Physics: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 [math]\displaystyle{ (X,Y) }[/math] and the relative coordinates [math]\displaystyle{ (R_x,R_y) }[/math].

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

Much like acting on a wave function [math]\displaystyle{ f(q) }[/math] of a one-dimensional quantum particle by the operators [math]\displaystyle{ e^{iaP} }[/math] and [math]\displaystyle{ e^{ibQ} }[/math] 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
[math]\displaystyle{ e^{i(p_x X + p_y Y)}, }[/math]
for any pair of numbers [math]\displaystyle{ (p_x, p_y) }[/math].

The magnetic translation operators corresponding to two different pairs [math]\displaystyle{ (p_x,p_y) }[/math] and [math]\displaystyle{ (p'_x,p'_y) }[/math] do not commute.