Orbital angular momentum of free electrons
s in free space can carry quantized orbital angular momentum 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 where is the reduced Planck constant, is the single-electron wave function, its mass, the position vector, and is time.This equation is a type of wave equation and when written in the Cartesian coordinate system, the solutions are given by a linear combination of plane waves, in the form of where is the linear momentum and is the electron energy, given by the usual dispersion relation. 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 of the electrons is fixed to a certain degree of precision.
When the Schrödinger equation is written in the cylindrical coordinate system, the solutions are no longer plane waves, but instead are given by Bessel beams, solutions that are a linear combination of that is, the product of three types of functions: a plane wave with momentum in the -direction, a radial component written as a Bessel function of the first kind, where is the linear momentum in the radial direction, and finally an azimuthal component written as where is the magnetic quantum number related to the angular momentum in the -direction. Thus, the dispersion relation reads . By azimuthal symmetry, the wave function has the property that is necessarily an integer, thus is quantized. If a measurement of is performed on an electron with selected energy, as does not depend on, it can give any integer value. It is possible to experimentally prepare states with non-zero by adding an azimuthal phase to an initial state with ; 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.
Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of can also be shown in the spherical coordinate system, where the wave function reduces to a product of spherical Bessel functions and spherical harmonics.