The motion of an electron in a magnetic field on a plane is described by the following four variables: guiding center coordinates and the relative coordinates .
The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy ,
where , which makes them mathematically similar to the position and momentum operators and in one-dimensional quantum mechanics.
Much like acting on a wave function of a one-dimensional quantum particle by the operators and generate the shift of momentum or position of the particle, for the quantum particle in 2D in magnetic field one considers the magnetic translation operators
for any pair of numbers .
The magnetic translation operators corresponding to two different pairs and do not commute.