Phase space correspondence between classical optics and quantum mechanics

Abstract

The paper scrutinizes both the similarities and the differences between the classical optics and quantum mechanical theories in phase space, especially between the Wigner distribution functions (WDF) defined in the respective phase spaces. Classical optics is able to provide an understanding of either the corpuscular or wave aspects of quantum mechanics, reflected in phase space through the classical limit of the quantum WDF or the WDF in classical optics, respectively. However, classical optics, as any classical theory, cannot mimic the w ave-particle duality that is at the heart of quantum mechanics. Moreover, it is never enough underlined that, although the mathematical phase space formalisms in classical optics and quantum mechanics are very similar, the main difference between these theories, evidenced in the results of measurements, is as deep as it can get even in phase space. On the other hand, the phase space treatment allows an unexpected similar treatment of interference phenomena, although quantum and classical superpositions of w avefunctions and fields, respectively, have a completely different behavior. This similarity originates in the bilinear character of the WDF in both quantum mechanics and classical optical wave theory. Actually, the phase space treatment of the quantum and classical wave theory is identical from the mathematical point of view, if the Planck's constant is replaced by the wavelength of light. Even WDFs of particular quantum states, such as the Schrödinger cat state, can be mimicked by classical optical means, but not the true quantum character, which resides in the probability significance of the wavefunction, in comparison to the physical 'realness' of classical wave fields. 2 1. INTRODUCTION Quantum mechanics was born from the need to quantify the energy of the emitted or absorbed electromagnetic radiation in order to explain the black body spectrum and the photoelectric effect. Light was thus considered, from the beginning of the quantum revolution, either as an extended wave or as a bunch of particles with definite energy. The subsequent evolution of quantum mechanics has not succeeded to settle the century old controversy regarding the corpuscular or wave-like nature of light, but rather deepened the mystery, extending the wave-corpuscle controversy to material particles. Quantum mechanics, and more recently quantum optics, is able to correctly calculate the essential characteristics of a quantum system, such as its energy levels, quantum statistics and so on, but still longs for a proper interpretation of its calculations. The widely accepted probabilistic interpretation of quantum mechanics is still an open question, the meaning and even existence of a wavefunction for photons, for example, being still subject to debate.One of the main reasons for this paradoxical situation of the most successful recent theory in physics is that although it has initially borrowed a lot of concepts from classical physics, in particular from classical optics, the subsequent evolu...

Nonclassicality of vortex Airy beams in the Wigner representation

“…According to Dragoman [25], classical optics is able to provide an understanding of the wave aspect of quantum mechanics through the WDF with a classical field function, but nonclassical features in the WDF cannot always be mimicked by a classical phase-space distribution. He noted that only the WDFs of certain quantum states, such as the Schrödinger cat, can be mimicked by classical optical fields.…”

“…The features can be extended to describe partial coherence of beams and their propagation in different media [22][23][24]. A good review on the classical quantum correspondence of the Wigner function has been given by Dragoman [25].…”

Symplectic evolution of an observed light bundle

“…The analogy between classical optics and quantum mechanics follows from the similarities between the Schrödinger equation of a particle wavefunction and the Maxwell equation of a complex amplitude of a classical wave in the scalar theory 19,20 . The → 0 limit of quantum mechanics is said to give classical mechanics, and λ → 0 limit of wave optics gives ray optics.…”

The Snell law for quaternionic potentials

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By using the analogy between optics and quantum mechanics, we obtain the Snell law for the planar motion of quantum particles in the presence of quaternionic potentials.

I.INTRODUCTIONAnalogies between quantum mechanics [1, 2] and optics [3,4] are well known. The possibility to propose optical experiments to study the fascinating behavior of quantum mechanical system is a subject of great interest in litterature [5,6]. The recent progress [7][8][9][10][11][12][13][14][15][16] in looking for quantitative and qualitative differences between complex and quaternionic quantum mechanics [17] surely have contributed to make the subject more useful to and accessible over the worldwide community of scientists interested in testing the existence of quaternionic potentials.

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