Translation separation in a magnetic field gradient

In a gauge with potentials independent of y, the canonical momentum $p_y$ commutes with the Hamiltonian. A simultaneous eigenbasis can be chosen as $e^{iky}{\Phi }_k(x)$, with $p_y=\hslash k$. Periodic length $L_y$ gives $k=2\pi n/L_y$. The kinetic y-momentum is instead $\hslash k-q{A}_y(x)$ and is not spatially constant. Confusing these two momenta loses the magnetic contribution to the velocity.