The declared prescribed dipole-drive model and its long-wavelength regime

(a) Adopted readings. Adopt the state and composition postulate P1, the observable and probability postulate P2 and the autonomous-dynamics postulate P4 (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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) and the static finite-level dipole postulate Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (i)-(iv), in the state and outcome 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 and Assigned sharp observables and the outcome law of the framework and in the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Read the finite-level matter data of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clause (b): the subspace $W\simeq {\mathbb{C}}^d$, the compressed matter energy $H_a$ (joules) and the compressed dipole matrices $D_1^W,D_2^W,D_3^W$ ($\mathrm{C}\mathrm{m}$).

(b) Prescribed field and its regime. Fix a nonempty compact interval $I=[t_0,t_1]\subset \mathbb{R}$ of physical times (seconds) and a prescribed electric field family $t\mapsto {\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(t)=(E_{\mathrm{e}\mathrm{x}\mathrm{t},1}(t),E_{\mathrm{e}\mathrm{x}\mathrm{t},2}(t),E_{\mathrm{e}\mathrm{x}\mathrm{t},3}(t))$, a continuous map $I\to {\mathbb{R}}^3$ whose components are read in $\mathrm{V}{\mathrm{m}}^{-1}$. The field is prescribed: no Maxwell equation, source model, propagation, back-reaction or feedback of the matter on the source is assumed, and no claim is made that a laboratory arrangement realizes it.

(c) The drive Hamiltonian. The declared dipole-drive model is the finite-dimensional family $H_{\mathrm{d}}(t):=H_a-{\sum }_{j=1}^3{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t)D_j^W,t\in I,$ read as the model's energy in joules, on the matter factor $W\simeq {\mathbb{C}}^d$; its control is the bounded self-adjoint family $V(t):=-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t)D_j^W$ with $\mathrm{\parallel }V(t)\mathrm{\parallel }\le ({\sum }_{j=1}^3\mathrm{\mid }{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}({\sum }_{j=1}^3\mathrm{\parallel }{D}_j^W{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}<\mathrm{\infty },$ so $t\mapsto H_{\mathrm{d}}(t)$ is norm-continuous on $I$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, A bounded linear operator between normed spaces). The model therefore declares the driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clause (b)(i) in its bounded norm-continuous class, with propagator $U(t,s)$ the unique norm-continuous solution of the integral equation $U(t,s)=I-\frac{i}{\mathrm{\hslash }}{\int }_s^t{H}_{\mathrm{d}}(r)U(r,s)dr,U(s,s)=I,$ constructed by the Dyson series of Norm Dyson series and uniqueness. If $t_0<t_1$, it satisfies $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)=H_{\mathrm{d}}(t)U(t,s)$ in operator norm, with one-sided derivatives at the endpoints. If $t_0=t_1$, the evolution declaration is only $U(t_0,t_0)=I$ and asserts no time derivative; state and outcome readings are those of clauses (d)-(e) of the driven model, with level populations read through the eigenbasis of $H_a$.

(d) Long-wavelength regime. Declare a characteristic matter length $L>0$ (metres), a reference point $x_0\in {\mathbb{R}}^3$ (metres), a wavevector $k\in {\mathbb{R}}^3$ (inverse metres), and a dimensionless tolerance $0<\delta <1$ as model data. A prescribed spatial phase template for the field of clause (b) is the continuous map ${\mathbf{E}}^{\mathrm{s}\mathrm{p}}:{\mathbb{R}}^3\times I\to {\mathbb{R}}^3,{\mathbf{E}}^{\mathrm{s}\mathrm{p}}(x,t):=\mathrm{Re}({\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(t)e^{ik\cdot (x-x_0)}),$ with components in $\mathrm{V}{\mathrm{m}}^{-1}$, so ${\mathbf{E}}^{\mathrm{s}\mathrm{p}}(x_0,t)={\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(t)$. This template specifies a spatial phase without assuming Maxwell dynamics or a dispersion relation. The long-wavelength (electric-dipole) regime at tolerance $\delta$ means $\mathrm{\mid }k\mathrm{\mid }L\le \delta$: for $\mathrm{\mid }x-x_0\mathrm{\mid }\le L$, the phase obeys $\mathrm{\mid }k\cdot (x-x_0)\mathrm{\mid }\le \delta$. The model adopts the neglect of this spatial phase and uses the reference-point field in the interaction $-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}(t)D_j^W$ of clause (c). The case $k=0$ is the spatially uniform template. The length $L$ is a declared scale, not a compact-support assertion for the matter wavefunctions. The notation $\mathrm{\mid }k\mathrm{\mid }L\ll 1$ denotes choosing a small declared tolerance; no universal tolerance or error estimate for the induced quantum dynamics is asserted. This regime is an adopted model assumption, and identifications of $L$, $k$ and the field with laboratory data are tested identifications with uncertainty under P7.

(e) Scope. Clauses (b)-(d) are adopted model data and readings of a declared finite-dimensional matter model with a prescribed classical field. The model yields stimulated transition calculations through clause (c) and the mathematics of the driven model; it contains no quantized field, no photon number, no vacuum state and no detector, and a prescribed field of this kind does not by itself supply spontaneous emission: the prescribed source is unaffected by the matter's transitions, in contrast to the quantized light-matter model of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, where the field's vacuum and the matter's change of level are one coupled model. No rate, lifetime, linewidth, decay or empirical result is asserted.