Orbital angular momentum of free electrons
Electrons in free space can carry quantized orbital angular momentum (OAM) projected along the direction of propagation. This orbital angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle. Electron beams with quantized orbital angular momentum are also called electron vortex beams. Theory. An electron in free space travelling at non-relativistic speeds, follows the Schrödinger equation for a free particle, that is formula_1 where formula_2 is the reduced Planck constant, formula_3 is the single-electron wave function, formula_4 its mass, formula_5 the position vector, and formula_6 is time . This equation is a type of wave equation and when written in the Cartesian coordinate system (formula_7,formula_8,formula_9), the solutions are given by a linear combination of plane waves, in the form of formula_10 where formula_11 is the linear momentum and formula_12 is the electron energy, given by the usual dispersion relation formula_13. By measuring the momentum of the electron, its wave function must collapse and give a particular value. If the energy of the electron beam is selected beforehand, the total momentum (not its directional components) of the electrons is fixed to a certain degree of precision. When the Schrödinger  equation is written in the cylindrical coordinate system (formula_14,formula_15,formula_9), the solutions are no longer plane waves, but instead are given by Bessel beams, solutions that are a linear combination of formula_17 that is, the product of three types of functions: a plane wave with momentum formula_18 in the formula_9-direction, a radial component written as a Bessel function of the first kind formula_20, where formula_21 is the linear momentum in the radial direction, and finally an azimuthal component written as formula_22 where formula_23 (sometimes written formula_24) is the magnetic quantum number related to the angular momentum formula_25 in the formula_9 -direction. Thus, the dispersion relation reads formula_27. By azimuthal symmetry, the wave function has the property that formula_28 is necessarily an integer, thus formula_29 is quantized. If a measurement of formula_25 is performed on an electron with selected energy, as formula_31 does not depend on formula_23, it can give any integer value. It is possible to experimentally prepare states with non-zero formula_23 by adding an azimuthal phase to an initial state with formula_34; experimental techniques designed to measure the orbital angular momentum of a single electron are under development. Simultaneous measurement of electron energy and orbital angular momentum is  allowed because the Hamiltonian commutes with the angular momentum operator related to formula_25. Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of formula_36 can also be shown in the spherical coordinate system, where the wave function reduces to a product of spherical Bessel functions and spherical harmonics. Preparation. There are a variety of methods to prepare an electron in an orbital angular momentum state. All methods involve an interaction with an optical element such that the electron acquires an azimuthal phase. The optical element can be material, magnetostatic, or electrostatic . It is possible to either directly imprint an azimuthal phase, or to imprint an azimuthal phase with a holographic diffraction grating, where grating pattern is defined by the interference of the azimuthal phase and a planar or spherical carrier wave. Applications. Electron vortex beams have a variety of proposed and demonstrated applications, including for mapping magnetization, studying chiral molecules and chiral plasmon resonances, and identification of crystal chirality. Measurement. Interferometric methods borrowed from light optics also work to determine the orbital angular momentum of free electrons in pure states. Interference with a planar reference wave, diffractive filtering and self-interference  can serve to characterize a prepared electron orbital angular momentum state. In order to measure the orbital angular momentum of a superposition or of the mixed state that results from interaction with an atom or material, a non-interferometric method is necessary. Wavefront flattening, transformation of an orbital angular momentum state into a planar wave, or cylindrically symmetric Stern-Gerlach-like measurement is necessary to measure the orbital angular momentum mixed or superposition state.