Maxwell's equations are valid in both the classical and the quantum realm. In the Heisenberg representation of Quantum Mechanics, the equations of the E and B operators are precisely Maxwell's equations. Of course since the fields are quantum operators, there are many aspects which differ from the classical fields. For example, the E field acts like the momentum conjugate to the spatial components of the vector potential A. This of course leads to many aspects of the quantum electromagnetic field with differ from them as classical fields but they still obey the same evolution equations as the classical field does.