The Schrödinger formalism of electromagnetism and other classical waves—How to make quantum-wave analogies rigorous

Giuseppe De Nittis; Max Lein

doi:10.1016/j.aop.2018.02.019

The Schrödinger formalism of electromagnetism and other classical waves—How to make quantum-wave analogies rigorous

Giuseppe De Nittis, Max Lein

Advanced Institute for Materials Research (AIMR)

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

This paper systematically develops the Schrödinger formalism that is valid also for gyrotropic media where the material weights W=εχχ^∗μ≠W¯are complex. This is a non-trivial extension of the Schrödinger formalism for non- gyrotropic media (where W=W¯) that has been known since at least the 1960s (Wilcox, 1966; Kato, 1967). Here, Maxwell's equations are rewritten in the form i∂_tΨ=MΨ where the selfadjoint (hermitian) Maxwell operator M=W⁻¹Rot|_ω≥0=M^∗ takes the place of the Hamiltonian and Ψ is a complex wave representing the physical field (E,H)=2ReΨ. 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.

Original language	English
Pages (from-to)	579-617
Number of pages	39
Journal	Annals of Physics
Volume	396
DOIs	https://doi.org/10.1016/j.aop.2018.02.019
Publication status	Published - 2018 Sept

Keywords

Maxwell equations
Maxwell operator
Quantum-wave analogies
Schrödinger equation

ASJC Scopus subject areas

Physics and Astronomy(all)

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10.1016/j.aop.2018.02.019

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title = "The Schr{\"o}dinger formalism of electromagnetism and other classical waves—How to make quantum-wave analogies rigorous",

abstract = "This paper systematically develops the Schr{\"o}dinger formalism that is valid also for gyrotropic media where the material weights W=εχχ∗μ≠W¯are complex. This is a non-trivial extension of the Schr{\"o}dinger formalism for non- gyrotropic media (where W=W¯) that has been known since at least the 1960s (Wilcox, 1966; Kato, 1967). Here, Maxwell's equations are rewritten in the form i∂tΨ=MΨ where the selfadjoint (hermitian) Maxwell operator M=W−1Rot|ω≥0=M∗ takes the place of the Hamiltonian and Ψ is a complex wave representing the physical field (E,H)=2ReΨ. Writing Maxwell's equations in Schr{\"o}dinger 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.",

keywords = "Maxwell equations, Maxwell operator, Quantum-wave analogies, Schr{\"o}dinger equation",

author = "{De Nittis}, Giuseppe and Max Lein",

note = "Funding Information: Assumption 2.3 Current Density Only Excites States Supported by the Medium Funding Information: G.D. research is supported by the grant Iniciaci{\'o}n en Investigaci{\'o}n 2015 — No. 11150143 funded by FONDECYT. M.L. has been supported by JSPS through a WAKATE B grant (grant number 16K17761) and a Fusion grant from the WPI-AIMR. M.L. is indebted to Kostya Bliokh for the discussions which initiated this work and appreciates his comments on the first version of the manuscript. The authors would also like to thank Ilya Dodin for very useful references, and generous explanations about the applicability of our framework to problems from plasma physics. Publisher Copyright: {\textcopyright} 2018 Elsevier Inc.",

year = "2018",

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doi = "10.1016/j.aop.2018.02.019",

language = "English",

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AU - De Nittis, Giuseppe

AU - Lein, Max

N1 - Funding Information: Assumption 2.3 Current Density Only Excites States Supported by the Medium Funding Information: G.D. research is supported by the grant Iniciación en Investigación 2015 — No. 11150143 funded by FONDECYT. M.L. has been supported by JSPS through a WAKATE B grant (grant number 16K17761) and a Fusion grant from the WPI-AIMR. M.L. is indebted to Kostya Bliokh for the discussions which initiated this work and appreciates his comments on the first version of the manuscript. The authors would also like to thank Ilya Dodin for very useful references, and generous explanations about the applicability of our framework to problems from plasma physics. Publisher Copyright: © 2018 Elsevier Inc.

PY - 2018/9

Y1 - 2018/9

N2 - This paper systematically develops the Schrödinger formalism that is valid also for gyrotropic media where the material weights W=εχχ∗μ≠W¯are complex. This is a non-trivial extension of the Schrödinger formalism for non- gyrotropic media (where W=W¯) that has been known since at least the 1960s (Wilcox, 1966; Kato, 1967). Here, Maxwell's equations are rewritten in the form i∂tΨ=MΨ where the selfadjoint (hermitian) Maxwell operator M=W−1Rot|ω≥0=M∗ takes the place of the Hamiltonian and Ψ is a complex wave representing the physical field (E,H)=2ReΨ. 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.

AB - This paper systematically develops the Schrödinger formalism that is valid also for gyrotropic media where the material weights W=εχχ∗μ≠W¯are complex. This is a non-trivial extension of the Schrödinger formalism for non- gyrotropic media (where W=W¯) that has been known since at least the 1960s (Wilcox, 1966; Kato, 1967). Here, Maxwell's equations are rewritten in the form i∂tΨ=MΨ where the selfadjoint (hermitian) Maxwell operator M=W−1Rot|ω≥0=M∗ takes the place of the Hamiltonian and Ψ is a complex wave representing the physical field (E,H)=2ReΨ. 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.

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The Schrödinger formalism of electromagnetism and other classical waves—How to make quantum-wave analogies rigorous

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