It is shown how the quantum mechanical mass flux and the electromagnetic fields of an electron Bessel vortex mode generate its intrinsic linear momentum and angular momentum properties. Although the corresponding volume density vectors due to the mass flux contain transverse vector components, their volume integrals are shown by explicit analysis to yield null results. The total linear and angular momenta are thus purely axial vectors. There are additional contributions associated with the vortex electric and magnetic fields and these too are shown to be purely axial vectors. Order of magnitude estimates are made in the context of a suggested experiment on the rotation of an optically levitated nanoparticle subject to an electron vortex.
It is shown how the quantum mechanical mass flux and the electromagnetic fields of an electron Bessel vortex mode generate its intrinsic linear momentum and angular momentum properties. Although the corresponding volume density vectors due to the mass flux contain transverse vector components, their volume integrals are shown by explicit analysis to yield null results. The total linear and angular momenta are thus purely axial vectors. There are additional contributions associated with the vortex electric and magnetic fields and these too are shown to be purely axial vectors. Order of magnitude estimates are made in the context of a suggested experiment on the rotation of an optically levitated nanoparticle subject to an electron vortex.
Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. It is wildly believed that spin is a purely quantum mechanical concept and has no classical analogue. In fact, elementary particles are conceived as point objects which have no axis to “spin” around. Therefore, there is no explaining how spin arises at the fundamental level, why particles have the values they do, and what underpins the Pauli Exclusion principle and Bose-Einstein behavior. However, spin is like a vector quantity; it has a definite magnitude, and it has a “direction”, in order to spin should be composite. In this paper we propose a physical explanation for spin of the electron at the sub-particle level, relying on the vortex model of the electron. The electron is described as a superfluid frictionless vortex which has a mass, angular momentum and spin to provide a complete explanation of all properties of the electron: it composite, spinning around its own axis, produces a tiny magnetic fields independent of those from its orbital motions. The classical hydrodynamic laws are used to describe the quantum properties of the electron, such as spin, angular momentum, magnetic momentum and a magnetic dipole. The circulation in the vortex is constant, and the angular momentum of the vortex is conserved and has the same value of Planck constant. The direction of the angular momentum of a spinning electron vortex is along the axis of rotation and determined by the direction of spin. The spin quantum number 1/2 has a fixed value which represents the gap between the circulation rate of the core of the vortex and the boundaries of the vortex. The changeable values +1/2 “spin-up” or -1/2 “spin-down” indicate the direction of the magnetic dipole of the vortex. The relation between spin and Planck constant is discussed and the origin h/4pi angular momentum units are revealed.
We obtain general analytic expressions for the linear and angular momentum density of an elliptically polarized paraxial laser beam. As a partial case, we obtain expressions for the linear and angular momentum density of a paraxial elliptically polarized optical vortex. It is shown that for an arbitrary vortex field with rotational symmetry and with circular polarization the effect of the ``angular tractor'' takes place, which means that the flux of light energy rotates around the optical axis clockwise or counterclockwise at different radii in the beam cross section. It is also shown that the axial component of the angular momentum density of the vortex light field changes its sign at the same radii in the beam cross section. Microparticles trapped in the vortex Bessel beam at different radii are experimentally shown to rotate simultaneously clockwise and counterclockwise.
Optical vortices became a hot topic since almost two decades ago, when it was recognized that Laguerre-Gaussian laser modes carry orbital angular momentum [Allen et al. Phys Rev A 45, 8185 (1992)] related with a screw phase dislocation, and different from the spin angular momentum associated to circular polarization. In 1995, this dynamical quantity was transferred to matter in an optical micromanipulation system for the first time [H. He, et al., Phys. Rev. Lett. 75, 826 (1995)], and since then, a number of studies on angular momentum of light have unveiled different interesting aspects on the subject. However, there are still open questions, which have arisen together with the generation of novel light beams, such as vector vortices, for instance. In contrast with scalar vortices, with usual polarization states (linear, circular, elliptical), the orientation and magnitude of the electric field of vector vortices (solutions of the vector wave equation) is a function of space and time. In this work, we present an experimental study of the local angular momentum density of a Bessel vector vortex of first order by means of an optical trap. For this purpose, we used different probe particles in order to sense the local contribution to the optical angular momentum in each region of the beam. But optical fields are not the only wave fields that may exhibit phase dislocations or singularities. There are close analogies between light and sound fields that can be exploited in order to get a better understanding of common phenomena and study new aspects in both branches of physics. Here we also present the first experimental demonstration and theoretical analysis of acoustical vortices in free field, with similar properties to those of the optical vortices, including the angular momentum that can be transferred to matter. The corresponding analogies and differences with the optical case turn out to be very enlightening for the understanding of the phenomenon of angular momentum in wave fields.
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.
In this paper we present a gauge-invariant formalism for describing the spin and orbital angular momenta of scalar and vector optical vortices. The correction term, arising from consistent incorporation of the transverse vector potential's nonlocality, restores gauge invariance and makes it possible to separate the total angular momentum into spin and orbital components. We propose a method for calculating the orbital and spin angular momenta of multimode vector vortex fields and show that the orbital angular momentum of such beams contains two qualitatively different components. The polarization-independent contribution to the orbital angular momentum is related to the topological charge that determines the order of the phase singularity, while the polarization-dependent one is proportional to the helical topological charge that determines the spatial inhomogeneity of the polarization. We also discuss the quantization procedure for fields with spatially inhomogeneous polarization in terms of modes with a helical topological charge, and examine the limits of applicability of the proposed formalism.
A particle moving along a straight-line has a linear momentum mv. A revolving particle has angular momentum due to its mass and angular velocity. In classical mechanics, orbital angular momentum is the vector product of position (r) and linear momentum (p): L = r× p . (11.1) Let us imagine the electron as a small ball, revolving about some axis. It possesses angular momentum. Precisely, the angular momentum vector is a vector pointing along the axis of rotation with magnitude proportional to the speed of rotation. A vector which changes sign under space inversion (parity operation) is called a polar vector . r and p are polar vectors as they change sign. Since L = (−r)× (−p), L does not change sign under parity operation. Such vectors which do not change sign under parity operation are called axial vectors or psuedo-vectors. Classically, the angular momentum of the electron could have any magnitude. In quantum mechanics the orbital angular momentum operator is obtained by replacing p by −i~∇. The angular position of a particle is the angle at which it is located in two-dimensional polar coordinates.