The Gaussian ground state: position and momentum laws and the saturated bound

Given: the model of The quadratic-potential oscillator model of the framework with parameters $m,\omega ,\mathrm{\hslash }>0$, length $\mathrm{\ell }=\sqrt{\mathrm{\hslash }\mathrm{/}(m\omega )}$, energy operator $H_1$, eigenbasis $({\psi }_n)$ and ground vector ${\psi }_0$; the operators $Q$ and $P$ with their domains.

The ground vector from the scaling theorem. With $g_0(y)={\pi }^{-1\mathrm{/}4}{e}^{-y^2\mathrm{/}2}$ and $D$ the scaling unitary of Oscillator scaling and exact realization, one has ${\psi }_0=D{g}_0$, ${\psi }_0(x)={\mathrm{\ell }}^{-1\mathrm{/}2}{g}_0(x\mathrm{/}\mathrm{\ell })={\mathrm{\ell }}^{-1\mathrm{/}2}{\pi }^{-1\mathrm{/}4}{e}^{-x^2\mathrm{/}(2{\mathrm{\ell }}^2)}$, $\mathrm{\parallel }{\psi }_0\mathrm{\parallel }=1$, and $H_1{\psi }_0=\frac{\mathrm{\hslash }\omega }{2}{\psi }_0$; moreover ${\psi }_0={\mathrm{\Psi }}_0$ is the coherent state of label $0$ (Oscillator scaling and exact realization clauses (a), (b); Hermite recurrence clause (a); The coherent states of the oscillator model clause (c); The energy observable of the oscillator model: spectrum, eigenbasis and outcome law clause (a)).

The position law. $(Q,\mathrm{m})$ is an assigned sharp observable of the model with $Q=M_x$ on $D(Q)=\{f:xf\in L^2\}$ and law ${\mu }_{\psi }^Q(B)=\mathrm{\parallel }{E}_Q(B)\psi {\mathrm{\parallel }}^2$; for the real multiplier $m(x)=x$ on the $\sigma$-finite space $(\mathbb{R},\text{Lebesgue})$ the assignment $E(B)=M_{{\mathbf{1}}_{x^{-1}(B)}}$ is the spectral projection-valued measure of $M_x$ and $\mathrm{\parallel }E(B)\psi {\mathrm{\parallel }}^2={\int }_{x^{-1}(B)}\mathrm{\mid }\psi {\mathrm{\mid }}^{2}dx$ (The quadratic-potential oscillator model of the framework clause (b); Assigned sharp observables and the outcome law of the framework clause (b); Spectral measure of a real L2 multiplication operator clauses (b), (c)).

Heat kernel and its moments. For $t>0$ and $n=1$, $\mathrm{\Gamma }(x,t)=(4\pi t{)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}(4t)}$ satisfies $\int \mathrm{\Gamma }(x,t)dx=1$, and all first and second moments are absolutely integrable with $\int x\mathrm{\Gamma }(x,t)dx=0$ and $\int x^2\mathrm{\Gamma }(x,t)dx=2t$ (The heat kernel on ${\mathbb{R}}^n$ and its causal extension, Normalisation, parabolic scaling, heat equation and derivative bounds for the heat kernel clause (i), First and second Gaussian heat-kernel moments).

The momentum law. ${\mathcal{F}}_2(P\psi )(\xi )=2\pi \mathrm{\hslash }\xi {\mathcal{F}}_2\psi (\xi )$ for $\psi \in D(P)$, and $D(P)=\{\psi :\xi {\mathcal{F}}_2\psi \in L^2\}$; with $g(\xi ):=2\pi \mathrm{\hslash }\xi$ and $M_g$ the multiplication operator on $L^2(\mathbb{R},d\xi )$ one has $P={\mathcal{F}}_2^{-1}{M}_g{\mathcal{F}}_2$ with $D(P)={\mathcal{F}}_2^{-1}D(M_g)$, so by the transport lemma the spectral projection-valued measure of $P$ is $E_P(B)={\mathcal{F}}_2^{-1}{E}_{M_g}(B){\mathcal{F}}_2$ and, for a unit $\psi$, ${\mu }_{\psi }^P(B)=\mathrm{\parallel }{E}_{M_g}(B){\mathcal{F}}_2\psi {\mathrm{\parallel }}^2$; the multiplication lemma applied to $M_g$ gives $E_{M_g}(B)=M_{{\mathbf{1}}_{g^{-1}(B)}}$ and ${\mu }_{\psi }^P(B)={\int }_{g^{-1}(B)}\mathrm{\mid }{\mathcal{F}}_2\psi (\xi ){\mathrm{\mid }}^{2}d\xi$ (Componentwise position and momentum operators on the component-valued wave-function space clause (b) at $d=1$, $r=1$; Transport of spectral projection-valued measures along isometric bijections clauses (b), (c); Spectral measure of a real L2 multiplication operator clauses (b), (c)).

Gaussian Fourier transform and linear change of variables. ${\mathcal{F}}_2(e^{-\pi t{x}^2})(\xi )=t^{-1\mathrm{/}2}{e}^{-\pi {\xi }^2\mathrm{/}t}$ for $t>0$ (Euclidean Gaussian transform with the 2π normalization); and for the $C^1$ diffeomorphism $T(\xi )=2\pi \mathrm{\hslash }\xi$ of $\mathbb{R}$ and $h\in L^1$, ${\int }_{\mathbb{R}}h(p)dp={\int }_{\mathbb{R}}h(2\pi \mathrm{\hslash }\xi )2\pi \mathrm{\hslash }d\xi$, so ${\int }_{\mathbb{R}}h(2\pi \mathrm{\hslash }\xi )d\xi =(2\pi \mathrm{\hslash }{)}^{-1}{\int }_{\mathbb{R}}h(p)dp$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

Means, variances and spreads. For an assigned sharp observable $(A,u_A)$ and a unit vector $\psi \in D(A)$ the outcome-law mean and second moment exist and equal $⟨A\psi ,\psi ⟩$ and $\mathrm{\parallel }A\psi {\mathrm{\parallel }}^2$, and the preparation spread is the square root of the outcome-law variance $\int (\lambda -m{)}^{2}d{\mu }_{\psi }^A$; for the coherent preparations of the model the identity $\mathrm{\Delta }Q\mathrm{\Delta }P=\mathrm{\hslash }\mathrm{/}2$ is known (Mean, second moment and variance of an assigned observable in a preparation clauses (a), (b); Preparation variance and spread of an assigned sharp observable clauses (b), (c); Fluctuations of the oscillator coherent states and the position-momentum relation of the model clause (a)).