Malus and Jones ratios of the declared analyser: the same cos^2 from the Born law and from a classical transverse amplitude

Adopt the state and composition postulate P1, the observable and probability postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes), the radiation extension postulate clause (i) (Radiation extension of the framework: transverse mode and photon assignments) and the declared fixed-mode polarisation model The fixed-mode polarisation model of the framework with polarisation space $V$, orthonormal basis $(e_1,e_2)$ and analyser projections $P_{\theta }$. Assume the Axioms of Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The Axiom of Choice). Fix an angle $\theta \in \mathbb{R}$. This example compares two declared models of the same analyser:

(a) The quantised-model ratio. Consider the pure one-photon polarisation preparation of the ray $[e_1]$ and the declared ideal analyser at angle $\theta$. By The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio the outcome law of the analyser is $p_{+1}=\mathrm{\parallel }{P}_{\theta }{e}_1{\mathrm{\parallel }}^2={\cos }^2\theta ,p_{-1}={\sin }^2\theta ,$ so the transmitted Born probability is ${\cos }^2\theta$ in the unit $1$ of dimensionless probabilities.

(b) The declared classical transverse-amplitude model. Declare the following elementary model, with no claim that it is Maxwell theory: its field data are complex vectors $E\in V$ whose coordinates are dimensionless amplitude ratios relative to one fixed reference amplitude $E_{\mathrm{r}\mathrm{e}\mathrm{f}}$; no absolute field strength is assigned. The analyser at angle $\theta$ acts on the field data by the orthogonal projection $E⟼P_{\theta }E;$ the model's declared dimensionless relative intensity of a field datum is $I(E):=\mathrm{\parallel }E{\mathrm{\parallel }}^2,$ and the transmitted intensity fraction is $I(P_{\theta }E)\mathrm{/}I(E)$ for $E\mathrm{\ne }0$. This is a declared idealisation of the classical description of a polariser as projecting the transverse field amplitude (the Jones-vector reading); it imports no Maxwell equation and makes no detector claim.

The example computes both ratios for the same declared data and compares them.

• What is compared. Only the two declared models on the declared two-dimensional space are compared. No claim is made that the elementary classical model reproduces the photon counting statistics or the non-classical states of the radiation sector. Quantum count laws are computed in Number statistics of a single declared mode: sharp, Poisson and geometric preparations; A two-bin comparison of a declared classical field model and the quantised model: identical normalised laws do not discriminate states the additional classical count law, preparation and detector declarations required for a count-statistics comparison.
• Sources and status. The coincidence of the classical transmitted fraction with the quantum transmitted probability is explicitly noted in the cited source; the classical model here is an elementary declared idealisation constructed for the comparison and is not a derivation from Maxwell theory.