Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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target="_blank" rel="nofollow" href="#ulink_a8d11189-5151-51bc-bce1-275865b30e7f">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

       β. Condensed bosons

      As μ gets closer to zero, the population N₀ of the ground state becomes:

(63)

This population diverges in the limit μ = 0 and, when |μ| gets small enough, it can become arbitrarily large. It can, for example, become proportional⁵ to the volume , in which case it adds a finite contribution to the particle numerical density (particle number per unit volume) as .

This particularity is limited to the ground state, which, in this case, plays a very different role than the other levels. Let us show, for example, that the first excited state population does not yield a similar effect. Assuming the system to be contained in a cubic box⁶ of edge length L, the population of the first excited energy level e₁ ~ π²ħ²/(2mL² ) can be written as:

      (64)

      (we assume the box to be large enough so that L ≫ λ_T, which means βe₁ ≪ 1); this population can therefore be proportional only to the square of L, i.e. to the volume to the power 2/3. It shows that this first excited level cannot make a contribution to the particle density in the limit L → ∞; the same is true for all the other excited levels whose contributions are even smaller. The only arbitrary contribution to the density comes from the ground state.

This arbitrarily large value as μ → 0 obviously does not appear in relation (59), which predicts that the density is always less than a finite value as shown by (62). This is not surprising: as the population varies radically from the first energy level to the next, we can no longer compute the average particle number by replacing in (55) the discrete summation by an integral and a more precise calculation is necessary. Actually, only the ground state population must be treated separately, and the summation over all the excited states (of which none contributes to the density divergence) can still be replaced by an integral as before. Consequently, to get the total population of the physical system we simply add the integral on the right-hand side of (57) to the contribution N₀ of the ground level:

(65)

where N₀ is defined in (63).

      As μ → 0, the total population of all the excited levels (others than the ground level) remains practically constant and equal to its upper limit (62); only the ground state has a continuously increasing population N₀, which becomes comparable to the total population of all the excited states when the right-hand sides of (63) and (62) are of the same order of magnitude:

(66)

(μ being of course always negative). When this condition is satisfied, a significant fraction of the particles accumulates in the individual ground level, which is said to have a “macroscopic population” (proportional to the volume). We can even encounter situations where the majority of the particles all occupy the same quantum state. This phenomenon is called “Bose-Einstein condensation” (it was predicted by Einstein in 1935, following Bose’s studies of quantum statistics applicable to photons). It occurs when the total density ntot reaches the maximum predicted by formula (62), that is:

      (67)

      This condition means that the average distance between particles is of the order of the thermal wavelength λ_T.

      Initially, Bose-Einstein condensation was considered to be a mathematical curiosity rather than an important physical phenomenon. Later on, people realized that it played an important role in superfluid liquid Helium 4, although this was a system with constantly interacting particles, hence far from an ideal gas. For a dilute gas, Bose-Einstein condensation was observed for the first time in 1995, and in a great number of later experiments.

5. Equation of state, pressure

      The “equation of state” of a fluid at thermal equilibrium is the relation that links, for a given particle number N, its pressure P, volume V, and temperature T = 1/kBβ. We have just studied the variations of the total particle number. We shall now examine the pressure of a fermion or boson ideal gas.

5-a. Fermions

The grand canonical potential of a fermion ideal gas is given by (9). Equation (14) indicates that, for a system at thermal equilibrium, this grand potential is equal to the opposite of the product of the volume and the pressure P. We thus have:

      (68)

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