Abstract
We consider the quantum theory of paraxial nonrelativistic electron beams in nonuniform magnetic fields, such as the Glaser field. We find the wave function of an electron from such a beam and show that it is a joint eigenstate of two ($z$-dependent) commuting gauge-independent operators. This generalized Laguerre-Gaussian vortex beam has a phase that is shown to consist of two parts, each being proportional to the eigenvalue of one of the two conserved operators and each having different symmetries. We also describe the dynamics of the angular momentum and cross-sectional area of any mode and how a varying magnetic field can split a mode into a superposition of modes. By a suitable change in the frame of reference, all of our analysis also applies to an electron in a quantum Hall system with a time-dependent magnetic field.
Highlights
The field of electron optics [1,2] was pioneered by Glaser [3,4], based on similarities to light optics and the possibility of using electromagnetic fields as electron lenses
We have derived the wave function of a paraxial electron beam moving in a nonuniform (z-dependent) magnetic field
One angular momentum and the other denoted by I (z) of Eq (10), that are conserved as functions of z
Summary
The field of electron optics [1,2] was pioneered by Glaser [3,4], based on similarities to light optics and the possibility of using electromagnetic fields as electron lenses. Be viewed as living in a Hilbert space of square-integrable functions on the two-dimensional (x, y) plane and physical quantities are represented as self-adjoint operators acting on that Hilbert space Such an operator description based on the paraxial approximation is well known in optics [39,40] and we show it is a convenient description for electron beams as well, yielding exact solutions of the paraxial equation. The theoretical descriptions of these systems are, by design, identical, but what one can measure in practice varies from one system to another
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