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Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence

Physical scope

The electric-dipole matrix elements of the declared hydrogenic model: the uncompressed relative-coordinate dipole components -q_el Q_q, their bound-state matrix elements, the necessary angular conditions l' = l + 1 with m' = m + q or l >= 1 with l' = l - 1 and m' = m + q, the accompanying opposite-parity (Laporte) condition, and the transfer of these vanishing conditions to compressed dipole matrices when the declared finite-level subspace is spanned by a declared finite subfamily of bound states and its declared orthonormal basis is chosen from that same subfamily. The theorem states necessary conditions only: it asserts no transition rate, oscillator strength, line intensity, observed spectrum or 'allowed transition' beyond the non-vanishing of the matrix element.

Statement

Adopt the declared charged pair of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (i)-(ii): the relative-coordinate factor $L^{2} (R^{3}; C)$ 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), the Cartesian coordinate multipliers $X_{j}$ (the operators denoted $Q_{j}$ in that supplier), with maximal domains $D (X_{j}) = {ψ : ξ_{j} ψ \in L^{2}}$ and Cartesian dipole components $D_{j} = - q_{e l} X_{j}$ , where $q_{e l} > 0$ is the charge magnitude in coulombs. Let $(ψ_{n l m})$ be the bound eigenfunctions of the Coulomb Hamiltonian of The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum with their angular labels $l \geq 0$ , $m \in {- l, \dots, l}$ . Write $Q_{0} : = X_{3}$ , $Q_{\pm} : = (X_{1} \pm i X_{2}) / \sqrt{2}$ and $D_{q} : = - q_{e l} Q_{q}$ for $q \in {- 1, 0, 1}$ (the dimensionless spherical-component index), with $D (Q_{q}) : = {u \in L^{2} : Q_{q} (x) u (x) \in L^{2}}$ . Thus $D (Q_{0}) = D (X_{3})$ and $D (Q_{\pm}) = D (X_{1}) \cap D (X_{2})$ ; $D (D_{q}) = D (Q_{q})$ since $q_{e l} > 0$ . Let $(Y_{l m})$ be the spherical harmonics and $u_{n l}$ the reduced radial functions of Atomic dipole matrix-element domain. Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) (domain and finiteness) every bound state satisfies $ψ_{n l m} \in D (X_{j})$ for $j = 1, 2, 3$ , and for all labels and every $q \in {- 1, 0, 1}$ the matrix element $⟨ ψ_{n^{'} l^{'} m^{'}}, D_{q} ψ_{n l m} ⟩$ is absolutely convergent;

(b) (factorisation and angular condition) for all labels and every $q \in {- 1, 0, 1}$ , $⟨ ψ_{n^{'} l^{'} m^{'}}, D_{q} ψ_{n l m} ⟩ = - q_{e l} (\int_{0}^{\infty} r u_{n^{'} l^{'}} (r) \overline{u_{n l} (r)} d r) (\int_{S^{2}} Y_{l^{'} m^{'}} (Ω) \overline{Q_{q} (Ω) Y_{l m} (Ω)} d σ (Ω)),$ and the angular factor vanishes unless $l^{'} = l + 1 and m^{'} = m + q, or l \geq 1, l^{'} = l - 1 and m^{'} = m + q;$

(c) (necessary conditions only) consequently a nonvanishing matrix element requires $m^{'} - m = q \in {0, \pm 1}$ and $∣ l^{'} - l ∣ = 1$ ; these are necessary conditions on the matrix element: the radial factor may vanish for an allowed angular pair, and no transition probability, rate, oscillator strength, line intensity or claim that an "allowed" transition occurs is asserted;

(d) (parity consequence) let $(Π_{p a r} ψ) (x) : = ψ (- x)$ on $L^{2} (R^{3}; C)$ ; then $Π_{p a r} ψ_{n l m} = (- 1)^{l} ψ_{n l m}$ and $Π_{p a r}$ is a unitary involution with $Π_{p a r} D (Q_{q}) = D (Q_{q})$ , and $Π_{p a r} Q_{q} Π_{p a r} = - Q_{q}$ as operators on $D (Q_{q})$ for $q \in {- 1, 0, 1}$ ; hence a nonvanishing matrix element $⟨ ψ_{n^{'} l^{'} m^{'}}, D_{q} ψ_{n l m} ⟩$ requires $(- 1)^{l^{'}} = - (- 1)^{l}$ , i.e. opposite parity of the two bound states (the Laporte condition), consistently with $∣ l^{'} - l ∣ = 1$ ;

(e) (transfer to the compressed model matrices) if the declared finite-dimensional subspace $W$ of The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clause (b) is the span of a declared finite subfamily of the bound states, with an orthonormal basis chosen from that subfamily, define the spherical compression $D_{q}^{W} : = Π_{W} D_{q} ∣_{W}$ ; then for all $w, w^{'} \in W$ $⟨ w^{'}, D_{q}^{W} w ⟩ = ⟨ w^{'}, D_{q} w ⟩,$ so the model's compressed dipole matrices inherit every vanishing condition of clauses (b)-(d) entry by entry;

(f) (scope) clauses (a)-(e) are conditional statements about the declared Coulomb pair and its compressions. They say nothing about transition rates, lineshapes, observed spectra or selection-rule violations by higher-order terms; the neglected next-order terms of the long-wavelength expansion and the magnetic-moment interaction are outside this theorem, and no empirical spectrum is claimed.

Facts & Assumptions

Given: the declared Coulomb pair with charge magnitude $q_{e l} > 0$ and relative-coordinate factor $L^{2} (R^{3}; C)$ , the Cartesian coordinate multipliers $X_{j}$ and spherical components $Q_{q}$ , the dipole components $D_{q} = - q_{e l} Q_{q}$ , the bound states $ψ_{n l m} = u_{n l} (r) Y_{l m} (Ω) / r$ , the spherical harmonics $(Y_{l m})$ , and, for clause (e), a declared finite subfamily spanning $W$ with the declared orthonormal basis chosen from that subfamily, together with the parity map $Π_{p a r}$ .

[F1]

Coordinate domains and matrix elements. Each Cartesian $X_{i}$ is self-adjoint on its maximal domain, every bound state lies in every $D (X_{i})$ and hence every spherical $D (Q_{q})$ , and $⟨ ψ_{n^{'} l^{'} m^{'}}, Q_{q} ψ_{n l m} ⟩$ has the displayed absolutely convergent radial/angular factorisation. Since the physical scalar $- q_{e l}$ is real, the dipole matrix element is $- q_{e l}$ times that coordinate matrix element even in the linear-first convention (Atomic dipole matrix-element domain clauses (a),(b) and its explicit spherical maximal domains; The space $L^{p} (μ)$ as the quotient by null functions, Real and complex inner product spaces, with the inner product linear in the first argument, Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (i),(ii)). Only those domain and factorisation clauses are used here; the separate angular-selection claim of that supplier is not used.

[F2]

Harmonic blocks, angular weight, and polynomial calculus. For each $l, m$ , the polynomial representative $h_{l m}$ is homogeneous harmonic of degree $l$ and restricts to $Y_{l m}$ ; these restrictions span the degree- $l$ block $V_{l}$ , and distinct $V_{l}$ are orthogonal; put $V_{- 1} : = {0}$ . Use the chart $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ ; there $Q_{0} = \cos θ$ and $Q_{\pm} = \sin θ (\cos φ \pm i \sin φ) / \sqrt{2}$ . The operator $L_{z}$ is the self-adjoint diagonal operator on the full Hilbert space $L^{2} (S^{2}, σ; C)$ with domain $D (L_{z}) = {F : \sum_{l, m} m^{2} ∣ ⟨ F, Y_{l m} ⟩ ∣^{2} < \infty}$ and $L_{z} Y_{l m} = ℏ m Y_{l m}$ ; its algebraic finite span $H = span {Y_{l m}}$ is a core contained in $D (L_{z})$ , and on $H$ its chart differential expression is $- i ℏ \partial_{φ}$ (Sphere Laplacian diagonal domain clauses (a)-(c)). On the complexified homogeneous polynomial spaces, $Δ = \partial_{1}^{2} + \partial_{2}^{2} + \partial_{3}^{2}$ acts complex-linearly; partial derivatives commute on polynomials termwise and obey the product rule, applied componentwise to real and imaginary parts. If $p$ is homogeneous of degree $d$ , then $x \cdot \nabla p = d p$ , by termwise differentiation of its monomials. In particular, for harmonic homogeneous $p$ of degree $d$ , $Δ (r^{2} p) = (4 d + 6) p$ ; this follows from $Δ r^{2} = 6$ , $\nabla r^{2} = 2 x$ , and the product rule (Harmonic polynomial dimension clause (c), Associated Legendre function normalization clauses (c),(d), Sphere Laplacian diagonal domain clauses (a),(c), The derivatives of sine and cosine are cosine and minus sine, The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ , homogeneous polynomial and homogeneous ideal, 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 $g (c) \neq 0$ ).

[F3]

Parity of the spherical harmonics. Each $Y_{l m}$ has a polynomial representative $h_{l m}$ that is homogeneous of degree $l$ ; hence $Y_{l m} (- Ω) = (- 1)^{l} Y_{l m} (Ω)$ , and for the polynomial $Q_{q}$ of degree one, $Q_{q} (- x) = - Q_{q} (x)$ (Harmonic polynomial dimension clause (c)).

[F4]

Compression and linear-first coordinates. For a closed subspace $W \subseteq D (A)$ and its orthogonal projection $Π_{W}$ , $Π_{W} A ∣_{W}$ has matrix entries $⟨ A w_{j}, w_{i} ⟩$ in an orthonormal basis $(w_{i})$ , with input column $j$ and output row $i$ , because $Π_{W} A w_{j} = \sum_{i} ⟨ A w_{j}, w_{i} ⟩ w_{i}$ . Separately, the valid reversed pairing is $⟨ w^{'}, Π_{W} A w ⟩ = ⟨ w^{'}, A w ⟩$ for $w, w^{'} \in W$ ; it is the conjugate of the corresponding coordinate entry. This follows from $A w - Π_{W} A w \in W^{⊥}$ and the linear-first inner product (The Hilbert orthogonal projection onto a closed subspace, Real and complex inner product spaces, with the inner product linear in the first argument, Orthonormal families, complete orthonormal systems and Hilbert bases). The bound states of the declared Coulomb model are those of The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum with $u_{n l} \in L^{2} (0, \infty)$ .

[F5]

Framework reading. The model's dipole components, its finite-level data and its units are those of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field clauses (ii),(iii), The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector clause (b) and Dimensions, SI unit conventions and quantity types of the framework; the compression is the model's declared replacement of the unbounded dipole, and the bound states are read through the model's observable vocabulary Assigned sharp observables and the outcome law of the framework under P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems). The symbol $q_{e l}$ here denotes the positive charge magnitude called $q$ in the postulate; it is independent of the spherical index $q \in {- 1, 0, 1}$ , and every $D_{q}$ retains dipole units $C m$ .

[F6]

Registered reflection premise. Reflection $x \mapsto - x$ preserves Lebesgue measurability, null sets and Lebesgue measure on $R^{3}$ (For a nonzero real $c$ , dilation by $c$ multiplies Lebesgue outer measure by $∣ c ∣^{n}$ , and reflection in the origin preserves it clauses (1)-(3), $n = 3$ , $c = - 1$ ). Thus pullback by reflection is well defined on almost-everywhere classes and preserves their $L^{2}$ inner products; its square is the identity. The weighted norm calculation for the unbounded coordinate domains is supplied below.

Proof

technique · direct; use the cited domain and matrix-element factorisation, prove the angular degree and weight restrictions from the harmonic-polynomial decomposition and $L_z$ action, transport homogeneity to the parity condition, and evaluate the compression on the selected subfamily

Given: The data of the statement and [F1]-[F6].

1.1F1algebra

[Domain and factorisation] Clause (a) is [F1]: each bound state lies in every $D (X_{i})$ , so $D_{q} ψ_{n l m} = - q_{e l} Q_{q} ψ_{n l m}$ is defined and the matrix element is absolutely convergent. For clause (b), [F1] gives the factorisation $⟨ ψ_{n^{'} l^{'} m^{'}}, D_{q} ψ_{n l m} ⟩ = - q_{e l} ⟨ ψ_{n^{'} l^{'} m^{'}}, Q_{q} ψ_{n l m} ⟩$ with the displayed radial and angular factors.

2.1F1F2step 1.1algebra

[Angular degree and weight] Let $H : = h_{l m}$ , the homogeneous harmonic representative of $Y_{l m}$ from [F2]. Write $Q_{q} = \sum_{i = 1}^{3} a_{q, i} x_{i}$ , where $a_{0, i} = δ_{i 3}$ and $a_{\pm, i} = (δ_{i 1} \pm i δ_{i 2}) / \sqrt{2}$ ; these coefficients match exactly $Q_{\pm} = (x_{1} \pm i x_{2}) / \sqrt{2}$ . If $l \geq 1$ , put $G_{q} : = (\sum_{i} a_{q, i} \partial_{i} H) / (2 l + 1)$ ; since $H$ is harmonic, partial derivatives commute with $Δ$ , so $G_{q}$ is harmonic homogeneous of degree $l - 1$ . If $l = 0$ , put $G_{q} : = 0$ . The product rule gives $Δ (Q_{q} H) = 2 \sum_{i} a_{q, i} \partial_{i} H$ . For $l \geq 1$ , [F2] gives $Δ (r^{2} G_{q}) = (4 (l - 1) + 6) G_{q} = (4 l + 2) G_{q} = 2 \sum_{i} a_{q, i} \partial_{i} H$ ; for $l = 0$ both sides are zero. Thus $R_{q} : = Q_{q} H - r^{2} G_{q}$ is harmonic homogeneous of degree $l + 1$ . Restricting to $S^{2}$ , where $r^{2} = 1$ , yields $Q_{q} Y_{l m} = R_{q} ∣_{S^{2}} + G_{q} ∣_{S^{2}}$ when $l \geq 1$ and only the first term when $l = 0$ . By [F2], this lies in $V_{l + 1} \oplus V_{l - 1}$ (with $V_{- 1} = {0}$ ), so its pairing with $Y_{l^{'} m^{'}}$ vanishes if $l^{'} \notin {l - 1, l + 1}$ . The product $F : = Q_{q} Y_{l m}$ is consequently in the finite span $H$ . On the chart, [F2] and the product rule give $\partial_{φ} F = i q F + i m F = i (m + q) F$ : indeed $\partial_{φ} Q_{q} = i q Q_{q}$ , while $\partial_{φ} Y_{l m} = i m Y_{l m}$ follows from $L_{z} Y_{l m} = ℏ m Y_{l m}$ and $L_{z} = - i ℏ \partial_{φ}$ . Hence $L_{z} F = ℏ (m + q) F$ . If $m^{'} \neq m + q$ , self-adjointness of $L_{z}$ and $L_{z} Y_{l^{'} m^{'}} = ℏ m^{'} Y_{l^{'} m^{'}}$ imply $(m + q - m^{'}) ℏ ⟨ F, Y_{l^{'} m^{'}} ⟩ = 0$ , so $⟨ F, Y_{l^{'} m^{'}} ⟩ = 0$ . Since the reversed angular pairing in the Statement is its complex conjugate, it vanishes under the same conditions. This proves the claimed degree and component-dependent weight restrictions.

3.1step 2.1algebraF2

[Necessary conditions] Clause (c) restates step 2.1: $m^{'} - m = q \in {0, \pm 1}$ and $l^{'} - l \in {- 1, + 1}$ are necessary for a nonvanishing matrix element; the radial factor is not controlled by [F2], so no nonvanishing statement is made for an allowed angular pair, and no probability or rate is defined.

3.2F1F3F6step 2.1algebra

[Parity on the actual coordinate domains] By [F6], $Π_{p a r}$ is a unitary involution on $L^{2}$ . For any $u \in L^{2}$ , reflection invariance and $Q_{q} (- x) = - Q_{q} (x)$ from [F3] give $\int ∣ Q_{q} (x) ∣^{2} ∣ u (- x) ∣^{2} d x = \int ∣ Q_{q} (- x) ∣^{2} ∣ u (x) ∣^{2} d x = \int ∣ Q_{q} (x) ∣^{2} ∣ u (x) ∣^{2} d x$ , allowing infinite values. Consequently $Π_{p a r} D (Q_{q}) = D (Q_{q})$ and, on that domain, $Π_{p a r} Q_{q} Π_{p a r} = - Q_{q}$ . Also [F3] gives $Π_{p a r} ψ_{n l m} = (- 1)^{l} ψ_{n l m}$ because its radial factor depends on $∣ x ∣$ only. Now let $u \in D (Q_{q})$ and $v \in L^{2}$ satisfy $Π_{p a r} u = σ u$ , $Π_{p a r} v = σ^{'} v$ with real $σ, σ^{'} \in {- 1, 1}$ . Unitarity and the defined-domain identity give $⟨ v, Q_{q} u ⟩ = ⟨ Π_{p a r} v, Π_{p a r} Q_{q} u ⟩ = - σ σ^{'} ⟨ v, Q_{q} u ⟩$ . A nonzero pairing therefore forces $σ σ^{'} = - 1$ . The charge $q_{e l} > 0$ preserves nonvanishing when $Q_{q}$ is replaced by $D_{q} = - q_{e l} Q_{q}$ . Applying this to the bound states, whose input domains are supplied by [F1], gives $(- 1)^{l^{'}} = - (- 1)^{l}$ and proves (d).

4.1F1F4step 1.1step 2.1step 3.2algebra

[Compression] The finite span $W$ of the selected bound states lies in every $D (X_{i})$ and hence in every $D (Q_{q}) = D (D_{q})$ by [F1] and linearity of those domains. Thus $D_{q}^{W} = Π_{W} D_{q} ∣_{W}$ is defined on all $W$ . For $w, w^{'} \in W$ , the reversed-pairing identity of [F4] gives $⟨ w^{'}, D_{q}^{W} w ⟩ = ⟨ w^{'}, D_{q} w ⟩$ , which is exactly the valid display of (e). In the selected orthonormal basis the actual coordinate entry is $⟨ D_{q} w_{j}, w_{i} ⟩$ , the conjugate of $⟨ w_{i}, D_{q} w_{j} ⟩$ ; either vanishes exactly when the other does. Therefore each matrix entry inherits the uncompressed angular and parity vanishing conditions from steps 2.1 and 3.2, including the complex spherical components, proving (e).

5.1F5step 1.1step 2.1step 3.1step 3.2step 4.1∎

[Scope and assembly] Steps 1.1 and 2.1 prove (a) and (b), step 3.1 proves (c), step 3.2 proves (d) and step 4.1 proves (e). Only matrix elements of the declared model's dipole components entered; no transition probability, rate, lineshape or observed spectrum was formed, and the neglected higher-order terms are recorded as outside the theorem, which is (f).

Remarks

Necessary, not sufficient. A vanishing angular factor forbids an electric-dipole matrix element; a nonvanishing angular factor does not force a nonvanishing matrix element, because the radial integral may vanish. The theorem is therefore a statement about the model's dipole matrices, not about observed line strengths.
Why the parity clause is included. The angular rule $∣ l^{'} - l ∣ = 1$ already implies opposite parity for the hydrogenic bound states; clause (d) records the parity mechanism explicitly because it is the form of the rule that transfers to any finite-level subspace whose basis states have definite parity.

Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence

Statement

Facts & Assumptions

Proof

Remarks

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