Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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

      with:

      (70)image

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

      with:

      (73)image

      (74)image

      (75)image

      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 BXVI 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 .

      5 

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