Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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the second equality is valid in the limit of large volumes). Simplifying by , we get the pressure of a fermion system contained in a box of macroscopic dimension:

(69)

      with:

      (70)

where x has been defined in (53).

To obtain the equation of state, we must find a relation between the pressure P, the volume , and the temperature T of the physical system, assuming the particle number to be fixed. We have, however, used the grand canonical ensemble (cf. Appendix VI), where the temperature is determined by the parameter β and the volume is fixed, but where the particle number can vary: its average value is a function of a parameter, the chemical potential μ (for fixed values of μ and ). Mathematically, the pressure P appears as a function of , T and μ and not as the function of , T and the particle number we were looking for. We can nevertheless vary μ, and obtain values of the pressure and particle number of the system and consequently explore, point by point, the equation of state in this parametric form. To obtain an explicit form of the equation of state would require the elimination of the chemical potential using both (47) and (69); there is generally no algebraic solution, and people just use the parametric form of the equation of state, which allows computing all the possible state variables. There also exists a “virial expansion” in powers of the fugacity eβμ, which allows the explicit elimination of μ at all the successive orders; its description is beyond the scope of this book.

5-b. Bosons

The pressure of an ideal boson gas is derived from the grand potential (12), taking into account its relation (14) to the pressure P and volume :

(71)

      (the second relation being valid in the limit of large volumes). This leads to:

(72)

      with:

      (73)

As μ → 0, the contribution P₀ of the ground level to the pressure written in (71) is:

      (74)

When the chemical potential tends towards zero as in (66), it leads to:

      (75)

which therefore goes to zero in the limit of large volumes. For a large system, the ground level contribution to the pressure remains negligible compared to that of all the other individual energy levels, whose number gets bigger as the system gets larger. Contrary to what we encountered for the average total particle number, the condensed particles’ contribution to the pressure goes to zero in the limit of large volumes.

As we have seen for fermions, the equation of state must be obtained by eliminating the chemical potential μ between equations (72) yielding the pressure and (65) yielding the total particle number. As opposed to an ideal fermion gas, whose particle number and pressure increase without limit as μ and the density increase, the pressure in a boson system is limited. As soon as the system condenses, only the particle number in the individual ground state continues to grow, but not the pressure. In other words, the physical system acquires an infinite compressibility, and becomes a “marginally pathological” system (a system whose pressure decreases with its volume is unstable). This pathology comes, however, from totally neglecting the bosons’ interactions. As soon as repulsive interactions are introduced, no matter how small, the compressibility will take on a finite value and the pathology will disappear.

      This complement is a nice illustration of the simplifications incurred by the systematic use, in the calculations, of the creation and annihilation operators. We shall see in the following complements that these simplifications still occur when taking into account the interactions, provided we stay in the framework of the mean field approximation. Complement B_XVI will even show that for an interacting system studied without using this approximation, the ideal gas distribution functions are still somewhat useful for expressing the average values of various physical quantities.

      1 ¹ A physical observable is said to have a well defined value in a given quantum state if, in this state, its root mean square deviation is small compared to its average value.

      2 ² The subscript 3/2 refers to the subscript used for more general functions fm (z), often called the Fermi functions in physics. They are defined by , where z is the “fugacity” z = eβμ. Expanding in terms of z the function 1/[1 + z–1 ex] = ze–x/[1 + ze–x] and using the properties of the Euler Gamma function, it can be shown that I3/2(βμ) = f3/2(z).

      3 ³ Defining other boundary conditions on the box walls will lead in general to a non-zero ground state energy; choosing that value as the common origin for the energies and the chemical potential will leave the following computations unchanged.

      4 ⁴ The subscript 3/2 refers to the subscript used for the functions gm(z), often called, in physics, the Bose functions (or the polylogarithmic functions). They are defined by the series . The exact value of the number ζ defined in (61) is thus given by the series .

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