Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

Quantum Mechanics, Volume 3

number N.

C-5-a. Two terms in the matrix elements

Consider a physical process (schematized in Figure 1) where, in a system of N identical particles, an interaction produces a transfer from the two states |ukα) and |ukβ〉 towards the two states |ukγ〉 and |ukδ〉; we assume that the four states we are dealing with are different. In the summation over i, j, k, l of (C-16), the only terms involved in this process are those where the bra contains either i = kγ and j = kδ, or the opposite i = kδ and j = kγ; as for the ket, it must contain either k = kα and l = kβ, or the opposite k = kβ and l = kα. We are then left with four terms:

(C-21)

However, since the numbers used to label the particles are dummy variables, the first two matrix elements shown in (C-21) are equal and so are the last two. In addition, the product of creation and annihilation operators obey the following relations, for bosons (η = 1) as well as for fermions (η = —1):

(C-22)

These relations are obvious for bosons since we only commute either creation operators or annihilation operators. For fermions, as we assumed all the states were different, the anticommutation of operators a or of operators a^† leads to sign changes; these may cancel out depending on whether the number of anticommutations is even or odd. If we now double the sum of the first and last term of (C-21), we obtain the final contribution to (C-16):

(C-23)

Hence we are left with two terms whose relative sign depends on the nature (bosons or fermions) of the identical particles. They correspond to a different “switching point” for the incoming and outgoing individual states (Fig. 2).

For bosons, the product of the 4 operators in (C-23) acting on an occupation number ket introduces the square root:

(C-24)

For large occupation numbers, this square root may considerably increase the value of the matrix element. For fermions, however, this amplification effect does not occur. Furthermore, if the direct and exchange matrix elements of are equal, they will cancel 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

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