We develop a paraxial theory of spin dynamics for twisted electrons traversing the entrance or exit fringe of an axially symmetric magnetic coil. Starting from the Foldy-Wouthuysen reduction, we identify a transverse Pauli coupling proportional to the fringe-field gradient. The scalar orbital dynamics is treated exactly through the Ermakov mapping, while the transverse spin coupling is included perturbatively, yielding explicit first-order transition amplitudes for general solenoidal profiles. Axial symmetry requires each spin flip to be accompanied by a compensating change of orbital angular momentum, while the coordinate-linear perturbation restricts each incoming oscillator mode to at most two neighboring radial sidebands. For 100-keV electrons in a 1-T field, the low-order examples considered give integrated spin-flip probabilities of order ${10}^{-9}$ per edge. The framework therefore quantifies mixing between spin and orbital angular momentum (OAM) in realistic magnetic lenses and provides a basis for controlling spin-OAM conversion using engineered sequences of magnetic-field edges.

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• Cold atoms & matter waves
• Dirac equation
• Perturbation theory
• Spin dynamics
• Linear accelerators
• Angular momentum
• Spin
• Electron microscopy