  This paper systematically develops the Schr\"odinger formalism that is valid
also for gyrotropic media where the material weights $W = \left (
\begin{smallmatrix} \varepsilon & \chi \chi^* & \mu \end{smallmatrix} \right )
\neq \overline{W}$ are complex. This is a non-trivial extension of the
Schr\"odinger formalism for non-gyrotropic media (where $W = \overline{W}$)
that has been known since at least the 1960s. Here, Maxwell's equations are
rewritten in the form $\mathrm{i} \partial_t \Psi = M \Psi$ where the
selfadjoint (hermitian) Maxwell operator $M = W^{-1} \, \mathrm{Rot} \, \big
|_{\omega \geq 0} = M^*$ takes the place of the Hamiltonian and $\Psi$ is a
complex wave representing the physical field $(\mathbf{E},\mathbf{H}) = 2
\mathrm{Re} \, \Psi$. Writing Maxwell's equations in Schr\"odinger form gives
us access to the rich toolbox of techniques initially developed for quantum
mechanics and allows us to apply them to classical waves. To show its utility,
we explain how to identify conserved quantities in this formalism. Moreover, we
sketch how to extend our ideas to other classical waves.
