Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2, the ideal conditional-measurement postulate P3 and the preparation and apparatus discipline P7 (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, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), the finite-level cutoff dipole model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments and the declared ideal counting and energy-window detection models The declared ideal counting, energy-window and analyser detection models of the radiation sector, in the readings of The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity, Assigned sharp observables and the outcome law of the framework and Declared ideal measurements and instruments of the framework, with the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Declared level pair and transition energy. A declared level pair $(a,b)$ of the model is an ordered pair of orthonormal eigenvectors $v_a,v_b\in {\mathbb{C}}^d$ of the compressed matter energy $H_a$ (Orthonormal families, complete orthonormal systems and Hilbert bases, Real and complex inner product spaces, with the inner product linear in the first argument). Its transition energy is $\mathrm{\Delta }:=E_a-E_b$, read in joules; the pair is upward when $\mathrm{\Delta }>0$, meaning level $a$ lies above level $b$ in the model's energy ordering.

(c) Dipole coefficients and single-vertex photon density. For an upward declared level pair $(a,b)$ and the couplings $D_j^{\mathrm{c}\mathrm{h}},g_j,{\eta }_0$ of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clauses (c),(d), the model's dipole coefficients are the complex numbers $c_j:=⟨D_j^{\mathrm{c}\mathrm{h}}{v}_a,v_b{⟩}_{{\mathbb{C}}^d}(j=1,\dots ,J),$ the model's single-vertex mode vector is $M:={\sum }_{j=1}^J{c}_j{g}_j\in \mathfrak{h},$ and the model's single-vertex photon density is the finite Borel measure on $\mathbb{R}$ ${\nu }_b(B):={\int }_{{\mathbb{R}}^3}{\mathbf{1}}_B(\epsilon (k))\mathrm{\parallel }M(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2{d}^{3}k,B\subseteq \mathbb{R}\text{ Borel},$ read in the model's units and supported in $(0,\mathrm{\Lambda }]$. The pairing is linear in its first argument and conjugate-linear in its second. Thus $c_j$ is the actual coefficient of $v_b$ in $D_j^{\mathrm{c}\mathrm{h}}{v}_a$, and Hermiticity gives $⟨D_j^{\mathrm{c}\mathrm{h}}{v}_b,v_a⟩=\overline{c_j}$. These are exactly the coefficients of Golden-rule coefficient approximate identity clause (c), with no conjugation or change of the transverse mode vectors; its mode vector and measure are the present $M$ and ${\nu }_b$.

(d) Single-vertex emission vector and coefficient. For $t\ge 0$ the model's single-vertex emission vector is the element of $\mathfrak{h}$ $A_b(\cdot ,t):k\mapsto -\frac{i{\eta }_0}{\mathrm{\hslash }}M(k){\int }_0^t{e}^{i(\epsilon (k)-\mathrm{\Delta })s\mathrm{/}\mathrm{\hslash }}ds,$ and the model's single-vertex emission coefficient is ${\gamma }_b(t):=\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}_{\mathfrak{h}}^2\mathrm{.}$ The vector and the coefficient are defined model quantities: the model assigns them through this clause and through the cutoff dipole mathematics, and they are not asserted to be the exact probability of any measurement of the model. In particular ${\gamma }_b(t)$ is dimensionless and is not a rate; the reading of a rate coefficient requires a separate limit statement, which is not part of this definition.

(e) Normalised single-vertex photon-energy law and ideal detection. For $t>0$ and $M\mathrm{\ne }0$ the model's normalised single-vertex photon-energy law is the Borel probability measure on $\mathbb{R}$ $P_{b,t}(B):=\frac{{\int }_B{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)}{{\int }_{\mathbb{R}}{K}_t(E-\mathrm{\Delta })d{\nu }_b(E)},K_t(x):=\frac{1}{{\mathrm{\hslash }}^{2}t}\mid {\int }_0^t{e}^{ixs\mathrm{/}\mathrm{\hslash }}ds{\mid }^2,$ whose denominator is finite and strictly positive by The first-order emitted-photon energy law: factorised kernel, normalisation, concentration at the transition energy and positivity on positive-measure windows clause (b). This probability law is defined independently of ${\eta }_0$, including ${\eta }_0=0$. Its amplitude-based preparation reading additionally requires $A_b(\cdot ,t)\mathrm{\ne }0$. For $t>0$ and $M\mathrm{\ne }0$, this is equivalent to ${\eta }_0\mathrm{\ne }0$, since $\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}^2={\eta }_0^{2}t\int K_t(E-\mathrm{\Delta })d{\nu }_b(E)$ and the integral is positive. Under this condition the declared single-vertex one-photon preparation is the unit vector $v_b\otimes (A_b(\cdot ,t)\mathrm{/}\mathrm{\parallel }{A}_b(\cdot ,t){\mathrm{\parallel }}_{\mathfrak{h}})$ of ${\mathbb{C}}^d\otimes \mathfrak{h}\subseteq {\mathbb{C}}^d\otimes F(\mathfrak{h})$, and for a Borel window $B\subseteq [0,\mathrm{\infty })$ the declared ideal energy-window detector of window $B$ of The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (c) has the Born probability $P_{b,t}(B)$ in that preparation, because the energy law of a unit one-photon vector $f$ is the push-forward of $\mathrm{\parallel }f(k){\mathrm{\parallel }}_{{\mathbb{C}}^3}^2{d}^{3}k$ under $k\mapsto \epsilon (k)$ (The radiation sector of the framework: modes and the number, energy and momentum assignments clause (e)). When ${\eta }_0=0$, $A_b(\cdot ,t)=0$ and no preparation obtained by normalizing that emission amplitude is declared; the formal law $P_{b,t}$ remains the probability measure defined above. A detected photon energy is therefore a spectral reading of the assigned free-field energy observable $(H_f,\mathrm{J})$, not a sharp energy of a normalisable state and not a quantum number of the model's mode labels.

(f) Scope. Clauses (b)-(e) fix the model's spectral vocabulary. They assert no sharp spectral line: the energy law of clause (e) is spread over the support of ${\nu }_b$. A point mass at $\mathrm{\Delta }$ (the probability measure assigning ${\mathbf{1}}_B(\mathrm{\Delta })$ to a Borel set $B$) is the limiting idealisation of The first-order emitted-photon energy law: factorised kernel, normalisation, concentration at the transition energy and positivity on positive-measure windows clause (c) only under its hypotheses: ${\nu }_b=J_b\lambda$, where $\lambda$ is Lebesgue measure on $\mathbb{R}$ and $J_b$ is an integrable nonnegative density representative vanishing almost everywhere off $(0,\mathrm{\Lambda }]$, continuous at $0<\mathrm{\Delta }<\mathrm{\Lambda }$ and satisfying $J_b(\mathrm{\Delta })>0$. Under these hypotheses, $\int fd{P}_{b,t}\to f(\mathrm{\Delta })$ as $t\to \mathrm{\infty }$ for every bounded continuous $f:\mathbb{R}\to \mathbb{C}$; no such limit is asserted for other allowed level pairs or densities. In particular, if $\mathrm{\Delta }>\mathrm{\Lambda }$, the finite-time laws assign all mass to $(0,\mathrm{\Lambda }]$ and cannot converge to the point mass at $\mathrm{\Delta }$. They supply no spectrometer, no calibration, no detector response beyond the declared ideal window, no efficiency, no background, no sampling model, no count and no measurement record; the identification of a laboratory spectrometer with a declared window is a tested identification with uncertainty under P7. The single-vertex quantities of clauses (d)-(e) are conditional model data; their relation to the model's exact outcome probabilities is a separate proved statement.