The Schrödinger Formalism of Electromagnetism and Other Classical Waves --- How to Make Quantum-Wave Analogies Rigorous

Abstract

This paper systematically develops the Schrödinger formalism that is valid also for gyrotropic media where the material weights $W=\left(\begin{array}{ccc}\epsilon & \chi {\chi }^{\ast }& \mu \end{array}\right)\ne \overline{W}$ are complex. This is a non-trivial extension of the Schrödinger 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 $M=W^{-1}\mathrm{Rot}{|}_{\omega \ge 0}=M^{\ast }$ 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 Schrödinger 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.

58 pages, updated version incorporates suggestions from the community

References in corpus (8)

• Classification of topological insulators and superconductors in three spatial dimensions
• Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal
• Analogs of quantum Hall effect edge states in photonic crystals
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• On the K-theoretic classification of dynamically stable systems
• Transversality-Enforced Tight-Binding Models for 3D Photonic Crystals aided by Topological Quantum Chemistry
• Photonic quantum Hall effects