Scope and empirical status of the oscillator page

Adopted. The page adopts the framework postulates P1, P2 and P4 and the massive-particle postulates P5 in the readings fixed by the framework pages, and it adopts the quadratic-potential oscillator model of The quadratic-potential oscillator model of the framework as a model assumption: a declared spinless particle, a one-dimensional realization with state space $L^2(\mathbb{R},dx)$, an isotropic three-dimensional realization with state space $L^2({\mathbb{R}}^3,dx)$, model parameters $m>0$ (kg), $\omega >0$ (${\mathrm{s}}^{-1}$) and action constant $\mathrm{\hslash }>0$ ($\mathrm{J}\mathrm{s}$), the assigned energy observables $(H_1,\mathrm{J})$ and $(H_3,\mathrm{J})$ with the additive potential-energy constant $V(0)$ subtracted, and the assigned dimensionless excitation-number observable $(N_{\psi },1\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t})$. The coherent-state family of The coherent states of the oscillator model is a further declared one-parameter family of preparations of the one-dimensional realization.

Proved. The energy observable of the oscillator model: spectrum, eigenbasis and outcome law gives the exact eigenvalue list $\mathrm{\hslash }\omega (n+\frac{1}{2})$ under the potential-energy convention subtracting $V(0)$, with ground energy $\mathrm{\hslash }\omega \mathrm{/}2$, the one-dimensional eigenspaces $\mathbb{C}{\psi }_n$, completeness of the Hermite eigenbasis, the excitation operators and assigned number observable, the number and atomic energy outcome laws, and the isotropic three-dimensional spectrum $\mathrm{\hslash }\omega (M+\frac{3}{2})$ with multiplicities $(M+1)(M+2)\mathrm{/}2$; Fluctuations of the oscillator coherent states and the position-momentum relation of the model computes the coherent position and momentum spreads $\mathrm{\ell }\mathrm{/}\sqrt{2}$ and $\mathrm{\hslash }\mathrm{/}(\mathrm{\ell }\sqrt{2})$, proves the model's position-momentum relation $\mathrm{\Delta }Q\mathrm{\Delta }P\ge \mathrm{\hslash }\mathrm{/}2$ on the common domain and the saturation of the bound in every coherent preparation; and Evolution of the oscillator coherent states: classical mean motion and stationary fluctuations proves that the flow carries the coherent preparation $[{\mathrm{\Psi }}_z]$ to $[{\mathrm{\Psi }}_{z{e}^{-i\omega t}}]$, that the preparation means perform the classical harmonic motion of frequency $\omega$, that the spreads stay saturated, and that the coherent energy law is the affine image of the Poisson law of parameter $\mathrm{\mid }z{\mathrm{\mid }}^2$ under $n\mapsto \mathrm{\hslash }\omega (n+\frac{1}{2})$ with spread $\mathrm{\hslash }\omega \mathrm{\mid }z\mathrm{\mid }$. All of these are conditional model deductions from the adopted assumptions and the cited mathematics; they are not statements that any physical system is harmonic.

What is empirical here. Nothing on the page is a measured value. No numerical value of $m$, $\omega$ or $\mathrm{\hslash }$ is asserted beyond the defining constants of the SI recorded in Dimensions, SI unit conventions and quantity types of the framework, no spectral line or resonance is identified with an eigenvalue difference, and no laboratory preparation is identified with a coherent state. The adopted potential term is the leading quadratic term of $V(x)-V(0)$ for a $C^2$ real-valued potential-energy function $V:\mathbb{R}\to \mathbb{R}$ with a non-degenerate minimum at $0$; the approximation's range, the deviations at finite amplitude and the transition to anharmonic behaviour are not derived here. The identification of the model's observables with apparatus readings is a tested identification under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions); agreement or disagreement of a laboratory oscillator with the model's predictions would test the combined model and apparatus assumptions and would not settle the framework's postulates.

The Fourier-algebraic closure that this page consumes -- the diagonal operator, the Hermite family and its completeness, the scaling realization, the closed ladder domains and the coherent-vector convergence -- is authored on the mathematical page qm-math-oscillator. The one-dimensional non-normalizable formal solution of the ground-state equation and the ground-energy unit reading are authored on its examples companion as the formal solution example and the ground-energy unit example; this page links to them rather than duplicating them. The displacement-operator presentation of coherent states, overcompleteness and resolutions of unity, driven and damped oscillators, time-dependent parameters, anharmonic corrections, coupling to electromagnetic fields or radiation and the identification of oscillator excitations with photons are separate model assumptions and are not developed here. Downstream pages that consume this page's items inherit the model scope stated in The quadratic-potential oscillator model of the framework clause (e) and the conditional character of every derived claim.

The mathematical suppliers used here are the items of the run's mathematical oscillator page qm-math-oscillator and the published framework and spectral items cited in each item's contract; the two local lemmas Eigenspaces of the diagonal operator and Transport of spectral projection-valued measures along isometric bijections are proved on this page. Every proof item records its exact hypotheses, and no claim of this page is unconditional on the adopted postulates and model assumptions.