We present a microscopic calculation of spontaneous photon emission by twisted (paraxial) electrons propagating through inhomogeneous, axisymmetric magnetic fields. We construct exact electron states that incorporate transverse mode structure and wavefront curvature by combining the Foldy-Wouthuysen transformation with a geometric framework based on Lewis-Ermakov invariants and metaplectic transformations. We show that the evolution of such structured states corresponds to an open path in the space of quadratic forms, giving rise to a geometric contribution to the emission amplitude that cannot be eliminated by gauge choice or adiabatic arguments. The inverse radius of curvature of the electron wavefront controls a finite-window boundary contribution to the emission amplitude in regions where the external magnetic field vanishes locally, this contribution vanishes when the observation window is extended to infinity. Presented mechanism generalizes Landau-level radiation to nonasymptotic, structured electron states and establishes a direct connection between noncyclic geometric evolution and photon emission.
We present a microscopic calculation of spontaneous photon emission by twisted (paraxial) electrons propagating through inhomogeneous, axisymmetric magnetic fields. We construct exact electron states that incorporate transverse mode structure and wavefront curvature by combining the Foldy-Wouthuysen transformation with a geometric framework based on Lewis-Ermakov invariants and metaplectic transformations. We show that the evolution of such structured states corresponds to an open path in the space of quadratic forms, giving rise to a geometric contribution to the emission amplitude that cannot be eliminated by gauge choice or adiabatic arguments. The inverse radius of curvature of the electron wavefront controls a finite-window boundary contribution to the emission amplitude in regions where the external magnetic field vanishes locally, this contribution vanishes when the observation window is extended to infinity. Presented mechanism generalizes Landau-level radiation to nonasymptotic, structured electron states and establishes a direct connection between noncyclic geometric evolution and photon emission.
We study spin effects in twisted-electron propagation through the entrance or exit region of an axially symmetric magnetic coil. Starting from the Foldy-Wouthuysen reduction of the Dirac equation, we derive a paraxial spinor equation in which the longitudinally varying solenoidal field produces, in addition to the usual diagonal Zeeman term, a transverse Pauli coupling proportional to the fringe-field gradient. The scalar transverse dynamics is treated exactly by the Ermakov mapping, which absorbs the longitudinally dependent focusing into a metaplectic scaling transformation and reduces the orbital evolution to that of a stationary two-dimensional oscillator. On this background, the transverse Pauli term is treated perturbatively and yields an explicit first-order correction for arbitrary realistic solenoidal profiles. Axial symmetry implies conservation of the total projection of angular momentum, so each spin flip is accompanied by a compensating one-quantum change of orbital angular momentum. In addition, the linear coordinate structure of the perturbation restricts the first-order dynamics to at most two neighboring radial sidebands for each incoming oscillator component. We derive the corresponding transition amplitudes and show how their phases are governed jointly by the Ermakov accumulation and the diagonal spin-orbital rotation. The resulting framework provides a direct way to quantify spin-orbit mixing of twisted electrons in realistic magnetic lenses and solenoidal beam-line elements, and it identifies a route toward controlled spin-OAM conversion in engineered sequences of magnetic-field edges.
With the extension of the work of the preceding paper, the relativistic front form for Maxwell's equations for electromagnetism is developed and shown to be particularly suited to the description of paraxial waves. The generators of the Poincar\'e group in a form applicable directly to the electric and magnetic field vectors are derived. It is shown that the effect of a thin lens on a paraxial electromagnetic wave is given by a six-dimensional transformation matrix, constructed out of certain special generators of the Poincar\'e group. The method of construction guarantees that the free propagation of such waves as well as their transmission through ideal optical systems can be described in terms of the metaplectic group, exactly as found for scalar waves by Bacry and Cadilhac. An alternative formulation in terms of a vector potential is also constructed. It is chosen in a gauge suggested by the front form and by the requirement that the lens transformation matrix act locally in space. Pencils of light with accompanying polarization are defined for statistical states in terms of the two-point correlation function of the vector potential. Their propagation and transmission through lenses are briefly considered in the paraxial limit. This paper extends Fourier optics and completes it by formulating it for the Maxwell field. We stress that the derivations depend explicitly on the "henochromatic" idealization as well as the identification of the ideal lens with a quadratic phase shift and are heuristic to this extent.
The relativistic vortex states of massive charged particles propagating in non-uniform axisymmetric electric field are studied. Starting from the stationary-state equation after the relativistic Foldy-Wouthuysen (FW) transformation and employing the paraxial approximation, the coupled evolution equations for the beam width, wavefront curvature, and Gouy phase are derived. The equations are solved numerically for a quadratic electrostatic potential, an immersion lens, and an einzel lens. The essential influence of the transverse field on beam evolution is demonstrated. The results provide a relativistic quantum framework for controlling accelerated vortex particle beams using electrostatic fields.
We investigate the effective dynamics of a particle confined near a space curve. In the strict thin-layer reduction of the nondegenerate transverse ground state, torsion does not enter the local effective Hamiltonian, which contains only the curvature-induced scalar geometric potential. In contrast, for a thin guide with finite transverse width, a leading-order adiabatic projection onto the twofold-degenerate first-excited transverse band renders the rotation of the Frenet normal frame dynamically relevant and generates a matrix-valued Abelian gauge potential. Using a projection-based derivation in a co-rotating Frenet-frame basis, we show that this effective gauge potential is directly determined by the local torsion of the curve. The resulting effective Hamiltonian takes a gauge-covariant form and produces two transverse-mode branches whose parabolic dispersions are shifted in opposite directions in momentum space. For closed curves, the associated holonomy is controlled by the integrated torsion and leads to geometric interference. These results provide a direct realization of a Wilczek--Zee-type connection induced purely by spatial geometry in curved quantum waveguides. We further construct a classical-wave analogue using the degenerate bending modes of an isotropic elastic rod, demonstrating that the same torsion-induced gauge structure appears in continuum wave physics.
Classically, a charged particle in a magnetic field emits radiation, losing momentum and experiencing the Abraham-Lorentz (AL) / Landau-Lifshitz (LL) radiation reaction (RR) force. However, at atomic scales and outside the range of their applicability, the AL/LL equations fail and RR destroys the coherent state of an electron-undermining the very concept of a RR force. This process can be described by the coupled Schrödinger-Maxwell (SM) system under appropriate limits, but the system's nonlinear complexity has long limited purely analytical studies. We present geometric structure-preserving algorithms for the SM system that preserve gauge invariance, symplecticity, and unitarity on the discrete space-time lattice, which are implemented in our Structure-Preserving scHrodINger maXwell (SPHINX) code. By constructing coherent states from the Landau levels, SPHINX simulates the fully-coupled nonlinear dynamics of an electron coherent state, the energy partition evolution, and decoherence/relaxation of the electron wave packet in time due to RR. These simulations indicate that, in an external magnetic field, an electron prepared in an atomic-scale coherent state can radiate strongly, rapidly losing coherence and dispersing into a decoherent wave packet. Additionally, we also present the fully-coupled nonlinear evolution of the non-degenerate ground- and first-excited Landau levels themselves to understand how the coupled SM system modifies the well-known ideal (i.e., Schrödinger-only) dynamics of the Landau Levels. With appropriate boundary conditions, simulations show that the Landau levels are renormalized into stationary dressed eigenstates with constant electromagnetic and kinetic energies. This opens a new computational window into RR physics and advances modeling of extreme-field phenomena in fusion plasmas, astrophysics, and next-generation laser experiments
We study relativistic twisted (orbital-angular-momentum) states of a massive charged particle propagating through an axially symmetric, longitudinally inhomogeneous solenoid field and a co-directed accelerating or decelerating electric field. Starting from the Dirac equation and using controlled spinless and paraxial approximations, we show that the transverse envelope obeys an effective nonstationary Schrödinger equation governed by a Caldirola--Kanai Hamiltonian. The longitudinal energy gain or loss encoded in f(z)=[E0−V(z)]2−m2 generates an effective gain or damping rate γ(z)=∂zf(z)/[2f(z)] and a z-dependent oscillator frequency ω(z)=p0Ω(z)/f(z). Exploiting the Ermakov mapping (unitary equivalence of Caldirola--Kanai systems), we obtain a closed-form propagated twisted wave function by transforming the stationary Landau basis. The transverse evolution is controlled by a single scaling function b(z) that satisfies a generalized Ermakov--Pinney equation with coefficients determined by Ez(z) and Bz(z). In the limiting cases of uniform acceleration with Bz=0 and of solenoid focusing with negligible acceleration, our solution reduces to previously known analytic results, providing a direct bridge to established models.