Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 104

Since:

(43)

the time variation of the density may be obtained by first multiplying (41) by φ*(r, t), then its complex conjugate by φ(r, t), and then adding the two results; the potential terms in V₁(r, t) and g n(r, t) cancel out, and we get:

(44)

Let us now define a vector J(r, t) by:

(45)

If we compute the divergence of this vector, the terms in ▽φ* · ▽φ cancel out and we are left with terms identical to the right-hand side of (44), with the opposite sign. This leads to the conservation equation:

(46)

J(r, t) is thus the probability current associated with our boson system. Integrating over all space, using the divergence theorem, and assuming φ(r, t) (hence the current) goes to zero at infinity, we obtain:

(47)

This shows, as announced earlier, that the Gross-Pitaevskii equation conserves the norm of the wave function describing the particle system.

We now set:

(48)

The gradient of this function is written as:

(49)

Inserting this result in (45), we get:

(50)

or, defining the particle local velocity v(r, t) as the ratio of the current to the density:

(51)

We have defined a velocity field, similar to the velocity field of a fluid in motion in a certain region of space; this field velocity is irrotational (zero curl everywhere).