reduced density operator. It permits generalizing to interacting particles a number of results obtained for an ideal gas (Complement BXV).
| GXV : FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM | GXV : The trial density operator at non-zero temperature can be optimized using a variational method. This leads to self-consistent Hartree-Fock equations, of the same type as those derived in Complement EXV. We thus obtain an approximate value for the thermodynamic potential. |
| HXV : APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURES (FERMIONS AND BOSONS) | HXV: This complement discusses various applications of the method described in the previous complement: spontaneous magnetism of an ensemble of repulsive fermions, equation of state for bosons and instability in the presence of attractive interactions. |
1 1 For a one-particle symmetric operator, which includes the sum of N terms, both the ket and bra contain N! terms. The matrix element will therefore involve N(N!)2 terms, a very large number once N exceeds a few units.
2 2 A commonly accepted but a somewhat illogical expression, since no new quantification comes in addition to that of the usual postulates of Quantum Mechanics; its essential ingredient is the symmetrization of identical particles.
3 3 Remember that, by convention, 0! = 1.
4 4 The direct sum of two spaces P (with dimension P) and Q (with dimension Q ) is a space P + Q with dimension P + Q, spanned by all the linear combinations of a vector from the first space with a vector from the second. A basis for P + Q may be simply obtained by grouping together a basis for P and one for Q. For example, vectors of a two-dimensional plane belong to a space that is the direct sum of the one-dimensional spaces for the vectors of two axes of that plane.
5 5 A similar notation was used for the harmonic oscillator.
6 6 In this relation, the first ps sums are identical, as are the next pt sums, etc.
7 7 The two-particle state space is the tensor product of the two spaces of individual states (see § F-4-b of Chapter II). In the same way, the space of operators acting on two particles is the tensor product of the spaces of operators acting separately on these particles. For example, the operator for the interaction potential between two particles can be decomposed as a sum of products of two operators: the first one is a function of the position of the first particle, and the second one of the position of the second particle.
Complement AXV Particles and holes
1 1 Ground state of a non-interacting fermion gas
2 2 New definition for the creation and annihilation operators
Creation and annihilation operators are frequently used in solid state physics where the notion of particle and hole plays an important role. A good example is the study of metals or semiconductors, where we talk about an electron-hole pair created by photon absorption. A hole means an absence of a particle, but it has properties similar to a particle, like a mass, a momentum, an energy; the holes obey the same fermion statistics as the electrons they replace. Using creation or annihilation operators allows a better understanding of the hole concept. We will remain in the simple framework of a free particle gas, but the concepts can be generalized to the case of particles placed in an external potential or a Hartree-Fock mean potential (Complement EXV).
1. Ground state of a non-interacting fermion gas
Consider a system of non-interacting fermions in their ground state. We assume for simplicity that they are all in the same spin state, and thus introduce no spin index (generalization to several spin states is fairly simple). As we showed in Complement CXIV, this system in its ground state is described by a state where all the occupation numbers of the individual states having an energy lower than the Fermi energy EF are equal to 1, and all the other individual states are empty. In momentum space, the only occupied states are all the individual states whose wave vector k is included in a sphere (called the “Fermi sphere”) of radius kF (the “Fermi radius”) given by1:
where we have used the notation of formula (7) in Complement CXIV: EF is the Fermi energy (proportional to the particle density to the power 2/3), and L the edge length of the cube containing the N particles. When the system is in its ground state, all the individual states inside the Fermi sphere are occupied, whereas all the other individual states are empty. Choosing for the individual states basis {|ui〉} the plane wave basis, noted {|uk〉} to explicit the wave vector ki, the occupation numbers are:
(2)
In a macroscopic system, the number of occupied states is very large, of the order of the Avogadro number (≃1023). The ground state energy is given by:
with:
The sum over ki in (3) must be interpreted as a sum over all the ki values that obey the boundary conditions in the box of volume L3, as well as the restriction on the length of the vector ki which must be smaller or equal to kF.
2. New definition for the creation and annihilation operators
We now consider this ground state as a new “vacuum”
