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arXiv:quant-ph/9502028 · Physical Review A

Classical and quantum Malus laws

Krzysztof Wódkiewicz

Published Apr 1, 1995• 25 citations
arXivHTMLPDF

Abstract

The classical and the quantum Malus laws for light and spin are discussed. It is shown that for spin 1/2, the quantum Malus law is equivalent in form to the classical Malus law provided the statistical average involves a quasidistribution function that can become negative. A generalization of Malus's law for arbitrary spin s is obtained in the form of a Feynman path-integral representation for the Malus amplitude. The classical limit of the Malus amplitude for s\ensuremath{\rightarrow}\ensuremath{\infty} is discussed.

DOI
10.1103/physreva.51.2785

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Quantum Malus Law: Negative Quasi-Distributions Unify Classical and Spin-1/2 Polarization — E8 Intelligence Research

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FINDING: Malus's law \(I = I_0 \cos^2\theta\) governs both classical wave polarization and quantum spin-1/2 measurements, with the quantum version requiring a quasi-distribution that can go negative. | MATH: Classical: \(I(\theta) = I_0 \cos^2\theta\). Quantum (spin-1/2): \(P(\theta) = \frac{1}{2}(1 + \cos\theta)\) for spin projection — but the arXiv abstract (quant-ph/9502028) shows the *form* of Malus's law is preserved if the statistical average uses a quasi-distribution \(W(\lambda)\) that can be negative: \(\langle \cos^2\theta \rangle = \int W(\lambda) \cos^2(\theta-\lambda) d\lambda\), with \(W(\lambda) < 0\) allowed. This is the Wigner-function-like negativity — a hallmark of non-classicality. | CONNECTION: **4-fold rotational symmetry** — \(\cos^2\theta\) has period \(\pi\), but the *intensity* pattern \(I(\theta)\) is invariant under \(\theta \to \theta + \pi/2\) (i.e., 90° rotation) because \(\cos^2(\theta+\pi/2) = \sin^2\theta\), and \(I_0\cos^2\theta + I_0\sin^2\theta = I_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Unifying Classical and Quantum Malus Laws via Cosine-Squared Intensity — E8 Intelligence Research

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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Nov 15, 1990•2 citations
The classical and the quantum Malus laws for light and spin are discussed. It is shown that for spin 1/2, the quantum Malus law is equivalent in form to the classical Malus law provided the statistical average involves a quasidistribution function that can become negative. A generalization of Malus's law for arbitrary spin s is obtained in the form of a Feynman path-integral representation for the Malus amplitude. The classical limit of the Malus amplitude for s\ensuremath{\rightarrow}\ensuremath{\infty} is discussed.
FINDING: Malus's law I=I0cos⁡2θI = I_0 \cos^2\thetaI=I0​cos2θ governs both classical wave polarization and quantum spin-1/2 measurements, with the quantum version requiring a quasi-distribution that can go negative. | MATH: Classical: I(θ)=I0cos⁡2θI(\theta) = I_0 \cos^2\thetaI(θ)=I0​cos2θ. Quantum (spin-1/2): P(θ)=12(1+cos⁡θ)P(\theta) = \frac{1}{2}(1 + \cos\theta)P(θ)=21​(1+cosθ) for spin projection — but the arXiv abstract (quant-ph/9502028) shows the *form* of Malus's law is preserved if the statistical average uses a quasi-distribution W(λ)W(\lambda)W(λ) that can be negative: ⟨cos⁡2θ⟩=∫W(λ)cos⁡2(θ−λ)dλ\langle \cos^2\theta \rangle = \int W(\lambda) \cos^2(\theta-\lambda) d\lambda⟨cos2θ⟩=∫W(λ)cos2(θ−λ)dλ, with W(λ)<0W(\lambda) < 0W(λ)<0 allowed. This is the Wigner-function-like negativity — a hallmark of non-classicality. | CONNECTION: **4-fold rotational symmetry** — cos⁡2θ\cos^2\thetacos2θ has period π\piπ, but the *intensity* pattern I(θ)I(\theta)I(θ) is invariant under θ→θ+π/2\theta \to \theta + \pi/2θ→θ+π/2 (i.e., 90° rotation) because cos⁡2(θ+π/2)=sin⁡2θ\cos^2(\theta+\pi/2) = \sin^2\thetacos2(θ+π/2)=sin2θ, and \(I_0\cos^2\theta + I_0\sin^2\theta = I_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Malus's Law — both classical (light polarization) and quantum (spin-1/2) — reduces to a cosine-squared intensity law, with the quantum case requiring a quasi-probability distribution that can go negative, revealing a deep algebraic identity between SU(2) spin projections and classical polarization geometry. | MATH: Classical: I(θ)=I0cos⁡2θI(\theta) = I_0 \cos^2\thetaI(θ)=I0​cos2θ (Malus, 1809). Quantum spin-1/2: for a spin state ∣+⟩n|+\rangle_n∣+⟩n​ measured along axis mmm, probability P=cos⁡2(θ/2)P = \cos^2(\theta/2)P=cos2(θ/2), where θ\thetaθ is the angle between axes nnn and mmm. Equivalently, P=12(1+cos⁡θ)=12(1+n⋅m)P = \frac{1}{2}(1 + \cos\theta) = \frac{1}{2}(1 + \mathbf{n}\cdot\mathbf{m})P=21​(1+cosθ)=21​(1+n⋅m). The quantum Malus law is formally identical to classical if one uses a Wigner-like quasi-distribution W(θ)W(\theta)W(θ) that can be negative; the classical case has W(θ)=δ(θ−θ0)W(\theta) = \delta(\theta - \theta_0)W(θ)=δ(θ−θ0​) (positive). Generalization to spin-jjj: Pj(θ)=[cos⁡2j(θ/2)]P_j(\theta) = \left[\cos^{2j}(\theta/2)\right]Pj​(θ)=[cos2j(θ/2)] for extreme eigenstates, with intermediate proj Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
This work is part of a reconstruction of quantum mechanics from scratch with the aim to understand what it means. In previous work we have shown that the Dirac equation can be derived by classical reasoning just using relativity and group theory. This implies that contrary to common belief quantum mechanics (QM) is not magical or radically different from classical mechanics. It is also not incompatible with the theory of relativity but completely part of it. As the experimental violations of the Bell inequalities seem to take exception with this general scheme for making sense of QM we have scrutinized the derivation of the Bell inequalities to figure out the limitations of our approach. We were able to show that the derivation contains a logical error based on wrong modelling. In the present paper we complete this investigation by showing that Malus' law can be derived by classical reasoning using group representation theory, both for electrons and photons. This is a further, be it somewhat more indirect proof that the derivation of the Bell inequalities contains an error. But more importantly it shows that the philosophy of our approach remains intact, by proving that we can also understand Malus' law with our methods and thereby validating our approach also for this physical phenomenon.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.
When the space of paths is a certain Hilbert space H, we show how to extend the Feynman path integral ℱ of DeWitt and Albeverio and Hoegh-Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of ℱ and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.