Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 70

When the parameter μ takes on a sufficiently negative value (much lower than the opposite of the individual energy e₁ of the first excited level), the function in the summation (55) is sufficiently regular for the discrete summation to be replaced by an integral (in the limit of large volumes). The average particle number is then written as:

(57)

with:

(58)

Performing the same change of variables as above, this expression becomes:

(59)

with⁴:

(60)

The variations of as a function of μ are shown in Figure 3. Note that the total particle number tends towards a limit as tends towards zero through negative values, where ζ is the number:

(61)

As the function increases with μ, we can write:

(62)

There exists an insurmountable upper limit for the total particle number of a non-condensed ideal Bose gas.

Figure 3: Variations of the total particle number in a non-condensed ideal Bose gas, as a function of μ and for fixed β = 1/(kBT). The chemical potential is always negative, and the figure shows curves corresponding to several temperatures T = T₁ (thick line), T = 2 T₁ and T = 3T₁. Units on the axes are the same as in Figure 2: the thermal energy kBT₁ associated with curve T = T₁, and the particle number , where λ_T1 is the thermal wavelength for this same temperature T₁. As the chemical potential tends towards zero, the particle numbers tend towards a finite value. For T = T₁, this value is equal to ζN1 (shown as a dot on the vertical axis), where ζ is given by (61). This figure was kindly contributed by Geneviève Tastevin