Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

The physical interpretation of (C-28) is simple: the average interaction energy is equal to the sum over all the particles’ pairs of the interaction energy W_int(r₁, r₂) of a pair, multiplied by the probability of finding such a pair at points r₁ and r₂ (the factor 1/2 avoids the double counting of each pair).

       β. Specific case: the Fock states

      Let us assume the state |Φ〉 is a Fock state, with specified occupation numbers ni:

      (C-30)

      We can compute explicitly, as a function of the ni, the average values:

      (C-31)

contained in (C-29). We first notice that to get a non-zero result, the two operators a^† must create particles in the same states from which they were removed by the two annihilation operators a. Otherwise the action of the four operators on the ket |Φ〉 will yield a new Fock state orthogonal to the initial one, and hence a zero result. We must therefore impose either i = k and j = l, or the opposite i = l and j = k, or eventually the special case where all the subscripts are equal. The first case leads to what we call the “direct term”, and the second, the “exchange term”. We now compute their values.

(i) Direct term, i = k and j = l, shown on the left diagram of Figure 3. If i = j = k = l, the four operators acting on |Φ〉 reconstruct the same ket, multiplied by the factor ni (ni — 1); this yields a zero result for fermions. If i ≠ j, we can move the operator ak = ai just to the right of the first operator to form the particle number operator . This permutation in the operators’ order does not change anything: for bosons, we are moving commuting operators, and for fermions, two anticommutations introduce two minus signs, which cancel each other. The same goes for the operators with subscript j, leading to the particle number . Finally, the direct term is equal to:

      (C-32)

Figure 3: Schematic representation of a direct term (left diagram where each particle remains in the same individual state) and an exchange term (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 starting from the product , we just have to permute the two central operators ; when i ≠ j this operation is of no consequence for bosons, but introduces a change of sign for fermions (anticommutation). The exchange term can therefore be written as , with:

      (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 A_XVI.

      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 D_XV) 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;