In this gauge, the Hamiltonian is H ^ ^ = p ^ ^ x 2 2 m + 1 2 m ( p ^ ^ y − − q B x ^ ^ ) 2 + p ^ ^ z 2 2 m . The operator p ^ ^ y commutes with this Hamiltonian, since the operator ŷ is absent by the choice of gauge. Thus the operator p ^ ^ y can be replaced by its eigenvalue ħk y. Since z ^ ^ does not appear in the Hamiltonian and only the z-momentum appears in the kinetic energy, this motion along the z-direction is a free motion.