Ultrarelativistic electron optics in a weak magnetic field

Within an unprecedented analytical formulation, we find an approximate relationship for the ultrarelativistic velocity of electrons in the presence of a weak, time-independent and uniform magnetic field acting perpendicularly to the trajectory of the electrons. We also determine the corresponding velocity quantum operator whose eigenvalues are also determined as well as their corresponding Landau states. In addition, the corresponding synchrotron radiation losses are calculated.