8.4: Bose-Einstein and Fermi-Dirac Statistics
( \newcommand{\kernel}{\mathrm{null}\,}\)
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[eμβˆn−βH],
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⟩=−1β∂lnΞ∂Ej.
For the simple case where H=∑jEjˆnj and the creation and annihilation operators obey the commutator algebra, the exponent can be written as
exp[β∑j(μ−Ej)ˆa†jˆaj]=⨂j∞∑nj=0eβ(μ−Ej)nj|nj⟩⟨nj|,
and the trace becomes
Ξ=∏j11−eβ(μ−Ej).
The average photon number for energy Ej is
⟨nj⟩=−1β∂lnΞ∂Ej=−1βΞ∂Ξ∂Ej=1e−β(μ−Ej)−1.
This is the Bose-Einstein distribution for particles with energy Ej. It is shown for increasing Ej in Fig. 4 on the left.

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(μ−Ej)],
and the average number of particles with energy Ej becomes
⟨nj⟩=−1βΞ∂Ξ∂ℏωj=1e−β(μ−Ej)+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.
- Calculate the Slater determinant for three electrons and show that no two electrons can be in the same state.
- Particle statistics.
- What is the probability of finding n bosons with energy Ej in a thermal state?
- What is the probability of finding n fermions with energy Ej in a thermal state?
- 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.
- Calculate ⟨n0⟩ and increase the chemical potential to μ→E0 (e.g., by lowering the temperature). What happens when μ passes E0?
- What is the behaviour of ⟨nthermal ⟩≡∑∞j=1⟨nj⟩ as μ→E0? Sketch both ⟨n0⟩ and ⟨nthermal ⟩ as a function of μ. What is the fraction of particles in the ground state at μ=E0?
- What physical process does this describe?
- The process U=exp(rˆa†1ˆa†2−r∗ˆa1ˆa2) with r∈C creates particles in two systems, 1 and 2, when applied to the vacuum state |Ψ⟩=U|∅⟩.
- Show that the bosonic operators ˆa†1ˆa†2 and ˆa1ˆa2 obey the algebra
[K−,K+]=2K0 and [K0,K±]=±K±,
with K+=K†−.
- For operators obeying the algebra in (a) we can write
erK+−r∗K−=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.
- 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].
- What is the probability of finding n particles in system 1?
- Show that the bosonic operators ˆa†1ˆa†2 and ˆa1ˆa2 obey the algebra