Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 105

We now compute the time derivative of this velocity. Taking the derivative of (48), we get:

(52)

so that we can isolate the time derivative of α(r, t) by the following combination:

(53)

The left-hand side of this relation can be computed with the Gross-Pitaevskii equation (18) and its complex conjugate, as we now show. We first take the divergence of the gradient (49) to obtain the Laplacian:

(54)

We then insert the time derivative of φ(r, t) given by the Gross-Pitaevskii equation (18) in the left-hand side of relation (53), which becomes:

(55)

This result must be equal to the right-hand side of (53). We therefore get, after dividing both sides by —2n(r, t):

(56)

Using (51), we finally obtain the evolution equation for the velocity v(r, t):

(57)

This equation looks like the classical Newton equation. Its right-hand side includes the sum of the forces corresponding to the external potential V₁(r, t), and to the mean interaction potential with the other particles gn(r, t); the third term in the gradient is the classical kinetic energy gradient¹ (as in Bernoulli’s equation of classical hydrodynamics). The only purely quantum term is the last one, as shown by its explicit dependence on ħ². It involves spatial derivatives of n(r, t), and is only important if the relative variations δn/n of the density occur over small enough distances (for example, this term is zero for a uniform density). This term is sometimes called “quantum potential”, or “quantum pressure term” or, in other contexts, “Bohm potential”. A frequently used approximation is to consider the spatial variations of n(r, t) to be slow, which amounts to ignoring this quantum potential term: this is the so-called Thomas-Fermi approximation.

We have found for a system of N particles a series of properties usually associated with the wave function of a single particle, and in particular a local velocity directly proportional to its phase gradient². The only difference is that, for the N-particle case, we must add to the external potential V₁(r, t) a local interaction potential gn(r, t), which does not significantly change the form of the equations but introduces some nonlinearity that can lead to completely new physical effects.

Figure 2: A repulsive boson gas is contained in a toroidal box. All the bosons are supposed to be initially in the same quantum state describing a rotation around the Oz axis. As we explain in the text, this rotation can only slow down if the system overcomes a potential energy barrier that comes from the repulsive interactions between the particles. This prevents any observable damping of the rotation over any accessible time scale; the fluid rotates indefinitely, and is said to be superfluid.