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 ℓ B 2 ,
where ℓ B = ℏ / e B , which makes them mathematically similar to the position and momentum operators Q = q and P = − i ℏ d d q in one-dimensional quantum mechanics.

Much like acting on a wave function f ( q ) of a one-dimensional quantum particle by the operators e i a P and e i b Q 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.