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Atomic dipole matrix-element domain

Statement

Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $γ > 0$ , let $C = - Δ - γ / ∣ \cdot ∣$ on $H^{2} (R^{3}; C)$ be as above, let $(ψ_{n l m})$ be the bound eigenfunctions of Negative-subspace completeness and degeneracy, and for $i = 1, 2, 3$ define the coordinate multiplication operator $Q_{i} ψ : = (ξ \mapsto ξ_{i} ψ (ξ)), D (Q_{i}) : = {ψ \in L^{2} (R^{3}; C) : ξ_{i} ψ \in L^{2}} .$ Write also $Q_{0} : = x_{3}$ and $Q_{\pm} : = (x_{1} \pm i x_{2}) / \sqrt{2}$ . Then:

(a) each $Q_{i}$ is self-adjoint on $D (Q_{i})$ , the domain is a proper dense subspace of $L^{2}$ , and every bound state satisfies $ψ_{n l m} \in D (Q_{i})$ , with $∥ Q_{i} ψ_{n l m} ∥_{2}^{2} = \int_{0}^{\infty} r^{2} ∣ u_{n l} (r) ∣^{2} d r \cdot \int_{S^{2}} ∣ Ω_{i} Y_{l m} (Ω) ∣^{2} d σ (Ω) < \infty$ ;

(b) for all labels and each $q \in {- 1, 0, 1}$ the matrix element $⟨ ψ_{n^{'} l^{'} m^{'}}, Q_{q} ψ_{n l m} ⟩$ is absolutely convergent and factors as $⟨ ψ_{n^{'} l^{'} m^{'}}, Q_{q} ψ_{n l m} ⟩ = (\int_{0}^{\infty} r u_{n^{'} l^{'}} (r) \overline{u_{n l} (r)} d r) (\int_{S^{2}} Y_{l^{'} m^{'}} (Ω) \overline{Q_{q} (Ω) Y_{l m} (Ω)} d σ (Ω)),$ where $Q_{q} (Ω)$ denotes the restriction of the polynomial $Q_{q}$ to $S^{2}$ ;

(c) the angular factor vanishes unless $l^{'} = l + 1, m^{'} = m + q$ or $l \geq 1, l^{'} = l - 1, m^{'} = m + q$ ; these are necessary conditions only: no assertion is made that an allowed pair has nonzero radial integral, and no transition probability is asserted or computed;

(d) for the oscillator: fix real parameters $m > 0$ , $ω > 0$ , $ℏ > 0$ and put $ℓ : = \sqrt{ℏ / (m ω)} > 0$ . For the complete orthonormal family $(ψ_{n}^{o s c})$ of Oscillator scaling and exact realization and the position operator $X f (x) = x f (x)$ on $D (X) = {f \in L^{2} (R, d x) : x f \in L^{2}}$ (multiplication by $ℓ y$ after $x = ℓ y$ ), one has $X ψ_{n}^{o s c} = \frac{ℓ}{\sqrt{2}} (\sqrt{n} ψ_{n - 1}^{o s c} + \sqrt{n + 1} ψ_{n + 1}^{o s c}),$ so $⟨ ψ_{n^{'}}^{o s c}, X ψ_{n}^{o s c} ⟩$ vanishes unless $n^{'} = n \pm 1$ , with the displayed coefficients; in particular the oscillator position matrix elements are supported on a single step $∣ Δ n ∣ = 1$ , while the Coulomb selection rules constrain only $Δ l$ and $Δ m$ and leave the radial integral free.

This item concerns matrix elements of multiplication operators on $L^{2}$ and their finiteness and selection rules; it asserts no absorption, emission or transition rate and no physical interpretation.

Facts & Assumptions

Given: $γ > 0$ ; the integer bound/angular labels of the stated family; maximal domains $D (Q_{q}) = {f : Q_{q} f \in L^{2}}$ , so $D (Q_{0}) = D (Q_{3})$ and $D (Q_{\pm}) = D (Q_{1}) \cap D (Q_{2})$ because $∣ Q_{\pm} ∣^{2} = (x_{1}^{2} + x_{2}^{2}) / 2$ ; the bound states $ψ_{n l m} = u_{n l} (r) Y_{l m} (Ω) / r$ of the previous items; the coordinate multipliers $Q_{i}, Q_{q}$ ; the oscillator parameters $m, ω, ℏ > 0$ and $ℓ = \sqrt{ℏ / (m ω)} > 0$ .

[F1]

For a real finite-valued measurable $a$ the multiplier $M_{a} f = a f$ with $D (M_{a}) = {f : a f \in L^{2}}$ is self-adjoint (Real multiplication operators and free plus bounded-potential Schrödinger operators (a), Symmetric, self-adjoint and essentially self-adjoint operators); the coordinate functions $ξ_{i}$ are real finite-valued.

[F2]

Every bound state has $u_{n l} \in L^{2} (0, \infty)$ , $u_{n l} (r) = c r^{l + 1} e^{- k r} L_{n - l - 1}^{(2 l + 1)} (2 k r)$ up to a nonzero scalar with $k = γ / (2 n)$ , and $∥ ψ_{n l m} ∥_{2} = ∥ u_{n l} ∥_{2}$ (Negative-subspace completeness and degeneracy, Kummer recurrence and termination, Angular radial unitary reduction).

[F3]

Dipole angular selection rules: for the spherical components $Q_{0} = x_{3}$ , $Q_{\pm} = (x_{1} \pm i x_{2}) / \sqrt{2}$ and the family $(Y_{l m})$ , the angular matrix element $⟨ Y_{l^{'} m^{'}}, Q_{q} ∣_{S^{2}} Y_{l m} ⟩$ can be nonzero only if $l^{'} = l + 1, m^{'} = m + q$ or $l \geq 1, l^{'} = l - 1, m^{'} = m + q$ ; the rule is a necessary condition on the angular integrals only (degree and weight clauses (b), (c) of Dipole angular selection rules, Associated Legendre function normalization).

[F4]

Oscillator: for real $m, ω, ℏ > 0$ and $ℓ = \sqrt{ℏ / (m ω)} > 0$ , the scaled Hermite functions $ψ_{n}^{o s c} = ℓ^{- 1 / 2} g_{n} (x / ℓ)$ form a complete orthonormal family of $L^{2} (R, d x)$ with $x = ℓ y$ , and the Hermite recurrence gives $\sqrt{2} y g_{n} = \sqrt{n} g_{n - 1} + \sqrt{n + 1} g_{n + 1}$ with the lowering term omitted at $n = 0$ (equivalently define $g_{- 1} = 0$ and $ψ_{- 1}^{o s c} = 0$ ) (Oscillator scaling and exact realization, Hermite recurrence (d), (g), Orthonormal families, complete orthonormal systems and Hilbert bases, Parseval equivalences for an orthonormal family).

[F5]

Polar volume integration is $d x = r^{2} d r d σ$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma); nonnegative product integrals use Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, and absolutely convergent product integrals use Tonelli and Fubini for the completed product, with only almost-everywhere section measurability. For $a > 0$ and integer $p \geq 0$ , integration by parts starting with $p = 0$ gives $\int_{0}^{\infty} r^{p} e^{- a r} d r = p! / a^{p + 1}$ ; the boundary terms vanish since a sufficiently high exponential-series term bounds every fixed polynomial power (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , The exponential function is smooth and $(\exp)^{'} = \exp$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ). Cauchy-Schwarz for the $L^{2}$ and $L^{2} (S^{2}, σ)$ pairings (Cauchy-Schwarz inequality for $L^{2}$ , Real and complex inner product spaces, with the inner product linear in the first argument, The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions).

Proof

1.1F1F5givenalgebra

[Coordinate domains] Each real $Q_{i}$ is self-adjoint by [F1]. For $f \in L^{2}$ , $f_{N} = 1_{∣ x_{i} ∣ \leq N} f$ lies in $D (Q_{i})$ and converges in $L^{2}$ , proving density. Put $φ (t) = (1 + ∣ t ∣)^{- 1}$ ; it belongs to $L^{2} (R)$ whereas $t^{2} ∣ φ (t) ∣^{2} \geq 1 / 4$ for $∣ t ∣ \geq 1$ . For each $i$ , $f_{i} (x) = φ (x_{i}) \prod_{j \neq i} e^{- ∣ x_{j} ∣}$ belongs to $L^{2}$ but $x_{i} f_{i} \notin L^{2}$ by nonnegative product integration. Thus every domain is proper, including $i = 2, 3$ , not just the first-coordinate example.

2.1step 1.1F2F5algebra

[Bound-state domain membership] By [F2], $u_{n l}$ is polynomial-exponential with positive decay rate and is $O (r^{l + 1})$ at zero. Hence $\int_{0}^{\infty} r^{2} ∣ u_{n l} ∣^{2} d r < \infty$ : near zero the integrand is $O (r^{2 l + 4})$ , and all polynomial-exponential moments at infinity are finite by [F5]. Polar coordinates give exactly $∥ Q_{i} ψ_{n l m} ∥^{2} = \int r^{2} ∣ u_{n l} ∣^{2} d r \int ∣ Ω_{i} Y_{l m} ∣^{2} d σ$ , whose angular factor is at most one. Each bound state therefore lies in every $D (Q_{i})$ and in the stated complex component domains. This proves (a).

3.1step 2.1F2F5algebra

[Absolute convergence and factorization] Put $v = u_{n^{'} l^{'}}$ , $u = u_{n l}$ , $Y^{'} = Y_{l^{'} m^{'}}$ and $Y = Y_{l m}$ . Cauchy–Schwarz gives $\int_{0}^{\infty} r ∣ v \overline{u} ∣ d r \leq ∥ v ∥_{2} ∥ r u ∥_{2} < \infty$ and $\int_{S^{2}} ∣ Y^{'} \overline{Q_{q} Y} ∣ d σ \leq ∥ Y^{'} ∥_{2} ∥ Q_{q} Y ∥_{2} \leq 1$ . Tonelli and Fubini in [F5] justify absolute integration of the polar product and its factorization into $(\int_{0}^{\infty} r v \overline{u} d r) (\int_{S^{2}} Y^{'} \overline{Q_{q} Y} d σ)$ , exactly (b) for the linear-first pairing. Coordinate multiplication generally produces two angular blocks; no false single-block claim is used.

4.1step 3.1F3algebra

[All necessary angular branches] By [F3], $Q_{q} h_{l m} = R_{q} + r^{2} G_{q}$ has harmonic degrees $l + 1, l - 1$ , with $G_{q} = (\sum c_{q, i} \partial_{i} h_{l m}) / (2 l + 1)$ . At $l = 1$ its derivative term can be a nonzero constant in $V_{0}$ , so the downward branch $l^{'} = 0$ is retained. At $l = 0$ the derivatives vanish and only the upward branch remains. Degree orthogonality and the $L_{z}$ eigenvalue calculation give exactly $l^{'} = l + 1, m^{'} = m + q$ , or $l \geq 1, l^{'} = l - 1, m^{'} = m + q$ . These are necessary only; the radial factor is $\int_{0}^{\infty} r u_{n^{'} l^{'}} \overline{u_{n l}} d r$ and may vanish for an allowed angular pair. No transition probability is defined. This proves (c).

5.1step 4.1F4F5algebra

[The oscillator comparison] In the scaled variable $x = ℓ y$ multiplication by $x$ is $ℓ$ times multiplication by $y$ ; by [F4] the Hermite recurrence gives, on the complete orthonormal family $(ψ_{n}^{o s c})$ , $X ψ_{n}^{o s c} = ℓ y ψ_{n}^{o s c} = \frac{ℓ}{\sqrt{2}} (\sqrt{n} ψ_{n - 1}^{o s c} + \sqrt{n + 1} ψ_{n + 1}^{o s c})$ ; the lowering term being absent at $n = 0$ . The recurrence puts $X ψ_{n}^{o s c}$ in $L^{2}$ , proving membership in the maximal position domain; taking pairings with $ψ_{n^{'}}^{o s c}$ and using orthonormality [F4] and Cauchy-Schwarz [F5] gives $⟨ ψ_{n^{'}}^{o s c}, X ψ_{n}^{o s c} ⟩ = \frac{ℓ}{\sqrt{2}} (\sqrt{n} δ_{n^{'}, n - 1} + \sqrt{n + 1} δ_{n^{'}, n + 1})$ , so exactly the transitions $∣ Δ n ∣ = 1$ occur, with the displayed coefficients. This is (d).

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

[Conclusion] Clauses (a), (b), (c), (d) are steps 1.1 and 2.1, 3.1, 4.1, and 5.1.

Atomic dipole matrix-element domain

Statement

Facts & Assumptions

Proof

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