Fermions noninteracting

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

The next step is crucial. We have shown above that the exact wave functions of noninteracting fermions are Slater determinants.12 Thus, it will be possible to set up a noninteracting reference system, with a Hamiltonian in which we have introduced an effective, local potential Vs(r) ... 

Let us now consider a simple example four noninteracting fermions in a onedimensional box, x e [0, 7r], The wave function is a Slater determinant with two doubly occupied orbitals ... 

FIGURE 20.3 One-particle density for four noninteracting fermions in a one-dimensional box. The dots on the abscissa show different positions in which the density has the same value. 

Here, we should mention that there exists an extensive discussion in the literature on the capabilities of spin-DFT regarding, for instance, the question whether the Kohn-Sham spin density has to be equal to the spin density of the fully interacting system of electrons (and in the case of open-shell singlet broken-symmetry (BS) determinants (see below) for binuclear transition-metal clusters this is certainly not the case see Ref. (33) for a more detailed discussion). But the situation is much more subtle and one may basically set up the variational procedure in a Kohn-Sham framework such that the spin density of the Kohn-Sham system of noninteracting fermions represents the true spin density. However, the frame of this review is not sufficient to present all details on this matter (34,35). 

A Fermi liquid is a quantum-mechanical liquid of fermions at very low temperatures, with properties resembling those of a Fermi gas of noninteracting fermions. 

The second part of the renormalisation program, addressing the removal of the ultraviolet (UV) divergencies of QED, which result from the perturbative treatment of the interaction of the fermions with photons and the external field, is more involved. It is instructive to first consider noninteracting fermions in a given external potential,... 

In the special case that the wave function T is a Slater determinant, i.e., the wave function of N noninteracting fermions, the single-particle density matrix can be written in terms of the orbitals comprising the determinant,... 

Figure 9. Eigenvalue spectrum of noninteracting fermions in attractive atomic potential.

The fact that allowed fermion states have to be antisymmetric, i.e., changed in sign by any odd permutation of the fermions, leads to an interesting result concerning the allowed states. Let us write a state wavefunction for a system of n noninteracting fermions as... 

In a uniform electron gas (UEG) of noninteracting fermions with density p, the local kinetic energy per fermion is a constant ... 

As a result of the impossibility for two hydrogen atoms to occupy the same site, the expression for Q is the same as that for a gas of noninteracting fermions... 

The graph in fig. 3 is a plot of the specific heat y versus the low-temperature susceptibility x(0) for a number of mixed-valence and heavy-fermion materials (Jones 1985). For noninteracting fermions, we see from eqs. (2) and (4) that the ratio of these two quantities contains only fundamental constants, and this is indicated by the solid line in the graph. Many-body interactions will modify the specific heat and the susceptibility, but in different ways. In particular, exchange interaction between pairs of fermions enhances /(O) but not y, and this would put the actual points to the right of the solid line. The deviation from the solid line, then, is a measure of... 

For the benefit of beginners, we will start the discussion from the most primitive level. This will also serve the purpose of defining a set of basic quantities which will be useful later. Let us consider a band of noninteracting fermions with mass m. The kinetic energy Cn is related to the wave vector k by... 

The parameters a and have the same meaning in all cases. If the dilute occupation approximation cannot be used, Eqs. (25.2-26) and (25.2-29) can be rewritten for noninteracting fermions and bosons ... 

This simple theory is based on the assumption that the mobile electrons in a solid can be represented as a gas of noninteracting fermions. The orbitals for the electrons are free-particle wave functions like those of Eq. (15.3-41) ... 

Accounting for the ground state of the noninteracting fermion system, the occupation number density in the wavenumber space. 

Fermi-Dirac statistics Form of quantum statistics that must be used to describe a gaseous assembly of noninteracting fermions at low temperatures. 

---