# See the Math Superfluid 4Helium

Superfluid flow can be irrotational, involving density fluctuation, or rotational. When there are no density fluctuations, London, Onsager, and Feynman showed that the condensate motion responsible for superfluid flow is described by a wave function.

where the ri denote the positions of the particles, is the ground state wavefunction, and S(ri) denotes its position-dependent phase. The superfluid velocity is given by and is irrotational, since curl vs=0. As Feynman and Onsager first showed, as a result, the superfluid velocity, vs, possesses a fundamental property, characteristic of all superfluid systems; its circulation, C, along any closed curve is quantized, and equal to an integral multiple of h/m. To see this, we write:

where the integral is around any closed curve, and not that the condensate wavefunction must be single-valued, which means that S can change only by a multiple of 2 when one goes around a closed circuit—hence C must be an integral multiple of h/m.

It follows that if a vessel containing the superfluid is set in rotation, vortex lines, around which the circulation is quantized, will form in the fluid to enable it to follow the flow. Figure 12 illustrates the geometry of a single straight vortex line for the case of a fluid rotating about, say, the z axis. The superfluid velocity at some point M is tangential and depends only on the distance r between M and the axis of the vortex, with

where n is an integer giving the number of quanta of circulation in the vortex line. For a rotating superfluid, then, its superfluid velocity can only change through the motion of its quantized vortex lines.