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 wave-front 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 wave front 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. The mechanism presented generalizes Landau-level radiation to nonasymptotic, structured electron states and establishes a direct connection between noncyclic geometric evolution and photon emission.

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• Berry curvature
• Cold atoms & matter waves
• Landau levels
• Quantum optics
• Spontaneous emission
• Electron microscopy
• Geometry
• Integrable systems