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.