- References
Magnetic translations are naturally defined operators acting on wave function on a two-dimensional particle in a magnetic field.
The motion of an electron in a magnetic field on a plane is described by the following four variables:[1] 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.
References
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