The coherent states of the oscillator model

(a) Setting. Adopt the state and composition postulate P1 in the reading of The state space of the framework: preparations, pure rays and density operators (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and the one-dimensional oscillator model of The quadratic-potential oscillator model of the framework clauses (a), (b), (c), with parameters $m>0$, $\omega >0$, action constant $\mathrm{\hslash }>0$ and length scale $\mathrm{\ell }=\sqrt{\mathrm{\hslash }\mathrm{/}(m\omega )}$. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $(g_n{)}_{n\in \mathbb{N}}$ be the Hermite family of $L^2(\mathbb{R},dy)$, let $D:L^2(\mathbb{R},dy)\to L^2(\mathbb{R},dx)$ be the scaling unitary and $({\psi }_n)$ the scaled eigenbasis of Oscillator scaling and exact realization clauses (a), (b), with ${\psi }_n=D{g}_n$; let $a_{\psi }=Da{D}^{-1}$ be the transported annihilation operator of Closed ladder domains, and let $N_{\psi }=DN{D}^{-1}$ be the model's assigned excitation-number observable of The quadratic-potential oscillator model of the framework clause (c), with its stated domain and unit $1$.

(b) The family. For $z\in \mathbb{C}$ let ${\psi }_z^{(y)}$ be the coherent vector of Coherent-state convergence clause (a), ${\psi }_z^{(y)}:=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{n=0}^{\mathrm{\infty }}\frac{z^n}{\sqrt{n!}}{g}_n\in L^2(\mathbb{R},dy),$ the series converging in $L^2(\mathbb{R},dy)$, and define the model's coherent state with label $z$ by ${\mathrm{\Psi }}_z:=D{\psi }_z^{(y)}=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_{n=0}^{\mathrm{\infty }}\frac{z^n}{\sqrt{n!}}{\psi }_n\in {\mathcal{H}}_1\mathrm{.}$ The family $({\mathrm{\Psi }}_z{)}_{z\in \mathbb{C}}$ is the model's coherent-state family, and $z$ is its amplitude-phase label.

(c) Basic properties. The following properties of the family are those established in the cited items and are recorded here as part of the definition's reading: the series for ${\mathrm{\Psi }}_z$ converges and $\mathrm{\parallel }{\mathrm{\Psi }}_z\mathrm{\parallel }=1$, so by P1 the ray $[{\mathrm{\Psi }}_z]$ is a pure preparation of the model when it is declared as one; ${\mathrm{\Psi }}_z\in D(N_{\psi }^k)$ for every $k\in \mathbb{N}$; $a_{\psi }{\mathrm{\Psi }}_z=z{\mathrm{\Psi }}_z$, so ${\mathrm{\Psi }}_z$ is a unit eigenvector of the annihilation operator with eigenvalue $z$; $⟨{\mathrm{\Psi }}_z,{\mathrm{\Psi }}_w⟩=e^{-(\mathrm{\mid }z{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2)\mathrm{/}2+z\bar{w}}$, with modulus $e^{-\mathrm{\mid }z-w{\mathrm{\mid }}^2\mathrm{/}2}$ for all $z,w$, so distinct labels define distinct preparations (their overlap modulus is less than $1$, whereas unit vectors on the same ray have overlap modulus $1$) and no two members of the family are orthogonal; and ${\mathrm{\Psi }}_0={\psi }_0$ is the ground eigenvector of the energy operator. (Coherent-state convergence clauses (a), (b), (c), (d), (e), transported by the unitary $D$ of Oscillator scaling and exact realization clause (a).)

(d) Reading of the label. The label $z$ is dimensionless (it is a complex number, not a quantity with a physical dimension). Its amplitude is $\mathrm{\mid }z\mathrm{\mid }$; when $z\mathrm{\ne }0$, its phase is the class $\arg z\in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, while $z=0$ has zero amplitude and no phase label. The mathematical coherent-state series gives the quadratic-form value $⟨N_{\psi }{\mathrm{\Psi }}_z,{\mathrm{\Psi }}_z⟩=\mathrm{\mid }z{\mathrm{\mid }}^2$ for the assigned, dimensionless excitation-number observable $(N_{\psi },1)$; this is the number coordinate intended by the model's P2 reading. Under the model's flow, Coherent evolution and variance clauses (c), (e), (f) gives $U(t){\mathrm{\Psi }}_z=e^{-i\omega t\mathrm{/}2}{\mathrm{\Psi }}_{z{e}^{-i\omega t}}$ and the position and momentum means $\sqrt{2}\mathrm{\ell }\mathrm{Re}(z{e}^{-i\omega t})$ and $\sqrt{2}\mathrm{\hslash }\mathrm{Im}(z{e}^{-i\omega t})\mathrm{/}\mathrm{\ell }$, respectively; thus the mean pair, in coordinates scaled by $\sqrt{2}\mathrm{\ell }$ and $\sqrt{2}\mathrm{\hslash }\mathrm{/}\mathrm{\ell }$, rotates with the phase label at angular frequency $\omega$. Clause (d) records this interpretation of the label, not an outcome of measuring the label itself.

(e) Scope. The definition declares a one-parameter family of preparations of the one-dimensional model and fixes its notation. It does not assert that the coherent states are the only preparations with time-independent position or momentum spreads, it does not assert minimality of any uncertainty product (that is a separate result for this family), it does not construct a displacement operator, an overcompleteness relation or a resolution of unity, and it makes no measurement, apparatus or empirical claim. The two presentations of the family -- as eigenvectors of the annihilation operator, as in clause (b), and as translated ground states -- are not identified here; only the former is defined.

• Why the label is not an observable. The label $z$ indexes the family; it is not the outcome of a measurement of any assigned observable of the model. Statements about the distribution of $z$ would require a separate preparation model for the labels, which this definition does not introduce.
• Not an orthonormal family. The inner products of clause (c) show that no two distinct coherent states are orthogonal, so the family is not an orthonormal basis and its only energy eigenstate is ${\mathrm{\Psi }}_0={\psi }_0$: for $z\mathrm{\ne }0$ the coefficients of both ${\psi }_0$ and ${\psi }_1$ are nonzero, whereas each energy eigenspace is a single basis ray by the scaling theorem and the pairwise distinct energy eigenvalues. The energy eigenbasis remains $({\psi }_n)$; the coherent states are superpositions with Poisson-type weights, a fact used by the fluctuation and evolution items of this page.