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Quantum Derivation of Rayleigh Scattering: Why the Sky is Blue and the Sunset is Red

Gennadirvich Andrey Ryabokon

Published Feb 24, 2026• 0 citations
Publisher

Abstract

This explanation is based on the phenomenological decoherence principle presented in https://doi.org/10.5281/zenodo.18640591. According to this principle, the probability of maintaining coherence (e.g., the absence of scattering) is given by an exponential of the integral of the squared effective environmental field: C = C_0 exp(-alpha integral F^2 dx). We show how this principle, combined with quantum mechanics, naturally yields the famous Rayleigh law (λ^(-4)) that explains the color of the sky. We present an alternative derivation of the Rayleigh scattering cross section starting from a microscopic model of light scattering by independent molecules. The exponential attenuation law (Bouguer–Lambert–Beer law) is linked to the sum of squared scattering amplitudes of individual particles, which follows directly from the phenomenological decoherence principle. Using a three-dimensional harmonic oscillator as a model of a molecule, we compute the elastic photon scattering amplitude to second order in perturbation theory within the dipole approximation. The resulting cross section exhibits the characteristic ω^4 dependence, directly accounting for the blue color of the sky and the redness of the Sun at sunset. No empirical data are used; all results follow from first principles of quantum mechanics and the proposed decoherence principle. This publication is part of a broader conceptual program aimed at providing a unified phenomenological description of decoherence and attenuation processes. The framework is fully consistent with standard quantum mechanics and should be viewed as an organizing perspective rather than a proposal of new dynamics.

DOI
10.5281/zenodo.18754554

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This explanation is based on the phenomenological decoherence principle presented in https://doi.org/10.5281/zenodo.18640591. According to this principle, the probability of maintaining coherence (e.g., the absence of scattering) is given by an exponential of the integral of the squared effective environmental field: C = C_0 exp(-alpha integral F^2 dx). We show how this principle, combined with quantum mechanics, naturally yields the famous Rayleigh law (λ^(-4)) that explains the color of the sky. We present an alternative derivation of the Rayleigh scattering cross section starting from a microscopic model of light scattering by independent molecules. The exponential attenuation law (Bouguer–Lambert–Beer law) is linked to the sum of squared scattering amplitudes of individual particles, which follows directly from the phenomenological decoherence principle. Using a three-dimensional harmonic oscillator as a model of a molecule, we compute the elastic photon scattering amplitude to second order in perturbation theory within the dipole approximation. The resulting cross section exhibits the characteristic ω^4 dependence, directly accounting for the blue color of the sky and the redness of the Sun at sunset. No empirical data are used; all results follow from first principles of quantum mechanics and the proposed decoherence principle. This publication is part of a broader conceptual program aimed at providing a unified phenomenological description of decoherence and attenuation processes. The framework is fully consistent with standard quantum mechanics and should be viewed as an organizing perspective rather than a proposal of new dynamics.
This is the second paper in a series on the phenomenological principle of decoherence. The first paper, "Phenomenological Principle of Decoherence Based on the Integral Impact of Environmental Fields" (Ryabokon, 2026), is available at DOI: 10.5281/zenodo.18640591. In the present work, we elaborate on the physical meaning and possible realizations of the effective environmental field F(x), which plays a central role in the decoherence principle: C/C0 = exp(-α ∫_γ F²(x) dx). While the original formulation allowed F to represent any local interaction, here we show that in many realistic environments F is best described as a random field with fractal (scale-invariant) correlations. For such fields, the integral ∫ F² dx scales with the system size L as L^β, leading to a stretched-exponential decay of coherence: C/C0 = exp(-α' L^β). We derive the relation between the exponent β and the spectral index p of the field: β = p-2 (valid for 1 < p < 3). This result is supported by known examples from atmospheric optics (Kolmogorov turbulence, β = 5/3), biological tissues, and cosmological density fluctuations. We also discuss how to experimentally determine β and outline the limits of applicability of the proposed scaling. The framework does not modify the original principle but extends its domain to a broad class of natural and artificial media, offering a unified language for decoherence in complex environments.This publication is part of a broader conceptual program aimed at providing a unified phenomenological description of decoherence and attenuation processes. The framework is fully consistent with standard quantum mechanics and should be viewed as an organizing perspective rather than a proposal of new dynamics.
“Why is the sky blue?” or “why are clouds white?” are examples of basic questions that always intrigued the mind of young, and not-so-young, students before their exposure to the fundamentals of light–matter interaction, a phenomenon that occurs anytime and anywhere in our universe and constitutes the main thrust in this chapter. It starts by highlighting the basic aspects of absorption and spontaneous and stimulated emission, described (in terms of quantum mechanics) by using Einstein coefficients and transition moments ; further along, the (classical) macroscopic connection with the Beer–Lambert law is derived. The journey continues by visiting elastic scattering and its two modalities, Rayleigh and Mie scattering , depending on the scattering particle size. In the final step, inelastic Brillouin and Raman scattering are discussed, placing emphasis on the laser-spectroscopic techniques underlying these types of photon–matter interactions. The chapter ends with a brief description how prehistoric humans already knew how to prepare colors for paintings (as exemplified in the Altamira cave in Cantabria, Spain); and a short exposé into state-of-the-art single-molecule spectroscopy using a single photon. This thread of spanning an arc between (historical) breakthroughs and cutting-edge research will be common to all chapters of the book.
This work proposes a phenomenological formulation of a decoherence principle that relates the degree of suppression of quantum coherence to the integral of the squared effective environmental field in the physical systems and experimental configurations analyzed in this paper. The environment is treated as a set of effective physical influences relevant to a specific experimental or physical configuration and acting on the system. The applicability of the proposed principle is analyzed using seventeen experiments and observations spanning microscopic, mesoscopic, macroscopic, and cosmological scales. Seven key experiments are analyzed quantitatively and in detail, while the remaining ones are presented as supporting the broad applicability of the proposed phenomenological approach. This publication is part of a broader conceptual program aimed at providing a unified phenomenological description of decoherence and attenuation processes. The framework is fully consistent with standard quantum mechanics and should be viewed as an organizing perspective rather than a proposal of new dynamics.
The absolute value of the transition probability of the Rayleigh scattering is computed for the first time and applied to the scattering of solar light with molecules in the atmosphere and to the laser scattering with nanopartilces. The probability has a new contribution of unique properties from long-range correlations specific to the quantum mechanics. The magnitude is sufficient to resolve longstanding puzzle on diffusion lights in the sky and anomalous photon spectrum in laser experiments. The earth's albedo from the new calculations on Rayleigh scattering agrees with observations with satelites.
The phenomenological decoherence principle introduced in Ryabokon (2026a) expresses coherence loss as an exponential of the integral of the squared effective field F. In this paper we analyze the statistical foundations of stretched-exponential decay, which arises when F possesses scale-invariant (fractal) statistics. Starting from a power-law spectrum Φ(k) ∝ k^{-p}, we derive explicit relations between the spectral index p and the stretched-exponential exponent β. For Gaussian fluctuations we obtain: β = p-1 for the field itself, and β = p-2 for integrated quantities (e.g., wave phase). The domain of validity and the role of non-Gaussian statistics are discussed. This publication is part of a broader conceptual program aimed at providing a unified phenomenological description of decoherence and attenuation processes. The framework is fully consistent with standard quantum mechanics and should be viewed as an organizing perspective rather than a proposal of new dynamics.
Abstract In this chapter we develop the stochastic dynamical theory of Rayleigh scattering, from the point of view of a random walk model. Physically this is appropriate to forward and backward scattering experiments where the scattering population is effectively constant over the observation period. From a theoretical point of view, the purpose of the chapter is to present a self-contained derivation that follows essentially from first principles. We shall see how the development, according to a random walk model pertaining to the components of the scattered field, leads to an elegant mathematical derivation of the Ornstein–Uhlenbeck process for the resultant amplitude. This derivation of the scattering dynamics, motivated by a physical model (the coherent superposition of a number of randomly phased scattered wave components) is of inherent pure mathematical interest, as well as being satisfying from a theoretical physics point of view.