Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 64

If i ≠ j, a calculation, similar to the one we just did, yields:

(35)

which differs in two ways from (34): the result now involves the Bose-Einstein distribution, and the exchange term is positive.

If i = j, only one individual state comes into a new calculation, which we now perform. Using for ρeq expression (5) we get, after summing as in (11) a geometric series:

(36)

The sum appearing in this equation can be written as:

(37)

The first order derivative term yields:

(38)

and the second order derivative term is:

(39)

Summing these two terms yields:

(40)

Multiplying by 1/[1 — e^{–β (ei – μ)}] the product at the end of the right-hand side of (36) yields the partition function Z, which cancels out the first factor 1/Z. We are then left with:

(41)

This result proves that (35) remains valid even in the case i = j.