What the worked oscillator examples establish

Ground state. The Gaussian ground state: position and momentum laws and the saturated bound fixes the explicit Gaussian representative ${\psi }_0(x)={\mathrm{\ell }}^{-1\mathrm{/}2}{\pi }^{-1\mathrm{/}4}{e}^{-x^2\mathrm{/}(2{\mathrm{\ell }}^2)}$ of the ground vector, its unit norm, the eigenvalue equation $H_1{\psi }_0=\frac{\mathrm{\hslash }\omega }{2}{\psi }_0$, the position law with density ${\mathrm{\ell }}^{-1}{\pi }^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}{\mathrm{\ell }}^2}$ and variance ${\mathrm{\ell }}^2\mathrm{/}2$, the momentum law with density $\frac{\mathrm{\ell }}{\mathrm{\hslash }\sqrt{\pi }}{e}^{-{\mathrm{\ell }}^2{p}^2\mathrm{/}{\mathrm{\hslash }}^2}$ and variance ${\mathrm{\hslash }}^2\mathrm{/}(2{\mathrm{\ell }}^2)$, and the saturation ${\mathrm{\Delta }}_{{\psi }_0}Q{\mathrm{\Delta }}_{{\psi }_0}P=\mathrm{\hslash }\mathrm{/}2$ of the model's position-momentum relation.

Ladder and excitation structure. Ladder norms and excitation statistics of the oscillator eigenstates computes the first ladder images and their norms, the ladder matrix elements $⟨{\psi }_m,a_{\psi }{\psi }_n⟩=\sqrt{n}{\delta }_{m,n-1}$ and $⟨{\psi }_m,a_{\psi }^{†}{\psi }_n⟩=\sqrt{n+1}{\delta }_{m,n+1}$, and the excitation-number and energy laws ${\mu }_{{\psi }_n}^{N_{\psi }}(\{n\})=1$, ${\mu }_{{\psi }_n}^{H_1}(\{E_n\})=1$ of the eigenstate preparations; the number coordinate is read by the assigned observable $(N_{\psi },1\mathrm{/}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t})$, and $a_{\psi }$ and $a_{\psi }^{†}$ are not assigned sharp observables.

Coherent motion. A coherent-state trajectory: period, phase-space ellipse and constant spreads shows that a nonzero-label coherent preparation is periodic with period $2\pi \mathrm{/}\omega$, that its mean pair traces the invariant phase-space ellipse $\frac{x^2}{2{\mathrm{\ell }}^2}+\frac{{\mathrm{\ell }}^2{p}^2}{2{\mathrm{\hslash }}^2}=\mathrm{\mid }z{\mathrm{\mid }}^2$ with the quarter-period values $x(\frac{\pi }{2\omega })=\frac{p_0}{m\omega }$, $p(\frac{\pi }{2\omega })=-m\omega x_0$, and that the position and momentum spreads and their product $\mathrm{\hslash }\mathrm{/}2$ are constant along the motion.

Every computation above is conditional on the adopted oscillator model of The quadratic-potential oscillator model of the framework and on the postulates P1, P2, P4 and P5 read through it; none of them reports a measurement. In particular: no numerical value of $m$, $\omega$ or $\mathrm{\hslash }$ is extracted; no laboratory state is identified with ${\psi }_0$, with an eigenstate ${\psi }_n$ or with a coherent state; the mean functions $x(t),p(t)$ are preparation means of the model's observables under the evolving preparation and are not the coordinates of a measured trajectory; and the periodicity of clause (a) of the coherent-trajectory example is a statement about the pure preparation (a ray), not about a measured signal. The harmonic approximation's range, anharmonic corrections, damping, driving, time-dependent parameters and coupling to electromagnetic fields or radiation are outside the examples, as is the displacement-operator presentation of the coherent family. The formal non-normalizable solution of the ground-state equation is authored on the mathematical companion examples page as its formal solution example, and is linked there rather than duplicated here.