Multipolar Representation of Maxwell and Schroedinger Equations: Lagrangian and Hamiltonian Formalisms: Examples

TL;DR

This paper generalizes Maxwell and Schrödinger equations using Lagrangian and Hamiltonian formalisms to include toroid polarizations, aiding the description of complex quantum systems like mesoscopic devices and materials.

Contribution

It introduces a novel electromagnetic framework incorporating toroid polarizations derived from quantum principles, expanding the modeling capabilities for complex quantum materials.

Findings

Electromagnetic equations are generalized with toroid polarizations.

Lagrangian and Hamiltonian formalisms are used to derive new equations.

Examples demonstrate electromagnetic properties described by toroid moments.

Abstract

Development of quantum engineering put forward new theoretical problems. Behavior of a single mesoscopic cell (device) we may usually describe by equations of quantum mechanics. However if experimentators gather hundreds of thousands of similar cells there arises some artificial medium that one already needs to describe by means of new electromagnetic equations. The same problem arises when we try to describe e.g. a sublattice structure of such complex substances like perovskites. It is demonstrated that the inherent primacy of vector potential in quantum systems leads to a generalization of the equations of electromagnetism by introducing in them toroid polarizations. To derive the equations of motion the Lagrangian and the Hamiltonian formalisms are used. Some examples where electromagnetic properties of molecules are described by the toroid moment are pointed out.