(right diagram where the particles exchange their individual states). As in Figure 2, the solid lines represent the particles free propagation, and the dashed lines their binary interaction.
where the second sum is zero for fermions (ni is equal to 0 or 1).
(ii) Exchange term, i = l and j = k, shown on the right diagram of Figure 3. The case where all four subscripts are equal is already included in the direct term. To get the operators’ product
(C-33)
Finally, the spatial correlation function (or double density) G2(r1 r2) is the sum of the direct and exchange terms:
(C-34)
where the factor η in front of the exchange term is 1 for bosons and –1 for fermions. The direct term only contains the product |ui(r1)|2 |uj(r2)|2 of the probability densities associated with the individual wave functions ui(r1) and uj(r2); it corresponds to non-correlated particles. We must add to it the exchange term, which has a more complex mathematical form and reveals correlations between the particles, even when they do not interact with each other. These correlations come from explicitly taking into account the fact that the particles are identical (symmetrization or antisymmetrization of the state vector). They are sometimes called “statistical correlations ” and their spatial dependence will be studied in more detail in Complement AXVI.
Conclusion
The creation and annihilation operators introduced in this chapter lead to compact and general expressions for operators acting on any particle number N. These expressions involve the occupation numbers of the individual states but the particles are no longer numbered. This considerably simplifies the computations performed on “N-body systems”, like N interacting bosons or fermions. The introduction of approximations such as the mean field approximation used in the Hartree-Fock method (Complement DXV) will also be facilitated.
We have shown the complete equivalence between this approach and the one where we explicitly take into account the effect of permutations between numbered particles. It is important to establish this link for the study of certain physical problems. In spite of the overwhelming efficiency of the creation and annihilation operator formalism, the labeling of particles is sometimes useful or cannot be avoided. This is often the case for numerical computations, dealing with numbers or simple functions that require numbered particles and which, if needed, will be symmetrized (or antisymmetrized) afterwards.
In this chapter, we have only considered creation and annihilation operators with discrete subscripts. This comes from the fact that we have only used discrete bases or {|ui〉} or {|vj〉} for the individual states. Other bases could be used, such as the position eigenstates {|r〉} of a spinless particle. The creation and annihilation operators will then be labeled by a continuous subscript r. Fields of operators are thus introduced at each space point: they are called “field operators” and will be studied in the next chapter.
COMPLEMENTS OF CHAPTER XV, READER’S GUIDE
| AXV: PARTICLES AND HOLES | In an ideal gas of fermions, one can define creation and annihilation operators of holes (absence of a particle). Acting on the ground state, these operators allow building excited states. This is an important concept in condensed matter physics. Easy to grasp, this complement can be considered to be a preliminary to Complement EXV. |
| BXV : IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS | Studying the thermal equilibrium of an ideal gas of fermions or bosons, we introduce the distribution functions characterizing the physical properties of a particle or of a pair of particles. These distribution functions will be used in several other complements, in particular GXV and HXV. Bose-Einstein condensation is introduced in the case of bosons. The equation of state is discussed for both types of particles. |
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Series of four complements, discussing the behavior of particles interacting through a mean field created by all the others. Important, since the mean field concept is largely used throughout many domains of physics and chemistry.
| CXV : CONDENSED BOSON SYSTEM, GROSSPITAEVSKII EQUATION | CXV : This complement shows how to use a variational method for studying the ground state of a system of interacting bosons. The system is described by a one-particle wave function in which all the particles of the system accumulate. This wave function obeys the Gross-Pitaevskii equation. |
| DXV : TIME-DEPENDENT GROSS-PITAEVSKII EQUATION | DXV : This complement generalizes the previous one to the case where the Gross-Pitaevskii wave function is time-dependent. This allows us to obtain the excitation spectrum (Bogolubov spectrum), and to discuss metastable flows (superfluidity). |
| EXV : FERMION SYSTEM, HARTREE-FOCK APPROXIMATION | EXV : An ensemble of interacting fermions can be treated by a variational method, the Hartree-Fock approximation, which plays an essential role in atomic, molecular and solid state physics. In this approximation, the interaction of each particle with all the others is replaced by a mean field created by the other particles. The correlations introduced by the interactions are thus ignored, but the fermions’ indistinguishability is accurately treated. This allows computing the energy levels of the system to an approximation that is satisfactory in many situations. |
| FXV : FERMIONS, TIME-DEPENDENT HARTREE-FOCK APPROXIMATION | FXV : We often have to study an ensemble of fermions in a time-dependent situation, as for example electrons in a molecule or a solid subjected to an oscillating electric field. The Hartree-Fock mean field method also applies to time-dependent problems. It leads to a set of coupled equations of motion involving a Hartree-Fock mean field potential, very similar to the one encountered for time-independent problems. |
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The mean field approximation can also be used to study the properties, at thermal equilibrium, of systems of interacting fermions or bosons. The variational method amounts
