Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

for the spatial range of the wave function transition regime. In the region where φ(r) is constant, relation (56) yields:

      (58)

Figure 1: Variation as a function of the position x of the wave function φ(x) in the vicinity of a wall (at x = 0) where it is forced to be zero. This variation occurs over a distance of the order of the healing length ξ defined in (61); the stronger the particle interactions, the shorter that length. As x increases, the wave function tends towards a constant plateau, of coordinate , represented as a dashed line.

      On the other hand, in the whole region where φ(r) has significantly decreased, and in particular close to the origin, we have:

      (59)

In one dimension⁴, we then get the differential equation:

      (60)

      whose solutions are sums of exponential functions e^±ix/ξ, with:

(61)

The solution that is zero for x = 0 is the difference between these two exponentials; it is proportional to sin(x/ξ), a function that starts from zero and increases over a characteristic length ξ. Figure 1 shows the wave function variation in the vicinity of the wall where it is forced to be zero.

      The stronger the interactions, the shorter this “healing length” ξ; it varies as the inverse of the square root of the product of the coupling constant g and the density n₀. From a physical point of view, the healing length results from a compromise between the repulsive interaction forces, which try to keep the wave function as constant as possible in space, and the kinetic energy, which tends to minimize its spatial derivative (while the wave function is forced to be zero at x = 0); ξ is equal (except for a 2π coefficient) to the de Broglie wavelength of a free particle having a kinetic energy comparable to the repulsion energy gn₀ in the boson system.

4-c. Another trial ket: fragmentation of the condensate

      We now show that repulsive interactions do stabilize a boson “condensate” where all the particles occupy the same individual state, as opposed to a “fragmented” state where some particles occupy a different state, which can be very close in energy. Instead of using a trial ket (7), where all the particles form a perfect Bose-Einstein condensate in a single quantum state |θ〉, we can “fragment” this condensate by distributing the N particles in two distinct individual states. Consequently, we take a trial ket where Na particles are in the state |θ_a〉 and Nb = N — Na in the orthogonal state |θ_b〉:

      (62)

We now compute the change in the average variational energy. In formula (29) giving the average kinetic energy, for the operator to yield a Fock state identical to , we must have either k = l = a, or k = l = b. This leads to:

      (63)

      The computation of the one-body potential energy is similar and leads to:

      (64)

      In both cases, the contributions of two populated states are proportional to their respective populations, as expected for energies involving a single particle.

As for the two-body interaction energy, we use again relation (32). It contains the operator , which will reconstruct the Fock state in the following three cases:

       - k = l = m = n = a or b yields the contribution:(65)

       - k = m = a and l = n = b, or k = m = b and l = n = a; these two possibilities yield the same contribution (since the W2 operator is symmetric), and the 1/2 factor disappears, leading to the direct term:(66)

       - Finally k = n = a and l = m = b, or k = n = b and l = m = a, yield two contributions whose sum introduces the exchange term (here again without the factor 1/2):

      (67)

      The direct and exchange terms have been schematized in Figure 3 in Chapter XV (replacing |u_i〉 by |θ_a〉, and |uj〉 by |θ_b〉), with the direct term on the left, and the exchange term on the right.

The variational energy can thus be written as:

      (68)

      As above, the interaction between particles in the same state |θ_a〉 contributes a term proportional to Na (Na — 1)/2, the number of pairs of particles in that state; the same is true for the interaction term between particles in the same state |θ_b〉. The direct term associated with the interaction between two particles in distinct states is proportional to NaNb, the number of such pairs. But to this direct term we must add an exchange term, also proportional to NaNb, corresponding to an additional interaction. This increased interaction is due to the bunching effect of two bosons in different quantum states, that will be discussed in more detail in § 3-b of Complement A_XVI. As they are indistinguishable, two bosons occupying individual orthogonal states show correlations in their positions; this increases the probability of finding them at the same point in space. This increase does not occur when the two bosons occupy the same individual quantum state.

      We now assume the diagonal matrix elements of [K₀ + V₁] between the two states |θ_a〉 and |θ_b〉 to be practically the same. For example, if these two states are the lowest energy levels of spinless particles in a cubic box of edge L, the corresponding energy difference is proportional to 1/L²