The angular momentum, creation, and significance of quantized vortices

Abstract

The flux of probability density corresponding to a complex wavefunction, ψ=ψr+iψi, can form vortices which rotate about a nodal region of ψ, and have integer values of the circulation numbers. For one particle wavefunctions (or for approximate natural spin molecular orbitals), the vortices are either axial or toroidal. The axial vortices have angular momentum dipole moments which are usually quantized only if the wave function is an eigenfunction of Lz. The axial vortices interact with homogeneous magnetic fields whereas the toroidal interact with inhomogeneous magnetic fields. Toroidal vortices are easily created or annihilated by a perturbation which moves the nodal surface of ψi relative to the nodal surface of ψr so that these surfaces intersect or become separated. Thus, toroidal vortices are unstable and may be important only near resonance. However, axial vortices are stable since their creation requires (a) photon absorption or emission, (b) a perturbation (such as a magnetic field) which converts a real into a complex wavefunction, or (c) conversion of a toroidal into an axial vortex by expanding the nodal loop until a part of it reaches the boundary of configuration space. Thus, axial vortices should play an important role in energy transfer, photochemical processes, etc., and their circulation numbers should be good quantum numbers. A method for determining generalized first-order density matrices and natural spin–orbitals for use in scattering problems is proposed.