Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 107

To prevent any confusion with the azimuthal angle φ we now call χ the Gross-Pitaevskii wave function. The time-independent Gross-Pitaevskii equation then becomes (in the absence of any potential except the wall potentials of the box):

(58)

We look for solutions of the form:

(59)

where l is necessarily an integer (otherwise the wave function would be multi-valued). Such a solution has an angular momentum with a well defined component along Oz, equal to lħ per atom. Inserting this expression in (58), we obtain the equation for ul(r, z):

(60)

which must be solved with the boundary conditions imposed by the torus shape to obtain the ground state (associated with the lowest value of μ). The term in l²ħ²/2mr² is simply the rotational kinetic energy around Oz. If the tore radius R is very large compared to the size of its cross-section, the term l²/r² may, to a good approximation, be replaced by the constant l²/R². It follows that the same solution of (60) is valid for any value of l as long as the chemical potential is increased accordingly. Each value of the angular momentum thus yields a ground state and the larger l, the higher the corresponding chemical potential. All the coefficients of the equation being real, we shall assume, from now on, the functions ul(r, z) to be real.

As the wave function is of the form (59), its phase only depends on φ and expression (51) for the fluid velocity is written as:

(61)

where e_φ is the tangential unit vector (perpendicular both to r and the Oz axis). Consequently, the fluid rotates along the toroidal tube, with a velocity proportional to l. As v is a gradient, its circulation along a closed loop “equivalent to zero” (i.e. which can be contracted continuously to a point) is zero. If the closed loop goes around the tore, the path is no longer equivalent to zero and its circulation may be computed along a circle where r and z remain constant, and φ varies from 0 to 2π; as the path length equals 2πr, we get:

(62)

(with a + sign if the rotation is counterclockwise and a — sign in the opposite case). As l is an integer, the velocity circulation around the center of the tore is quantized in units of h/m. This is obviously a pure quantum property (for a classical fluid, this circulation can take on a continuous set of values).

To simplify the calculations, we have assumed until now that the fluid rotates as a whole inside the toroidal ring. More complex fluid motions, with different geometries, are obviously possible. An important case, which we will return to later, concerns the rotation around an axis still parallel to Oz, but located inside the fluid. The Gross-Pitaevskii wave function must then be zero along a line inside the fluid itself, which thus contains a singular line. This means that the phase may change by 2π as one rotates around this line. This situation corresponds to what is called a “vortex”, a little swirl of fluid rotating around the singular line, called the “vortex core line”. As the circulation of the velocity only depends on the phase change along the path going around the vortex core, the quantization relation (62) remains valid. Actually, from a historical point of view, the Gross-Pitaevskii equation was first introduced for the study of superfluidity and the quantization of the vortices circulation.