Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji


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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)image

      The first order derivative term yields:

      (38)image

      and the second order derivative term is:

      (39)image

      Summing these two terms yields:

      (40)image

       β. Physical discussion: occupation number fluctuations

      For two different physical states i and j, the average value image for an ideal gas is simply equal to the product of the average values image and image; this is a consequence of the total absence of interaction between the particles. The same is true for the average value image.

      The square of the root mean square deviation Δni, is therefore given by:

      (42b)image

      To summarize, we can write in all cases:

      with:

      (44)image

      Complement CXVI will show how the Wick theorem allows generalizing these results to operators dealing with any number of particles.

      The operator image corresponding to the total number of particles is given by the sum over all the individual states:

      (45)image

      and its average value is given by:

      As increases as a function of μ, the total number of particles is controlled (for fixed β) by the chemical potential.


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