Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field

Adopt as a physical assumption of the pair, additional to P1, P2 and P4 and valid only for the declared finite-level model of this clause, the following static electric-dipole coupling. Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(i) Declared charged pair and relative coordinate. Take the declared two-body Coulomb model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (d): two distinguishable spinless particles with mass parameters $m_1,m_2>0$ and opposite charges $-q$ and $+q$, $q>0$ (coulombs), with the relative coordinate $r=x_1-x_2$ and the centre-of-mass/relative unitary $U_{\mathrm{\Phi }}$. Work in the relative-coordinate factor $L^2({\mathbb{R}}^3;\mathbb{C})$ of that reading, with Coulomb mass parameter $m=\mu :=m_1{m}_2\mathrm{/}(m_1+m_2)$, and with units as in Dimensions, SI unit conventions and quantity types of the framework.

(ii) Dipole operator. Let $Q_j$, $j=1,2,3$, be the coordinate multiplication operators on $L^2({\mathbb{R}}^3;\mathbb{C})$ with maximal domains $D(Q_j)=\{\psi :{\xi }_j\psi \in L^2\}$ of Atomic dipole matrix-element domain clause (a); they are self-adjoint and proper densely defined. The model's dipole operator is the triple $D_j:=-q{Q}_j(j=1,2,3),$ because the dipole of charges $-q$ at $x_1$ and $+q$ at $x_2$ is $-q{x}_1+q{x}_2=-qr$: it points from the negative to the positive charge, opposite to the declared relative coordinate $r=x_1-x_2$. Each $D_j$ is self-adjoint on $D(Q_j)$ and has unit $\mathrm{C}\mathrm{m}$. This is the dipole of the declared pair in its relative-coordinate reading; no multipole expansion, no field equation and no coupling to the centre-of-mass motion is assumed.

(iii) Finite-level truncation. Fix, as part of the model specification, a declared finite-dimensional subspace $W\subseteq D(H_{\kappa })\cap D(Q_1)\cap D(Q_2)\cap D(Q_3)$ of the relative-coordinate space with $D(H_{\kappa })=H^2({\mathbb{R}}^3;\mathbb{C})$ and $\dim W=d<\mathrm{\infty }$, together with an orthonormal basis of $W$. The model declares the compressed dipole matrix on $W$, $D_j^W:={\mathrm{\Pi }}_W{D}_j{\mid }_W,$ where ${\mathrm{\Pi }}_W$ is the orthogonal projection onto $W$ and the restriction to $W$ is read in the declared orthonormal basis; $D_j^W$ is the finite Hermitian $d\times d$ matrix of the dipole component, with unit $\mathrm{C}\mathrm{m}$, and it is bounded with norm at most ${\sup }_{\psi \in W,\mathrm{\parallel }\psi \mathrm{\parallel }=1}\mathrm{\parallel }{D}_j\psi \mathrm{\parallel }$. The truncation and the basis are model assumptions: they are not derived from the Coulomb model, and they replace the unbounded operator $D_j$ by its compression throughout this pair.

(iv) Static uniform field and interaction operator. The external datum is a uniform static electric field value ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\in {\mathbb{R}}^3$, read in $\mathrm{V}{\mathrm{m}}^{-1}$; it is prescribed, and no Maxwell equation, source model, propagation or back-reaction is assumed. The finite-level static Stark model is the finite-dimensional operator on $W$ $H_S({\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}):={\mathrm{\Pi }}_W{H}_{\kappa }{\mid }_W-{\sum }_{j=1}^3{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}{D}_j^W,$ where the energy-domain condition of (iii) makes the compression defined, and $H_{\kappa }=C_{\mu }$ is the one-centre Coulomb energy operator with mass parameter $m=\mu$ of the declared model, read on the relative-coordinate factor through the angular/radial identification used by the pair. The interaction term has unit $\mathrm{C}\mathrm{m}\cdot \mathrm{V}{\mathrm{m}}^{-1}=\mathrm{J}$ and is bounded Hermitian on $W$, so $H_S({\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}})$ is a bounded Hermitian $d\times d$ matrix and the Kato-Rellich and perturbation statements of the pair apply to it with a bounded perturbation; this identification of the matrix with the model's energy on the truncated subspace is a model assumption.

(v) Scope and explicit exclusion. Clauses (i)-(iv) are adopted model assumptions. They assert nothing about a laboratory atom, field or apparatus; no measured Stark shift is claimed; no transition, absorption or emission rate is defined; and no time dependence of the field is assumed. In particular the postulate deliberately does not treat the uncompressed linear potential $-{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{D}$ as a bounded perturbation of $H_{\kappa }$. For ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\ne }0$, define this composite interaction to be the maximal multiplication operator $M_{\mathbf{E}}\psi (\mathbf{r}):=q{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{r}\psi (\mathbf{r}),D(M_{\mathbf{E}}):=\{\psi \in L^2:({\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{r})\psi \in L^2\};$ it agrees with $-{\sum }_j{E}_{\mathrm{e}\mathrm{x}\mathrm{t},j}{D}_j$ on the common domain ${\bigcap }_{j}D(D_j)$ and is unbounded because its real multiplier is unbounded. The common operator domain $D(H_{\kappa })\cap D(M_{\mathbf{E}})$ contains $C_c^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{C})$. Its lack of a lower bound can be demonstrated without assuming a full energy-form domain: let ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\ne }0$, $\widehat{e}={\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{/}\mathrm{\mid }{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\mid }$, choose a fixed support radius $\rho >0$ in metres and normalized $\phi \in C_c^{\mathrm{\infty }}(B(0,\rho );\mathbb{C})$, and for each length $R>\rho$ put ${\psi }_R(\mathbf{r}):=\phi (\mathbf{r}+R\widehat{e})\mathrm{.}$ Then ${\psi }_R\in C_c^{\mathrm{\infty }}({\mathbb{R}}^3;\mathbb{C})\subset D(H_{\kappa })\cap D(M_{\mathbf{E}})\cap {\bigcap }_{j=1}^{3}D(Q_j)$ and $\mathrm{\parallel }{\psi }_R{\mathrm{\parallel }}_2=1$. On this common test domain define the quadratic expectation ${\mathcal{Q}}_{{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}}[\psi ]:=⟨H_{\kappa }\psi ,\psi ⟩+⟨M_{\mathbf{E}}\psi ,\psi ⟩\mathrm{.}$ With $\mathbf{s}=\mathbf{r}+R\widehat{e}$, the Coulomb action and the translation-invariant kinetic expectation give ${\mathcal{Q}}_{{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}}[{\psi }_R]=\frac{{\mathrm{\hslash }}^2}{2\mu }\int \mathrm{\mid }\mathrm{\nabla }\phi (\mathbf{s}){\mathrm{\mid }}^{2}d\mathbf{s}-\kappa \int \frac{\mathrm{\mid }\phi (\mathbf{s}){\mathrm{\mid }}^2}{\mathrm{\mid }\mathbf{s}-R\widehat{e}\mathrm{\mid }}d\mathbf{s}+q{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \int \mathbf{s}\mathrm{\mid }\phi (\mathbf{s}){\mathrm{\mid }}^{2}d\mathbf{s}-qR\mathrm{\mid }{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\mid }\mathrm{.}$ Here $-{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{D}=q{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{r}$ because $\mathbf{D}=-q\mathbf{r}$ in the relative-coordinate convention. For $R>\rho$, the Coulomb integral is finite and at most $1\mathrm{/}(R-\rho )$, the kinetic term is constant, and the other field term apart from $-qR\mathrm{\mid }{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\mathrm{\mid }$ is independent of $R$. Therefore ${\mathcal{Q}}_{{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}}[{\psi }_R]\to -\mathrm{\infty }$ as $R\to \mathrm{\infty }$. This proves unboundedness below of the displayed quadratic expectation on the common test domain, so no quadratic form bounded below there can agree with these values; no claim is made about a particular closed-form domain. The uncompressed static-field model is outside the scope of every statement of this pair. Only the compressed finite-level operator of clause (iv) is used.