Magnetic translation

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: guiding center coordinates $(X,Y)$ and the relative coordinates $(R_{x},R_{y})$.

The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy
$[X,Y]=-i\ell _{B}^{2}$,
where $\ell _{B}={\sqrt {\hbar /eB}}$, which makes them mathematically similar to the position and momentum operators $Q=q$ and $P=-i\hbar {\frac {d}{dq}}$ in one-dimensional quantum mechanics.

Much like acting on a wave function $f(q)$ of a one-dimensional quantum particle by the operators $e^{iaP}$ and $e^{ibQ}$ 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
$e^{i(p_{x}X+p_{y}Y)},$
for any pair of numbers $(p_{x},p_{y})$.

The magnetic translation operators corresponding to two different pairs $(p_{x},p_{y})$ and $(p'_{x},p'_{y})$ do not commute.