Magnetic translation operator
= Magnetic translation operator
{wiki=Magnetic_translation}
For a lattice vector $r$,
$$
\mathcal T_r=\exp\left(\frac i\hbar r\cdot K\right)
$$
acts in the symmetric gauge as
$$
(\mathcal T_r\psi)(x)
=\exp\left(\frac{ie}{2\hbar}r\cdot(B\times x)\right)\psi(x+r).
$$
It commutes with $(p-eA)^2/(2m)+V(x)$ when $V(x+r)=V(x)$.