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The declared two-body Coulomb model: reduced-mass decomposition, relative levels as thresholds and the absence of stationary total states

Physical scope

A conditional consequence of the declared two-body Coulomb model with masses m1, m2, opposite charge magnitudes q_- and q_+, and coupling kappa=q_-q_+/(4 pi epsilon_0), including its equal-magnitude neutral branch: the centre-of-mass/relative decomposition U_Phi H_2 U_Phi^(-1) = T_R tensor I + I tensor C_mu, the identification of the relative problem with the one-centre Coulomb model of mass mu, the relative gross-structure levels E_n^mu = -mu kappa^2/(2 hbar^2 n^2) with degeneracy n^2 within the relative problem, the fact that the total two-body Hamiltonian has no eigenvalues and continuous negative spectrum with the relative levels as channel thresholds, and the absence of normalisable total-momentum eigenstates on the composite L2(R6) state space. H2, T_R and C_mu are energy operators in joules; the lifted total-momentum components have unit kg m s^-1. The item asserts no measured level, no transition, no photon, no external-field or many-body effect and no empirical result; it does not model binding or formation dynamics and does not identify the parameters with measured masses or charges.

Statement

Adopt the framework postulates P1, P2, P4 and P5 and the readings of the declared two-body Coulomb model The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clauses (a), (b), (d) and (e) with declared masses $m_{1}, m_{2} > 0$ , opposite charge magnitudes $q_{-}, q_{+} > 0$ , and coupling $κ = q_{-} q_{+} / (4 π ε_{0})$ (the equal-magnitude branch $q_{-} = q_{+} = q$ has $κ = q^{2} / (4 π ε_{0})$ ), reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , the centre-of-mass/relative coordinate map $Φ$ , its unitary $U_{Φ}$ , the total operator $H_{2}$ on $H^{2} (R^{6}; C)$ , the centre-of-mass kinetic operator $T_{R}$ and the relative operator $C_{μ}$ , together with the one-centre results The one-centre Coulomb model: energy observable, flow, stationary states and spectral support and The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum applied with mass parameter $μ$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) the identity $U_{Φ} H_{2} U_{Φ}^{- 1} = T_{R} \otimes I + I \otimes C_{μ}$ holds on the tensor product, and $C_{μ}$ is exactly the one-centre Coulomb operator of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (b) with mass parameter $μ$ ; the model's relative problem is therefore the declared one-centre Coulomb model with parameters $(ℏ, μ, κ)$ ;

(b) the relative bound energies are $E_{n} : = - \frac{μ κ^{2}}{2 ℏ^{2} n^{2}} = - \frac{ℏ^{2}}{2 μ a_{μ}^{2} n^{2}}, a_{μ} : = \frac{ℏ^{2}}{μ κ} (n \geq 1),$ with model degeneracy $n^{2}$ within the relative problem and the same radial and position readings as in The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum and The one-centre Coulomb model: position and radial distributions of the bound states, with the ground-state density and its scales with $m$ replaced by $μ$ ;

(c) the total operator $H_{2}$ has no eigenvalues at all: $σ (H_{2}) = [- \frac{μ κ^{2}}{2 ℏ^{2}}, + \infty)$ , the whole negative part $[- \frac{μ κ^{2}}{2 ℏ^{2}}, 0)$ of the spectrum is continuous, and each relative bound energy $E_{n}$ is the lower endpoint of its continuous centre-of-mass channel $[E_{n}, + \infty)$ , hence a channel threshold rather than an eigenvalue of $H_{2}$ ;

(d) let $P_{R, j} : = - i ℏ \partial_{R_{j}}$ on the factor domain $D (P_{R, j}) = {φ \in L^{2} (R_{R}^{3}; C) : \partial_{R_{j}} φ \in L^{2} (R_{R}^{3}; C) as a weak derivative},$ and define the lifted total-momentum component on the composite state space by ${\tilde{P}}_{R, j} : = U_{Φ}^{- 1} (P_{R, j} \otimes I) U_{Φ}, D ({\tilde{P}}_{R, j}) = U_{Φ}^{- 1} {F \in L^{2} (R_{R}^{3} \times R_{r}^{3}; C) : \partial_{R_{j}} F \in L^{2} as a weak derivative};$ it is self-adjoint on this maximal weak-derivative domain, has momentum unit $k g m s^{- 1}$ , and has no nonzero eigenvectors in $L^{2} (R^{6}; C)$ . A state of definite total momentum would be a simultaneous eigenvector of the three lifted components, so the model has no such normalisable state. Since $H_{2}$ has no eigenvalues, for every $φ \in L^{2} (R_{R}^{3}; C)$ and nonzero relative bound vector $u$ with $C_{μ} u = E_{n} u$ , any nonzero tensor product $φ \otimes u$ in the transformed $(R, r)$ tensor space, equivalently $U_{Φ}^{- 1} (φ \otimes u)$ in the composite $L^{2} (R^{6})$ state space, is not an eigenvector; the stationary-state criterion therefore excludes its pure preparation as a stationary state;

(e) Reading. The reduced mass obeys $μ = \frac{m_{1}}{1 + m_{1} / m_{2}}$ and satisfies $μ \leq \min {m_{1}, m_{2}}$ with $μ \to m_{1}$ as $m_{2} \to \infty$ at fixed $m_{1}$ ; the model reads the one-centre model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (b) with mass $m_{1}$ as the fixed-nucleus limiting description, and the relative energies $E_{n}$ are strictly weaker in magnitude than the fixed-nucleus energies $- \frac{m_{1} κ^{2}}{2 ℏ^{2} n^{2}}$ ;

(f) Scope. The statement is conditional on P1, P2, P4, P5, the declared two-body model and the cited mathematics; it asserts no measured mass, charge or level, no transition, photon or radiation statement, no binding or formation dynamics, and no identification of the model with a laboratory atom beyond the tested identification of clause (e) of the model definition.

Facts & Assumptions

Given: The declared two-body Coulomb model with $m_{1}, m_{2}, ℏ > 0$ , opposite charge magnitudes $q_{-}, q_{+} > 0$ , coupling $κ = q_{-} q_{+} / (4 π ε_{0})$ , reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , the coordinate map $Φ$ , the unitary $U_{Φ}$ , the operators $T_{R}$ , $C_{μ}$ and $H_{2}$ ; the one-centre results with mass parameter $μ$ ; the Axiom of Countable Choice and the Axiom of Choice.

[F1]

Model data. $C_{μ} = - \frac{ℏ^{2}}{2 μ} Δ_{r} - \frac{κ}{∣ r ∣}$ on $H^{2} (R_{r}^{3}; C) = D (C_{μ})$ is exactly the one-centre Coulomb operator of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (b) with mass parameter $μ$ ; $H_{2}$ with $D (H_{2}) = H^{2} (R^{6}; C)$ is self-adjoint, and $Φ (x_{1}, x_{2}) = (R, r)$ is the centre-of-mass/relative coordinate map of clause (d) of the same definition (Two-body reduced mass).

[F2]

Unscaled relative Coulomb spectral representation. For $γ > 0$ , the mathematical operator $C_{γ} : = - Δ - γ / ∣ \cdot ∣$ on $H^{2} (R^{3}; C)$ is unitarily equivalent to the direct sum, over spherical channels $(l, m)$ , of its negative bound-state copies with values $- γ^{2} / (4 n^{2})$ for $n \geq l + 1$ and a continuum copy $L^{2} ((0, \infty), d λ)$ on which it is multiplication by $λ$ on the maximal domain ${h : λ h \in L^{2}}$ ; the negative family is complete and $0$ is not an eigenvalue (Onto radial continuum transform clause (d), Negative-subspace completeness and degeneracy).

[F3]

Spectral-measure transfer under positive scaling. If $A$ is self-adjoint with spectral PVM $E_{A}$ and $c > 0$ , the PVM of $c A$ is $E_{c A} (B) = E_{A} (c^{- 1} B)$ for every Borel $B$ (Positive scaling of a self-adjoint operator: the spectral projection-valued measure transforms by the preimage of the scaling map clauses (b), (c)).

[F4]

Dilation and center Fourier multiplier. For $c > 0$ , the map $D_{c} h (E) : = c^{- 1 / 2} h (E / c)$ is unitary from $L^{2} ((0, \infty), d λ)$ to the correspondingly scaled $L^{2} ((0, \infty), d E)$ ; explicitly $D_{c}^{- 1} g (λ) = c^{1 / 2} g (c λ)$ . The one-dimensional change-of-variables formula gives its norm identity, and direct substitution gives $D_{c} M_{f} D_{c}^{- 1} = M_{f (E / c)}$ for bounded Borel $f$ , where $M_{f}$ denotes the maximal multiplication operator (bounded here) by $f$ . On $L^{2} (R_{R}^{3})$ the $2 π$ -normalized Fourier transform sends $T_{R}$ to multiplication by $τ_{R} (ξ) : = 2 π^{2} ℏ^{2} ∣ ξ ∣^{2} / M$ and $P_{R, j}$ to multiplication by $2 π ℏ ξ_{j}$ . Their maximal multiplier domains are ${g : τ_{R} g \in L^{2}}$ and ${g : ξ_{j} g \in L^{2}}$ respectively; the corresponding real multiplication operators are self-adjoint on these maximal domains. Tensoring the transform with the identity gives the unitary partial Fourier transform on the composite $L^{2}$ space and the maximal weak-derivative domain for $P_{R, j} \otimes I$ . Spheres in $R_{ξ}^{3}$ are Lebesgue-null by the polar-coordinate formula; Tonelli applies to the $ξ$ and $E$ product measure (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions, Plancherel theorem, Distributional derivatives are polynomial Fourier multipliers, Weak derivative of a locally integrable function, Real multiplication operators and free plus bounded-potential Schrödinger operators, Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})).

[F5]

Two-body decomposition. $Φ$ is a linear isomorphism with $∣ \det Φ ∣ = 1$ , $U_{Φ}$ is unitary, and $U_{Φ} H_{2} U_{Φ}^{- 1} = T_{R} \otimes I + I \otimes C_{μ}$ under the identification $L^{2} (R^{6}) ≅ L^{2} (R_{R}^{3}) \hat{\otimes} L^{2} (R_{r}^{3})$ ; $H_{2}$ is self-adjoint on $H^{2} (R^{6}; C)$ and $C_{μ} = - (ℏ^{2} / 2 μ) Δ - κ / ∣ \cdot ∣$ is the one-centre Coulomb operator with mass parameter $μ$ (Two-body reduced mass clauses (a)-(c), Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b})).

[F6]

One-centre results with mass $μ$ . The one-centre Coulomb model with parameters $(ℏ, μ, κ)$ has bound energies $E_{n} = - \frac{μ κ^{2}}{2 ℏ^{2} n^{2}}$ , $n \geq 1$ , the eigenspace at $E_{n}$ having dimension $n^{2}$ and consisting of the transported Laguerre solutions, with the radial and position readings of the one-centre items (The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum clauses (a), (b), The one-centre Coulomb model: position and radial distributions of the bound states, with the ground-state density and its scales clauses (a), (c)).

[F7]

Stationary states. For an assigned energy observable, a nonzero vector is an eigenvector if and only if its pure preparation is stationary (Energy conservation, conserved sharp observables and the stationary eigenrays clause (c), Assigned energy observable, unitary flow and the two pictures of autonomous evolution).

Proof

technique · direct; transfer the exact unscaled relative PVM by the positive-scaling lemma and continuum dilation, derive the product-channel spectrum, and lift the centre-of-mass momentum through the coordinate unitary

Given: The data of the statement and [F1]-[F7].

1.1givenF1F5F6algebra

[Decomposition and relative problem] By [F1] and [F5], $U_{Φ} H_{2} U_{Φ}^{- 1} = T_{R} \otimes I + I \otimes C_{μ}$ and $C_{μ}$ is the one-centre Coulomb operator with mass parameter $μ$ ; with the model definition this proves (a). Applying [F6] at mass $μ$ gives the relative levels $E_{n} = - \frac{μ κ^{2}}{2 ℏ^{2} n^{2}} = - \frac{ℏ^{2}}{2 μ a_{μ}^{2} n^{2}}$ , $a_{μ} = ℏ^{2} / (μ κ)$ , degeneracy $n^{2}$ and the stated radial and position readings. This proves (b).

1.2F2F3F4algebra

[Scaled relative PVM and continuum dilation] Put $γ : = 2 μ κ / ℏ^{2} > 0$ and $c_{μ} : = ℏ^{2} / (2 μ) > 0$ , so $C_{μ} = c_{μ} C_{γ}$ . By [F2], a unitary spectral representation of $C_{γ}$ has bound channels with values $- γ^{2} / (4 n^{2})$ and continuum channels $L^{2} ((0, \infty), d λ)$ with maximal multiplier $M_{λ}$ . The PVM of the scaled operator is transferred by [F3], so on each continuum channel its projection for a Borel set $B$ is multiplication by $1_{c_{μ}^{- 1} B} (λ)$ . The unitary dilation of [F4], $(D_{c_{μ}} h) (E) : = c_{μ}^{- 1 / 2} h (E / c_{μ}),$ satisfies $D_{c_{μ}} M_{f} D_{c_{μ}}^{- 1} = M_{f (E / c_{μ})}$ for bounded Borel $f$ ; direct substitution on the maximal multiplier domain gives $D_{c_{μ}} (c_{μ} M_{λ}) D_{c_{μ}}^{- 1} = M_{E}$ , with domain ${g : E g \in L^{2} (d E)}$ . It sends the transferred projection $M_{1_{c_{μ}^{- 1} B} (λ)}$ to $M_{1_{B} (E)}$ . Each bound value scales to $c_{μ} (- γ^{2} / (4 n^{2})) = E_{n}$ . Hence the exact spectral representation of $C_{μ}$ consists of these bound channels and continuum copies with multiplier $E$ , with no zero eigenchannel. This explicitly transfers both the continuum representation and its PVM.

2.1step 1.1step 1.2F4F5algebra

[Product-channel representation and lower spectral bound] Apply the relative unitary of step 1.2 and the partial $2 π$ -normalized Fourier transform in $R$ to the tensor decomposition of [F5]. The transformed $H_{2}$ is multiplication on each bound channel $(n, l, m)$ by $τ_{R} (ξ) + E_{n}$ and on each continuum channel by $τ_{R} (ξ) + E$ , where $τ_{R} (ξ) : = 2 π^{2} ℏ^{2} ∣ ξ ∣^{2} / M \geq 0$ by [F4]. Since $E_{n} \geq E_{1} = - μ κ^{2} / (2 ℏ^{2})$ and continuum energies satisfy $E > 0$ , every channel multiplier is at least $E_{1}$ ; thus $H_{2} \geq E_{1} I$ .

2.2step 1.1F5algebra

[Reduced-mass factor] Algebra gives $μ = \frac{m_{1} m_{2}}{m_{1} + m_{2}} = \frac{m_{1}}{1 + m_{1} / m_{2}}$ , hence $μ < m_{1}$ and $μ < m_{2}$ , so $μ \leq \min {m_{1}, m_{2}}$ ; at fixed $m_{1}$ , $μ \to m_{1}$ as $m_{2} \to \infty$ . Comparing $E_{n} = - \frac{μ κ^{2}}{2 ℏ^{2} n^{2}}$ with the fixed-nucleus energy $- \frac{m_{1} κ^{2}}{2 ℏ^{2} n^{2}}$ , the relative binding magnitude is smaller by $μ / m_{1} = 1 / (1 + m_{1} / m_{2}) < 1$ . This proves (e).

3.1step 2.1F4F5algebra

[Every energy above threshold is spectral] Fix $Λ \geq E_{1}$ and choose $ξ_{0} \in R^{3}$ with $τ_{R} (ξ_{0}) = Λ - E_{1}$ ; such a point exists because $τ_{R}$ takes every value in $[0, \infty)$ . Choose normalized $χ_{k} \in C_{c}^{\infty} (R_{ξ}^{3})$ supported in balls of radius $1 / k$ about $ξ_{0}$ , and let $u_{1}$ be a normalized relative ground-state vector in the $(n, l, m) = (1, 0, 0)$ bound channel. Then $χ_{k} \otimes u_{1}$ lies in the product operator domain, has norm one, and $∥ ((H_{2} - Λ I) (χ_{k} \otimes u_{1})) ∥ \leq \sup_{∣ ξ - ξ_{0} ∣ \leq 1 / k} ∣ τ_{R} (ξ) - τ_{R} (ξ_{0}) ∣ ⟶ 0$ by continuity of $τ_{R}$ . If $H_{2} - Λ I$ had an everywhere-defined bounded inverse, then $1 \leq ∥ (H_{2} - Λ I)^{- 1} ∥ \cdot ∥ (H_{2} - Λ I) (χ_{k} \otimes u_{1}) ∥ \to 0$ , a contradiction. Thus every $Λ \geq E_{1}$ lies in $σ (H_{2})$ , and together with step 2.1 this gives $σ (H_{2}) = [E_{1}, \infty)$ .

4.1step 2.1step 3.1F4F5algebra

[No point spectrum and channel thresholds] Suppose an eigenvector at $Λ$ existed in the product representation. In a bound channel its Fourier component would vanish outside the level set ${ξ : τ_{R} (ξ) = Λ - E_{n}}$ ; this set is empty, a point, or a sphere, hence has Lebesgue measure zero by [F4]. In a continuum channel its component would be supported on ${(ξ, E) : E = Λ - τ_{R} (ξ)}$ ; every fixed- $ξ$ section in $E$ is empty or a singleton and has measure zero, so Tonelli [F4] makes this measurable graph null in the product measure. Every channel component is therefore zero, proving that $H_{2}$ has no eigenvalues. For real $Λ \in [E_{1}, 0)$ , the range of $H_{2} - Λ I$ is dense because its orthogonal complement is $\ker (H_{2} - Λ I)^{*} = \ker (H_{2} - Λ I) = {0}$ . If it were onto, the inverse map from $L^{2} (R^{6})$ to $D (H_{2})$ with the graph norm would have closed graph and would be bounded by Closed graph theorem; then $Λ$ would be in the resolvent as defined by Resolvent and spectrum of an unbounded operator, contradicting step 3.1. Thus $H_{2} - Λ I$ has dense but non-surjective range; this is the continuous spectrum of the self-adjoint operator, proving the continuous-spectrum clause of (c). Each relative bound energy $E_{n}$ is the lower endpoint of its bound channel $[E_{n}, \infty)$ , so it is a channel threshold and not a total eigenvalue. This completes (c).

5.1F4F5F7step 4.1algebra

[Lifted total momentum on the composite state space] In the partial Fourier representation of [F4], $P_{R, j}$ is the real multiplier $2 π ℏ ξ_{j}$ on its maximal domain. Therefore $P_{R, j} \otimes I$ is self-adjoint on the maximal weak-derivative domain in the product space, and conjugation by $U_{Φ}$ gives exactly ${\tilde{P}}_{R, j}$ and $D ({\tilde{P}}_{R, j})$ stated in (d). If ${\tilde{P}}_{R, j} Ψ = p Ψ$ , then for $F = U_{Φ} Ψ$ its partial Fourier transform satisfies $(2 π ℏ ξ_{j} - p) \hat{F} (ξ, r) = 0 for almost every (ξ, r) .$ It is supported on the hyperplane $ξ_{j} = p / (2 π ℏ)$ , which is null in $R_{ξ}^{3}$ ; Tonelli over the remaining momentum and relative variables gives $\hat{F} = 0$ , hence $Ψ = 0$ . The multiplier has momentum dimension $[ℏ] [ξ_{j}] = k g m s^{- 1}$ . This proves the total-momentum assertion of (d). Since step 4.1 excludes every eigenvector of $H_{2}$ , the stationary-state criterion [F7] also excludes a stationary preparation from any nonzero centre-of-mass/relative-bound product.

6.1step 1.1step 2.1step 3.1step 4.1step 5.1step 2.2∎

[Assembly and scope] Step 1.1 gives (a) and (b), steps 2.1-4.1 give (c), step 5.1 gives the total-momentum assertion of (d), and step 2.2 gives (e). Only the declared two-body model, P1, P2, P4, P5 and the cited mathematics were used, so (f) holds and no empirical identification, measured parameter, transition, photon or formation dynamics is asserted.

Remarks

Why the relative levels are not total energies. The centre-of-mass motion of an isolated two-body system is free in this model, so its operator has purely continuous spectrum; a relative bound state is an internal state and cannot be a stationary state of the total system. Any statement that a hydrogen atom "is" an eigenstate of a two-body Hamiltonian conflicts with this decomposition and is not made.
Threshold reading. The bottom of the continuous spectrum, $E_{1} = - \frac{μ κ^{2}}{2 ℏ^{2}}$ , coincides with the relative ground energy; the model reads it as the continuum threshold of the total two-body operator, and no ionisation or scattering process is described here.

The declared two-body Coulomb model: reduced-mass decomposition, relative levels as thresholds and the absence of stationary total states

Statement

Facts & Assumptions

Proof

Remarks

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