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:[1] guiding center coordinates ${\displaystyle (X,Y)}$ and the relative coordinates ${\displaystyle (R_{x},R_{y})}$.

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

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

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

References

1. Z.Ezawa. Quantum Hall Effect, 2nd ed, World Scientific. Chapter 28