EF whereas the others occupy the “conduction band”, separated from the previous band by an “energy gap”. Sending a photon with an energy larger than this gap allows the creation of an electron-hole pair, easily studied in the formalism we just introduced.
A somewhat similar case occurs when studying the relativistic Dirac wave equation, where two energy continuums appear: one with energies greater than the electron rest energy mc2 (where m is the electron mass, and c the speed of light), and one for negative energies less than —mc2 associated with the positron (the antiparticle of the electron, having the opposite charge). The energy spectrum is relativistic, and thus different from formula (4), even inside each of those two continuums. However, the general formalism remains valid, the operators
(ii) An arbitrary N-particle Fock state |Φ〉 does not have to be the ground state to be formally considered as a “quasi-particle vacuum”. We just have to consider any annihilation operator on an already occupied individual state as a creation operator of a hole (i.e. of an excitation); we then define the corresponding hole (or excitation) annihilation operators, which all have in common the eigenvector |Φ〉 with eigenvalue zero. This comment will be useful when studying the Wick theorem (Complement CXVI). In § E of Chapter XVII, we shall see another example of a quasi-particle vacuum, but where, this time, the new annihilation operators are no longer acting on individual states but on states of pairs of particles.
1 1 In Complement CXIV we had assumed that both spin states of the electron gas were occupied, whereas this is not the case here. This explains why the bracket in formula (1) contains the coefficient 6π2N instead of 3π2N.
Complement BXV Ideal gas in thermal equilibrium; quantum distribution functions
1 1 Grand canonical description of a system without interactions 1-a Density operator 1-b Grand canonical partition function, grand potential
2 2 Average values of symmetric one-particle operators 2-a Fermion distribution function 2-b Boson distribution function 2-c Common expression 2-d Characteristics of Fermi-Dirac and Bose-Einstein distributions
3 3 Two-particle operators 3-a Fermions 3-b Bosons 3-c Common expression
4 4 Total number of particles 4-a Fermions 4-b Bosons
5 5 Equation of state, pressure 5-a Fermions 5-b Bosons
This complement studies the average values of one- or two-particle operators for an ideal gas, in thermal equilibrium. It includes a discussion of several useful properties of the Fermi-Dirac and Bose-Einstein distribution functions, already introduced in Chapter XIV.
To describe thermal equilibrium, statistical mechanics often uses the grand canonical ensemble, where the particle number may fluctuate, with an average value fixed by the chemical potential μ (cf. Appendix VI, where you will find a number of useful concepts for reading this complement). This potential plays, with respect to the particle number, a role similar to the role the inverse of the temperature term β = 1/kBT plays with respect to the energy (kB is the Boltzmann constant). In quantum statistical mechanics, Fock space is a good choice for the grand canonical ensemble as it easily allows changing the total number of particles. As a direct application of the results of §§ B and C of Chapter XV, we shall compute the average values of symmetric one- or two-particle operators for a system of identical particles in thermal equilibrium.
We begin in § 1 with the density operator for non-interacting particles, and then show in §§ 2 and 3 that the average values of the symmetric operators may be expressed in terms of the Fermi-Dirac and Bose-Einstein distribution functions, increasing their application range and hence their importance. In § 5, we shall study the equation of state for an ideal gas of fermions or bosons at temperature T and contained in a volume
1. Grand canonical description of a system without interactions
We first recall how a system of non-interacting particles is described, in quantum statistical mechanics, by the grand canonical ensemble; more details on this subject can be found in Appendix VI, § 1-c.
1-a. Density operator
Using relations (42) and (43) of Appendix VI, we can write the grand canonical density operator ρeq (whose trace has been normalized to 1) as:
where Z is the grand canonical partition function:
In