A general problem of 2 → N f scattering is addressed with all the states being wave packets with arbitrary phases. Depending on these phases, one deals with coherent states in (3 + 1) D, vortex particles with orbital angular momentum, the Airy beams, and their generalizations. A method is developed in which a number of events represents a functional of the Wigner functions of such states. Using width of a packet σ p / p as a small parameter, the Wigner functions, the number of events, and a cross section are represented as power series in this parameter, the first non-vanishing corrections to their plane-wave expressions are derived, and generalizations for beams are made. Although in this regime the Wigner functions turn out to be everywhere positive, the cross section develops new specifically quantum features, inaccessible in the plane-wave approximation. Among them is dependence on an impact parameter between the beams, on phases of the incoming states, and on a phase of the scattering amplitude. A model-independent analysis of these effects is made. Two ways of measuring how a Coulomb phase and a hadronic one change with a transferred momentum t are discussed.

No abstract

The possibilities to accelerate vortex electrons with orbital angular momentum (OAM) to relativistic energies and to produce vortex ions, protons, and other charged particles crucially depend on whether the OAM is conserved during the acceleration and on how phase space of the wave packet evolves. We show that both the OAM and a mean emittance of the packet, the latter obeying the Schrödinger uncertainty relation, are conserved in axially symmetric fields of electric and magnetic lenses, typical for accelerators and electron microscopes, as well as in Penning traps. Moreover, a linear approximation of weakly inhomogeneous fields works much better for single packets than for classical beams. We analyze quantum dynamics of the packet’s rms radius ⟨ρ 2⟩, relate this dynamics to a generalized form of the van Cittert–Zernike theorem, applicable at arbitrary distances from a source and for non-Gaussian packets, and adapt the Courant–Snyder formalism to describe the evolution of the packet’s phase space. The vortex beams can therefore be accelerated, focused, steered, trapped, and even stored in azimuthally symmetric fields and traps, somewhat analogously to the classical angular-momentum-dominated beams. We also give a quantum version of the Busch theorem, which states how one can produce vortex electrons with a magnetized cathode during either field- or photoemission, as well as vortex ions and protons by using a magnetized stripping foil employed to change a charge state of ions. Spatial coherence of the packets plays a crucial role in these applications and we provide the necessary estimates for particles of different masses.

Due to the conservation of the canonical angular momentum, charged particle beams which are generated inside a solenoid field acquire a kinetic angular momentum outside of the solenoid field. The relation of kinetic orbital angular momentum to the field strength and the beam size on the cathode is called the Busch theorem. We formulate the Busch theorem in quantum mechanical form and discuss the generation of quantized vortex beams, i.e., beams carrying a quantized orbital angular momentum. Immersing a cathode in a solenoid field presents an efficient and flexible method for the generation of electron vortex beams, while, e.g., vortex ions can be generated by immersing a charge stripping foil in a solenoid field. Both techniques are utilized at accelerators for the production of nonquantized vortex beams. As a highly relevant use case we discuss in detail the conditions for the generation of quantized vortex beams from an immersed cathode in an electron microscope. General possibilities of this technique for the production of vortex beams of other charged particles are pointed out.

A plane-wave approximation in particle physics implies that a width of a massive wave packet σ ⊥ is much larger than its Compton wavelength λc = /mc. For Gaussian packets or for those with the non-singular phases (say, the Airy beams), corrections to this approximation are attenuated as λ 2 c /σ 2 ⊥ ≪ 1 and usually negligible. Here we show that this situation drastically changes for particles with the phase vortices associated with an orbital angular momentum ℓ . For highly twisted beams with |ℓ| ≫ 1, the non-paraxial corrections get |ℓ| times enhanced and |ℓ| can already be as large as 10 3 . We describe the relativistic wave packets, both for vortex bosons and fermions, which transform correctly under the Lorentz boosts, are localized in a 3D space, and represent a non-paraxial generalization of the massive Laguerre-Gaussian beams. We compare such states with their paraxial counterpart paying specific attention to relativistic effects and to the differences from the twisted photons. In particular, a Gouy phase is found to be Lorentz invariant and it generally depends on time rather than on a distance z. By calculating the electron packet's mean invariant mass, magnetic moment, etc., we demonstrate that the non-paraxial corrections can already reach the relative values of 10 −3 . These states and the non-paraxial effects can be relevant for the proper description of the spin-orbit phenomena in relativistic vortex beams, of scattering of the focused packets by atomic targets, of collision processes in particle and nuclear physics, and so forth.