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Explicit dipole angular factors of the hydrogenic model: the allowed 1s-to-2p entry, the forbidden 1s-to-2s entry and the oscillator contrast

Physical scope

Explicit angular factors of the declared hydrogenic model's dipole matrix elements: the values +-1/sqrt(3), 0 and their assignment to the angular pairs (1,0)->(0,0), (1,+-1)->(0,0), (0,0)->(0,0) and (0,0)->(1,0), read through the model's dipole components, together with the oscillator contrast Delta n = +-1 on a declared scaled oscillator family. The example computes angular factors only: no radial integral is evaluated, and no transition rate, oscillator strength, line intensity or observed spectrum is asserted.

Example

Adopt the declared Coulomb pair and the model dipole components $D_{q} = - q_{e l} Q_{q}$ with $q_{e l} > 0$ the charge magnitude with $Q_{0} = x_{3}$ and $Q_{\pm} = (x_{1} \pm i x_{2}) / \sqrt{2}$ of Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence, and use the explicit spherical harmonics of Associated Legendre function normalization in the angle coordinates $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ of Sphere integrals in angle coordinates, with $P_{0} = 1$ , $P_{1} (t) = t$ of Legendre Rodrigues equation: $Y_{00} = \frac{1}{\sqrt{4 π}}, Y_{10} = \sqrt{\frac{3}{4 π}} \cos θ, Y_{1, \pm 1} = \mp \sqrt{\frac{3}{8 π}} \sin θ e^{\pm i φ},$ so that $Q_{0} = \cos θ$ and $Q_{\pm} = \sin θ e^{\pm i φ} / \sqrt{2}$ on the chart. Then, with the inner product of $L^{2} (S^{2}, σ; C)$ linear in the first argument:

(a) (allowed $π$ entry) $⟨ Y_{10}, Q_{0} Y_{00} ⟩ = \frac{1}{\sqrt{3}}$ and $⟨ Y_{00}, Q_{0} Y_{10} ⟩ = \frac{1}{\sqrt{3}}$ ;

(b) (allowed $σ^{\pm}$ entries) $⟨ Y_{1, \pm 1}, Q_{\pm} Y_{00} ⟩ = \mp \frac{1}{\sqrt{3}}$ and $⟨ Y_{00}, Q_{\pm} Y_{1, \mp 1} ⟩ = \pm \frac{1}{\sqrt{3}}$ , all of modulus $\frac{1}{\sqrt{3}}$ ;

(c) (forbidden entries) $⟨ Y_{00}, Q_{0} Y_{00} ⟩ = 0$ , $⟨ Y_{00}, Q_{+} Y_{10} ⟩ = 0$ , $⟨ Y_{00}, Q_{0} Y_{11} ⟩ = 0$ , $⟨ Y_{1, \pm 1}, Q_{\mp} Y_{00} ⟩ = 0$ and $⟨ Y_{1, \pm 1}, Q_{\pm} Y_{1, \pm 1} ⟩ = 0$ ; each vanishing is the case $l^{'} = l$ or $m^{'} \neq m + q$ of the selection rules of Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence clause (b);

(d) (parity check) the nonzero entries of (a),(b) connect $l = 0$ with $l^{'} = 1$ , hence states of opposite parity, while the vanishing entries of (c) include the parity-preserving pair $l = l^{'} = 0$ and $l = l^{'} = 1$ ; this is the Laporte condition of clause (d) of the selection-rules theorem;

(e) (reading in the hydrogenic model) applied to the model's uncompressed bound-state matrix elements, the entries of (a),(b) give the angular factors of $⟨ ψ_{n^{'} 1 m^{'}}, D_{q} ψ_{n 00} ⟩ = - q_{e l} \cdot (radial factor) \cdot (the computed number)$ , and the entries of (c) show that the angular factor of $⟨ ψ_{n^{'} 00}, D_{0} ψ_{n 00} ⟩$ vanishes for every $n, n^{'}$ ; the radial factors are not evaluated, so no transition probability or intensity is computed;

(f) (oscillator contrast) on the declared scaled oscillator family $(ψ_{n}^{o s c})$ of Atomic dipole matrix-element domain clause (d) with scale $ℓ > 0$ , the position matrix elements are $⟨ ψ_{1}^{o s c}, X ψ_{0}^{o s c} ⟩ = ℓ / \sqrt{2}$ and $⟨ ψ_{2}^{o s c}, X ψ_{0}^{o s c} ⟩ = 0$ , so the oscillator's position selection rule is $∣ Δ n ∣ = 1$ rather than a rule on an angular quantum number;

(g) (scope) the example is a conditional computation of angular factors of declared model quantities. No rate, oscillator strength, line intensity, observed spectrum or laboratory identification is asserted, and no radial integral is evaluated.

Facts & Assumptions

Given: the explicit harmonics $Y_{00}, Y_{10}, Y_{1, \pm 1}$ , the chart forms of $Q_{0}, Q_{\pm}$ , the polar measure $σ$ , the declared model and oscillator data, and the cited suppliers.

[F1]

Explicit harmonics. For $0 \leq m \leq l$ , use $Y_{l m} = b_{l m} w_{l m} (\cos θ) e^{i m φ} / \sqrt{2 π}$ with $b_{l m} = (\int_{- 1}^{1} w_{l m}^{2} d t)^{- 1 / 2}$ ; for $1 \leq m \leq l$ , use $Y_{l, - m} = (- 1)^{m} \overline{Y_{l m}}$ , as in Associated Legendre function normalization clauses (c),(e). With $w_{l 0} = P_{l}$ , $P_{0} = 1$ , $P_{1} (t) = t$ and $w_{11} (t) = - (1 - t^{2})^{1 / 2}$ from the normalization supplier and Legendre Rodrigues equation clause (d), the norms are $b_{00} = 1 / \sqrt{2}$ , $b_{10} = \sqrt{3 / 2}$ and $b_{11} = \sqrt{3} / 2$ . Thus $Y_{00} = 1 / \sqrt{4 π}$ , $Y_{10} = \sqrt{3 / (4 π)} \cos θ$ , $Y_{1, 1} = - \sqrt{3 / (8 π)} \sin θ e^{i φ}$ and $Y_{1, - 1} = \sqrt{3 / (8 π)} \sin θ e^{- i φ}$ (The complex exponential by its power series, Sine and cosine defined by their real power series).

[F2]

Chart and sphere integrals. In the chart $Φ (θ, φ) = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ one has $Q_{0} = \cos θ$ and $Q_{\pm} = \sin θ e^{\pm i φ} / \sqrt{2}$ ; for Borel $F \geq 0$ , $\int_{S^{2}} F d σ = \int_{0}^{π} \int_{0}^{2 π} F (Φ (θ, φ)) \sin θ d φ d θ$ and $σ (S^{2}) = 4 π$ , whence $\int_{S^{2}} \cos^{2} θ d σ = 4 π / 3$ and $\int_{S^{2}} \sin^{2} θ d σ = 8 π / 3$ , while $\int_{0}^{2 π} e^{i k φ} d φ$ vanishes for every nonzero integer $k$ (Sphere integrals in angle coordinates, The polar surface set function on the unit sphere, 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).

[F3]

Selection rules. The angular matrix element $⟨ Y_{l^{'} m^{'}}, Q_{q} Y_{l m} ⟩$ can be nonzero only if $l^{'} = l + 1$ , $m^{'} = m + q$ , or $l \geq 1$ , $l^{'} = l - 1$ , $m^{'} = m + q$ , and a nonvanishing matrix element requires opposite parity of the two states (Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence clauses (b),(d); Associated Legendre function normalization).

[F4]

Factorization and oscillator. For the declared Coulomb pair the bound-state matrix elements of $D_{q} = - q_{e l} Q_{q}$ with $q_{e l} > 0$ the charge magnitude factor into the radial and angular integrals, and on the scaled oscillator family the position matrix elements are $⟨ ψ_{1}^{o s c}, X ψ_{0}^{o s c} ⟩ = ℓ / \sqrt{2}$ , $⟨ ψ_{2}^{o s c}, X ψ_{0}^{o s c} ⟩ = 0$ (Atomic dipole matrix-element domain clauses (b),(c),(d)).

[F5]

Model reading. The model's dipole components and finite-level data are those of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field and The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector, read under P1 with the units of Dimensions, SI unit conventions and quantity types of the framework (State and composition postulate P1: Hilbert spaces, preparations and composite systems).

Verification

technique · direct; compute the listed sphere integrals from the explicit harmonics and the chart form of the multipliers, and compare the vanishing patterns with the cited selection rules and the oscillator recurrence

Given: the explicit harmonics $Y_{00}, Y_{10}, Y_{1, \pm 1}$ , the chart forms of $Q_{0}, Q_{\pm}$ , the polar measure $σ$ , the declared model and oscillator data, and the cited suppliers.

1.1F1algebra

[Explicit harmonics] By [F1], $Y_{00} = b_{00} P_{0} / \sqrt{2 π} = 1 / \sqrt{4 π}$ , $Y_{10} = b_{10} \cos θ / \sqrt{2 π} = \sqrt{3 / (4 π)} \cos θ$ and $Y_{1, \pm 1} = \mp \sqrt{3 / (8 π)} \sin θ e^{\pm i φ}$ , the last from $w_{11} = - (1 - t^{2})^{1 / 2}$ , $b_{11} = \sqrt{3} / 2$ and the conjugation convention.

2.1F2step 1.1algebra

[Allowed integrals] By [F2], $⟨ Y_{10}, Q_{0} Y_{00} ⟩ = \frac{1}{\sqrt{4 π}} \sqrt{\frac{3}{4 π}} \int_{S^{2}} \cos^{2} θ d σ = \frac{1}{\sqrt{4 π}} \sqrt{\frac{3}{4 π}} \frac{4 π}{3} = \frac{1}{\sqrt{3}}$ and $⟨ Y_{1, \pm 1}, Q_{\pm} Y_{00} ⟩ = \mp \sqrt{\frac{3}{8 π}} \frac{1}{\sqrt{2}} \frac{1}{\sqrt{4 π}} \int_{S^{2}} \sin^{2} θ d σ = \mp \frac{1}{\sqrt{3}}$ ; the real $Q_{0}$ reverse entry is $1 / \sqrt{3}$ . Directly, $⟨ Y_{00}, Q_{+} Y_{1, - 1} ⟩ = + 1 / \sqrt{3}$ and $⟨ Y_{00}, Q_{-} Y_{1, + 1} ⟩ = - 1 / \sqrt{3}$ : substitution gives respectively positive and negative $\sin^{2} θ$ integrands, whose integral is $8 π / 3$ . These are consistent with $Q_{+}^{*} = Q_{-}$ and the explicit opposite-component signs of step 1.1.

2.2F2F3step 1.1algebra

[Forbidden integrals] In $⟨ Y_{00}, Q_{0} Y_{00} ⟩$ the integrand $\cos θ \sin θ$ is odd under $θ \mapsto π - θ$ ; in the entries $⟨ Y_{00}, Q_{+} Y_{10} ⟩$ , $⟨ Y_{00}, Q_{0} Y_{11} ⟩$ , $⟨ Y_{1, \pm 1}, Q_{\mp} Y_{00} ⟩$ and $⟨ Y_{1, \pm 1}, Q_{\pm} Y_{1, \pm 1} ⟩$ the $φ$ -integral of a character $e^{i k φ}$ with nonzero integer $k$ vanishes by [F2]; these are exactly the cases excluded by [F3].

3.1F3step 2.1step 2.2algebra

[Parity] The nonvanishing entries of step 2.1 connect $l = 0$ with $l^{'} = 1$ , i.e. an even representative with an odd one, while the vanishing entries of step 2.2 include $l = l^{'} = 0$ and $l = l^{'} = 1$ ; this is the Laporte condition recorded in [F3].

3.2F4F5step 2.1step 2.2algebra

[Model reading and oscillator] By [F4] the model's matrix elements factor into a radial factor and one of the angular numbers computed in steps 2.1 and 2.2, and the oscillator entries are $ℓ / \sqrt{2}$ and $0$ ; the radial factor is left unevaluated, so no product is formed.

4.1F5step 3.2∎

[Scope] The computation evaluated angular factors and one oscillator matrix element only; no radial integral, rate, intensity or observed spectrum was formed, which is clause (g).

Remarks

Angular nonvanishing is not transition occurrence. The 1s-to-2p angular factor $1 / \sqrt{3}$ is nonzero, but the matrix element also carries the radial factor, which this example does not compute; conversely the vanishing of the 1s-to-2s angular factor makes that electric-dipole matrix element vanish in the declared model, without asserting that no slower mechanism connects the levels.
Two different selection rules. The hydrogenic rule constrains the angular quantum numbers; the oscillator rule constrains the excitation number. Both are properties of the respective declared model's matrix elements.

Explicit dipole angular factors of the hydrogenic model: the allowed 1s-to-2p entry, the forbidden 1s-to-2s entry and the oscillator contrast

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

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