Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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each other in (C-23) and the corresponding transition amplitude of this process will be zero.

Figure 2: Two diagrams representing schematically the two terms appearing in equation (C-23); they differ by an exchange of the individual states of the exit particles. They correspond, in a manner of speaking, to a different “switching point” for the incoming and outgoing states. The solid lines represent the particles’ free propagation, and the dashed lines their binary interaction.

      C-5-b. Particle interaction energy, the direct and exchange terms

      Many physics problems involve computing the average particle interaction energy. For the sake of simplicity, we shall only study here spinless particles (or, equivalently, particles being in the same internal spin state so that the corresponding quantum number does not come into play) and assume their interactions to be binary. These interactions are then described by an operator , diagonal in the {|r₁, r₂, …r_N〉} basis (eigenstates of all the particles’ positions), which multiplies each of these states by the function:

      (C-25)

      In this expression, the function W₂(r_q, r_q′) yields the diagonal matrix elements of the operator associated with the two-particle (q, q′) interaction, where R_q is the quantum operator associated with the classical position r_q. The matrix elements of this operator in the |uk;ul〉 basis is simply obtained by inserting a closure relation for each of the two positions. This leads to:

(C-26)

       α. General expression:

Replacing in (C-16) operator by and taking (C-26) into account, we get:

      (C-27)

We can thus write the average value of the interaction energy in any normalized state |Φ〉 as:

(C-28)

      where G₂(r₁, r₂) is the spatial correlation function defined by:

(C-29)

      Consequently, knowing the correlation function G₂(r₁, r₂) associated with the quantum state |Φ〉 allows computing directly, by a double spatial integration, the average interaction energy in that state.

Actually, as we shall see in more detail in § B-3 of Chapter XVI, G₂(r₁, r₂) is simply the double density, equal to the probability density of finding any particle in r₁ and another one in r₂. 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

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