Vortices in an Imperfect Bose Gas. I. The Condensate

A theoretical study of rectilinear vortices in an imperfect Bose gas shows a close correspondence with classical hydrodynamics. The energy and momentum of a vortex pair in an unbounded fluid are calculated. The similarity between a vortex pair and a vortex ring leads to an estimate of the critical velocity $v_c$ of liquid He II in a tube of radius $R$ that includes the effect of the walls $v_c=\frac{C\hslash }{2mR}$, where $C$ is a constant of order unity. A variational treatment of a system of many identical vortices in a container shows that the energy is lowest for a uniform distribution, and that the number of vortices per unit area $\nu$ agrees with Feyman's result $\nu =\frac{2m\omega }{h}$. In the classical limit ($h\to 0$), the angular momentum and energy approach the values for solid-body rotation.