:

(32)

(33)

As for the second term (31), it is just the opposite of the first one. Consequently, we finally get:

(34)

The first term on the right-hand side is called the direct term. The second one is the exchange term, and has a minus sign, as expected for fermions.