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Physics LibreTexts

8.4: Bose-Einstein and Fermi-Dirac Statistics

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Finally, in this section we will derive the Bose-Einstein and Fermi-Dirac statistics. In particular, we are interested in the thermal equilibrium for a large number of (non-interacting) identical particles with some energy spectrum Ej, which my be continuous.

Since the number of particles is not fixed, we are dealing with the Grand Canonical Ensemble. Its partition function Ξ is given by

Ξ=Tr[eμβˆnβH],

where H is the many-body Hamiltonian, β=1/kBT and μ is the chemical potential. The average number of particles with single particle energy Ej is then given by

nj=1βlnΞEj.

For the simple case where H=jEjˆnj and the creation and annihilation operators obey the commutator algebra, the exponent can be written as

exp[βj(μEj)ˆajˆaj]=jnj=0eβ(μEj)nj|njnj|,

and the trace becomes

Ξ=j11eβ(μEj).

The average photon number for energy Ej is

nj=1βlnΞEj=1βΞΞEj=1eβ(μEj)1.

This is the Bose-Einstein distribution for particles with energy Ej. It is shown for increasing Ej in Fig. 4 on the left.

Screen Shot 2021-11-26 at 4.39.19 PM.png
Figure 4: Left: Bose-Einstein distribution for different temperatures (μ=0). The lower the temperature, the more particles occupy the low energy states. Right: Fermi-Dirac distribution for different temperatures and μ=1. The fermions will not occupy energy states with numbers higher than 1, and therefore higher energies are necessarily populated. The energy values Ej form a continuum on the horizontal axis.

Alternatively, if the creation and annihilation operators obey the anti-commutation relations, the sum over nj in Eq. (8.45) runs not from 0 to ∞, but over 0 and 1. The partition function of the grand canonical ensemble then becomes

Ξ=j[1+eβ(μEj)],

and the average number of particles with energy Ej becomes

nj=1βΞΞωj=1eβ(μEj)+1.

This is the Fermi-Dirac statistics for these particles, and it is shown in Fig. 4 on the right. The chemical potential is the highest occupied energy at zero temperature, and in solid state physics this is called the Fermi level. Note the sign difference in the denominator with respect to the Bose-Einstein statistics.

Exercises

  1. Calculate the Slater determinant for three electrons and show that no two electrons can be in the same state.
  2. Particle statistics.
    1. What is the probability of finding n bosons with energy Ej in a thermal state?
    2. What is the probability of finding n fermions with energy Ej in a thermal state?
  3. Consider a system of (non-interacting) identical bosons with a discrete energy spectrum and a ground state energy E0. Furthermore, the chemical potential starts out lower than the ground state energy μ<E0.
    1. Calculate n0 and increase the chemical potential to μE0 (e.g., by lowering the temperature). What happens when μ passes E0?
    2. What is the behaviour of nthermal j=1nj as μE0? Sketch both n0 and nthermal  as a function of μ. What is the fraction of particles in the ground state at μ=E0?
    3. What physical process does this describe?
  4. The process U=exp(rˆa1ˆa2rˆa1ˆa2) with rC creates particles in two systems, 1 and 2, when applied to the vacuum state |Ψ=U|.
    1. Show that the bosonic operators ˆa1ˆa2 and ˆa1ˆa2 obey the algebra

      [K,K+]=2K0 and [K0,K±]=±K±,

      with K+=K.

    2. For operators obeying the algebra in (a) we can write

      erK+rK=exp[r|r|tanh|r|K+]exp[2ln(cosh|r|)K0]×exp[r|r|tanh|r|K].

      Calculate the state |Ψ of the two systems.

    3. The amount of entanglement between two systems can be measured by the entropy S(r) of the reduced density matrix ρ1=Tr1[ρ] for one of the systems. Calculate S(r)=Tr[ρ1lnρ1].
    4. What is the probability of finding n particles in system 1?

This page titled 8.4: Bose-Einstein and Fermi-Dirac Statistics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform.

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