An electron in free space travelling at non-
relativistic speeds, follows the
Schrödinger equation for a
free particle, that is i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \frac{-\hbar^2}{2m}\nabla^2 \Psi(\mathbf{r},t), where \hbar is the reduced
Planck constant, \Psi(\mathbf r , t) is the single-electron
wave function, m its mass, \mathbf r the position vector, and t is time. This equation is a type of
wave equation and when written in the
Cartesian coordinate system (x,y,z), the solutions are given by a
linear combination of
plane waves, in the form of \Psi_{\mathbf p}(\mathbf{r},t)\propto e^{i(\mathbf{p}\cdot\mathbf{r}-E(\mathbf{p})t)/\hbar} where \mathbf{p} is the
linear momentum and E(\mathbf{p}) is the electron energy, given by the usual
dispersion relation E(\mathbf{p})=\frac{p^2}{2m}. 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 (\rho,\theta,z), the solutions are no longer plane waves, but instead are given by
Bessel beams, solutions that are a linear combination of \Psi_{p_\rho,\,p_z,\,\ell}(\rho,\theta,z)\propto J_\left(\frac{p_\rho \rho}{\hbar} \right)e^{i(p_zz-Et)/\hbar}e^{i\ell\theta}, that is, the product of three types of functions: a plane wave with momentum p_z in the z-direction, a radial component written as a
Bessel function of the first kind J_, where p_\rho is the linear momentum in the radial direction, and finally an azimuthal component written as e^{i\ell\theta} where \ell (sometimes written m_z) is the
magnetic quantum number related to the angular momentum L_z in the z-direction. Thus, the dispersion relation reads E=(p_z^2+p_\rho^2)/2m. By azimuthal symmetry, the wave function has the property that \ell=0,\pm 1,\pm2,\cdots is necessarily an
integer, thus L_z = \hbar\ell is quantized. If a measurement of L_z is performed on an electron with selected energy, as E does not depend on \ell, it can give any integer value. It is possible to experimentally
prepare states with non-zero \ell by adding an azimuthal phase to an initial state with \ell = 0; 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 L_z. Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of L_z 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 ==