Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji
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:
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:
This result proves that (35) remains valid even in the case i = j.
β. Physical discussion: occupation number fluctuations
For two different physical states i and j, the average value
Now if i = j, we note the factor 2 in relation (41). As we now show, this factor leads to the presence of strong fluctuations associated with the operator
The square of the root mean square deviation Δni, is therefore given by:
(42b)
The fluctuations of this operator are therefore larger than its average value, which implies that the population of each state |ui〉 is necessarily poorly defined1 at thermal equilibrium. This is particularly true for large
3-c. Common expression
To summarize, we can write in all cases:
with:
(44)
As shown in relation (C-19) of Chapter XV, this average value is simply the matrix element 〈1 : uk; 2 : ul
Complement CXVI will show how the Wick theorem allows generalizing these results to operators dealing with any number of particles.
4. Total number of particles
The operator
(45)
and its average value is given by:
As fβ increases as a function of μ, the total number of particles is controlled (for fixed β) by the chemical potential.
4-a. Fermions
For the sake of simplicity, we study the ideal gas properties without taking into account the spin, which assumes that all particles are in the same spin state (the spin can easily be accounted for by adding the contributions of the different individual spin states). For a large physical system, the energy levels are very close and the discrete sum in (46) can be replaced by an integral. This leads to: