Translation separation in a magnetic field gradient
= 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 $p_y=\hbar k$. Periodic length $L_y$ gives $k=2\pi n/L_y$. The kinetic y-momentum is instead $\hbar k-qA_y(x)$ and is not spatially constant. Confusing these two momenta loses the magnetic contribution to the velocity.