Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Superposition intensity and the coherent interference cross term

Physical scope

Deterministic superposition identities for two same-frequency ideal plane-wave solutions of the smooth source-free vacuum Maxwell branch in a named inertial chart with SI constants: the averaged energy density of the coherent sum with its interference cross term, the averaged Poynting vector, and the equal-wavevector specialization. Excluded: ensembles or random phases, partially coherent and finite sources, detection or sampling statistics, any empirical conclusion, and any claim discriminating classical from quantum descriptions.

Statement

Let $ω > 0$ and let $k_{1}, k_{2} \in R^{3} ∖ {0}$ satisfy $∣ k_{1} ∣ = ∣ k_{2} ∣ = ω / c$ . For $j = 1, 2$ let $e_{j} \in C^{3}$ with $k_{j} \cdot e_{j} = 0$ , put $b_{j} = k_{j} \times e_{j} / ω$ and let $(E_{j}, B_{j})$ be the plane-wave solutions of Dispersion, transversality and the two polarization conditions for plane-wave solutions given by Real plane-wave fields encoded by complex amplitudes. Let $a_{1}, a_{2} \in R$ and put $(E, B) : = a_{1} (E_{1}, B_{1}) + a_{2} (E_{2}, B_{2})$ . Write ${\hat{k}}_{j} = k_{j} / ∣ k_{j} ∣$ , $T = 2 π / ω$ , $∣ z ∣^{2} : = z \cdot \overline{z} = \sum_{ℓ = 1}^{3} ∣ z_{ℓ} ∣^{2}$ for $z \in C^{3}$ , and $⟨ f ⟩ (x) : = T^{- 1} \int_{0}^{T} f (x, s) d s$ componentwise. Then:

(a) [Superposition] $(E, B)$ is a smooth source-free solution of the vacuum Maxwell equations (Superposition for prescribed Maxwell fields and sources), and at each fixed $x$ it can be written with a single phase, $E (x, t) = Re (α (x) e^{i (k_{1} \cdot x - ω t)}), B (x, t) = Re (β (x) e^{i (k_{1} \cdot x - ω t)}),$ where $α (x) : = a_{1} e_{1} + a_{2} e_{2} e^{i δ (x)}$ , $β (x) : = a_{1} b_{1} + a_{2} b_{2} e^{i δ (x)}$ and $δ (x) : = (k_{2} - k_{1}) \cdot x$ ; constant relative phases are absorbed in $e_{2}$ .

(b) [Averaged energy density with the coherent cross term] For every $x$ , $⟨ u ⟩ (x) = \frac{ε_{0}}{4} ∣ α (x) ∣^{2} + \frac{1}{4 μ_{0}} ∣ β (x) ∣^{2} = \frac{ε_{0}}{2} (a_{1}^{2} ∣ e_{1} ∣^{2} + a_{2}^{2} ∣ e_{2} ∣^{2}) + \frac{ε_{0}}{2} a_{1} a_{2} Re (W (x)),$ where $W (x) : = (e_{1} \cdot {\overline{e}}_{2} + ({\hat{k}}_{1} \times e_{1}) \cdot ({\hat{k}}_{2} \times {\overline{e}}_{2})) e^{- i δ (x)}$ . The first two terms are the sum of the averaged energy densities of the two waves (Period-averaged energy density, Poynting vector and momentum flux of a plane wave); the third is the interference cross term, which may remain nonzero after time averaging at fixed relative phase.

(c) [Averaged flux and momentum; equal wavevectors] (with $u$ the energy density and $S$ the Poynting vector of Vacuum electromagnetic energy density and energy flux (the Poynting vector)) $⟨ S ⟩ (x) = \frac{1}{2 μ_{0}} Re (α (x) \times \overline{β (x)})$ . In particular, if $k_{1} = k_{2} = : k$ then the superposition is the single plane wave with amplitude $a_{1} e_{1} + a_{2} e_{2}$ , and $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ a_{1} e_{1} + a_{2} e_{2} ∣^{2}$ , $⟨ S ⟩ = \frac{ε_{0} c}{2} ∣ a_{1} e_{1} + a_{2} e_{2} ∣^{2} \hat{k}$ (Period-averaged energy density, Poynting vector and momentum flux of a plane wave).

(d) [No incoherent or statistical claim] The formulas are deterministic identities for the specified coherent superposition; they are not an ensemble average over phases, and no detection statistics, finite-sample visibility estimate or empirical conclusion is drawn. The computation is a conditional consequence of the adopted Maxwell postulate and does not by itself distinguish classical from quantum descriptions of light.

Facts & Assumptions

Given: The frequency $ω > 0$ ; wavevectors $k_{1}, k_{2} \neq 0$ with $∣ k_{1} ∣ = ∣ k_{2} ∣ = ω / c$ ; amplitudes $e_{j} \in C^{3}$ with $k_{j} \cdot e_{j} = 0$ and $b_{j} = k_{j} \times e_{j} / ω$ ; the plane-wave fields $(E_{j}, B_{j})$ with phases $θ_{j} (x, t) = k_{j} \cdot x - ω t$ ; real coefficients $a_{1}, a_{2}$ ; and the energy density $u$ and Poynting vector $S$ of Vacuum electromagnetic energy density and energy flux (the Poynting vector).

[F1]

Linear superposition (Superposition for prescribed Maxwell fields and sources(a)): for any finite list of field-source pairs satisfying one and the same branch of the vacuum Maxwell postulate on the same region and real coefficients, the corresponding linear combination of fields and sources satisfies the same branch with the same constants; pointwise in the smooth branch.

[F2]

Plane-wave solutions (Dispersion, transversality and the two polarization conditions for plane-wave solutions): for $∣ k ∣ = ω / c$ , $k \cdot e = 0$ and $b = k \times e / ω$ the plane-wave pair lies in the source-free smooth Maxwell branch, its fields are $C^{\infty}$ , and $B = \hat{k} \times E / c$ ; there are no sources.

[F3]

Plane-wave encoding (Real plane-wave fields encoded by complex amplitudes): $E_{j} = Re (e_{j} \exp (i θ_{j}))$ , $B_{j} = Re (b_{j} \exp (i θ_{j}))$ componentwise.

[F4]

Period averages (Period averages of products of real parts of complex exponentials): for $a, d \in C^{3}$ and $T = 2 π / ω$ , $\frac{1}{T} \int_{0}^{T} ⟨ Re (a e^{- i ω t}), Re (d e^{- i ω t}) ⟩ d t = \frac{1}{2} Re (⟨ a, \overline{d} ⟩)$ and $\frac{1}{T} \int_{0}^{T} Re (a e^{- i ω t}) \times Re (d e^{- i ω t}) d t = \frac{1}{2} Re (a \times \overline{d})$ componentwise.

[F5]

Energy density and Poynting vector (Vacuum electromagnetic energy density and energy flux (the Poynting vector)): $u = \frac{ε_{0}}{2} ⟨ E, E ⟩ + \frac{1}{2 μ_{0}} ⟨ B, B ⟩$ and $S = \frac{1}{μ_{0}} E \times B$ .

[F6]

Single-wave averages (Period-averaged energy density, Poynting vector and momentum flux of a plane wave): for a single transverse plane wave with amplitude $\tilde{e}$ , wavevector $\tilde{k}$ , $∣ \tilde{k} ∣ = ω / c$ , the period-averaged energy density and Poynting vector are $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ \tilde{e} ∣^{2}$ and $⟨ S ⟩ = \frac{ε_{0} c}{2} ∣ \tilde{e} ∣^{2} \hat{k}$ .

[F7]

Cross and inner product algebra (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ): both products are bilinear over $R$ and extend componentwise to complex entries as complex-bilinear maps, the cross product is alternating, and the polynomial identity $⟨ u \times v, w \times z ⟩ = ⟨ u, w ⟩ ⟨ v, z ⟩ - ⟨ u, z ⟩ ⟨ v, w ⟩$ holds; in particular $⟨ \hat{k} \times u, \hat{k} \times \overline{v} ⟩ = ⟨ u, \overline{v} ⟩$ whenever $⟨ \hat{k}, u ⟩ = ⟨ \hat{k}, v ⟩ = 0$ and $∣ \hat{k} ∣ = 1$ .

[F8]

Complex parts and conjugation (Real and imaginary parts, complex conjugation, and modulus): $Re$ is $R$ -linear, $\overline{z w} = \overline{z} \overline{w}$ , $z \overline{z} = ∣ z ∣^{2}$ , and the complex norm is $∣ z ∣ = (\sum_{ℓ} ∣ z_{ℓ} ∣^{2})^{1 / 2}$ ; for real vectors this is the Euclidean norm.

Proof

technique · direct

1.1givenF2F3

Setting. For $j = 1, 2$ the data satisfy $∣ k_{j} ∣ = ω / c$ , $k_{j} \cdot e_{j} = 0$ and $b_{j} = k_{j} \times e_{j} / ω$ , so by [F2] each $(E_{j}, B_{j})$ of [F3] is a $C^{\infty}$ source-free solution of the smooth vacuum Maxwell branch on $R^{3} \times R$ ; put $θ_{j} (x, t) = k_{j} \cdot x - ω t$ and $δ (x) : = (k_{2} - k_{1}) \cdot x = θ_{2} (x, t) - θ_{1} (x, t)$ , a function of $x$ alone.

2.1step 1.1F1F3F8algebra

Clause (a). The pairs $(E_{1}, B_{1}, ρ_{1}, J_{1})$ and $(E_{2}, B_{2}, ρ_{2}, J_{2})$ with $ρ_{1} = ρ_{2} = 0$ , $J_{1} = J_{2} = 0$ each satisfy the smooth branch on the same whole-space region by step 1.1, so F1 with $n = 2$ and the real coefficients $a_{1}, a_{2}$ gives that $(E, B) = a_{1} (E_{1}, B_{1}) + a_{2} (E_{2}, B_{2})$ satisfies the smooth source-free branch with the same constants, and the fields are $C^{\infty}$ as finite linear combinations of $C^{\infty}$ fields. The single-phase rewriting follows from $\exp (i θ_{2}) = \exp (i δ (x)) \exp (i θ_{1})$ and the $R$ -linearity of $Re$ of [F8]: componentwise $E = Re (e_{1} e^{i θ_{1}}) a_{1} + Re (e_{2} e^{i θ_{1}} e^{i δ}) a_{2} = Re ((a_{1} e_{1} + a_{2} e_{2} e^{i δ}) e^{i θ_{1}}) = Re (α e^{i θ_{1}})$ by the field arithmetic of $C$ , and identically for $B$ with $β = a_{1} b_{1} + a_{2} b_{2} e^{i δ}$ .

3.1step 2.1F4F5F7F8algebra

Clause (b). Fix $x$ and write $θ = θ_{1} (x, t) = k_{1} \cdot x - ω t$ , so that by step 2.1 $E = Re (A e^{- i ω t})$ and $B = Re (D e^{- i ω t})$ , where $A : = α (x) e^{i k_{1} \cdot x}$ and $D : = β (x) e^{i k_{1} \cdot x}$ . Applying [F4] to the scalar components (equivalently its vector form with one component retained) gives $⟨ E_{i} E_{j} ⟩ = \frac{1}{2} Re (A_{i} \overline{A_{j}}) = \frac{1}{2} Re (α_{i} \overline{α_{j}})$ , since $e^{i k_{1} \cdot x} e^{- i k_{1} \cdot x} = 1$ , hence $⟨ ∣ E ∣^{2} ⟩ = \frac{1}{2} ∣ α ∣^{2}$ , and likewise $⟨ ∣ B ∣^{2} ⟩ = \frac{1}{2} ∣ β ∣^{2}$ ; therefore $⟨ u ⟩ = \frac{ε_{0}}{4} ∣ α ∣^{2} + \frac{1}{4 μ_{0}} ∣ β ∣^{2}$ by [F5]. Expanding the squares with the complex bilinearity of [F7] and [F8] gives $∣ α ∣^{2} = a_{1}^{2} ∣ e_{1} ∣^{2} + a_{2}^{2} ∣ e_{2} ∣^{2} + 2 a_{1} a_{2} Re (e_{1} \cdot {\overline{e}}_{2} e^{- i δ})$ and $∣ β ∣^{2} = a_{1}^{2} ∣ b_{1} ∣^{2} + a_{2}^{2} ∣ b_{2} ∣^{2} + 2 a_{1} a_{2} Re (b_{1} \cdot {\overline{b}}_{2} e^{- i δ})$ . For the magnetic terms, $∣ b_{j} ∣^{2} = ω^{- 2} ⟨ k_{j} \times e_{j}, \overline{k_{j} \times e_{j}} ⟩ = ω^{- 2} ∣ k_{j} ∣^{2} ∣ e_{j} ∣^{2} = ∣ e_{j} ∣^{2} / c^{2}$ by [F7] and $k_{j} \cdot e_{j} = 0$ , and $b_{1} \cdot {\overline{b}}_{2} = ω^{- 2} (k_{1} \times e_{1}) \cdot \overline{(k_{2} \times e_{2})} = \frac{∣ k_{1} ∣ ∣ k_{2} ∣}{ω^{2}} ({\hat{k}}_{1} \times e_{1}) \cdot ({\hat{k}}_{2} \times {\overline{e}}_{2}) = \frac{1}{c^{2}} ({\hat{k}}_{1} \times e_{1}) \cdot ({\hat{k}}_{2} \times {\overline{e}}_{2})$ ; substituting and using $\frac{1}{4 μ_{0} c^{2}} = \frac{ε_{0}}{4}$ from the SI relation gives $⟨ u ⟩ = \frac{ε_{0}}{2} (a_{1}^{2} ∣ e_{1} ∣^{2} + a_{2}^{2} ∣ e_{2} ∣^{2}) + \frac{ε_{0}}{2} a_{1} a_{2} Re (W)$ with $W = (e_{1} \cdot {\overline{e}}_{2} + ({\hat{k}}_{1} \times e_{1}) \cdot ({\hat{k}}_{2} \times {\overline{e}}_{2})) e^{- i δ}$ , which is clause (b).

3.2step 2.1F4F5F6F7F8algebra

Clause (c). At the same fixed $x$ , the cross-product form of [F4] gives $⟨ E \times B ⟩ = \frac{1}{2} Re (A \times \overline{D}) = \frac{1}{2} Re (α \times \overline{β})$ , since the common spatial phase cancels, so $⟨ S ⟩ = \frac{1}{μ_{0}} ⟨ E \times B ⟩ = \frac{1}{2 μ_{0}} Re (α \times \overline{β})$ by [F5], which is the first display of clause (c). If $k_{1} = k_{2} = : k$ then $δ = 0$ , $α = a_{1} e_{1} + a_{2} e_{2} = : \tilde{e}$ and $β = a_{1} b_{1} + a_{2} b_{2}$ ; by the bilinearity of the cross product [F7], $β = k \times (a_{1} e_{1} + a_{2} e_{2}) / ω = k \times \tilde{e} / ω$ , and $k \cdot \tilde{e} = 0$ , so $(E, B)$ is exactly the single plane wave with amplitude $\tilde{e}$ and wavevector $k$ , of the class covered by [F6]; clause (a) of [F6] gives $⟨ u ⟩ = \frac{ε_{0}}{2} ∣ \tilde{e} ∣^{2}$ and clause (b) gives $⟨ S ⟩ = \frac{ε_{0} c}{2} ∣ \tilde{e} ∣^{2} \hat{k}$ . These agree with clauses (b) and the first display of (c): when $k_{1} = k_{2} = k$ the bracket in $W$ reduces to $2 e_{1} \cdot {\overline{e}}_{2}$ by the identity of [F7], and $∣ \tilde{e} ∣^{2} = a_{1}^{2} ∣ e_{1} ∣^{2} + a_{2}^{2} ∣ e_{2} ∣^{2} + 2 a_{1} a_{2} Re (e_{1} \cdot {\overline{e}}_{2})$ by [F8].

4.1step 2.1step 3.1step 3.2F5given∎

Conclusion and scope. Step 2.1 is clause (a), step 3.1 is clause (b), step 3.2 is clause (c), and clause (d) is a scope qualification on the same computation: every average above is the deterministic period average at a fixed point of the specified coherent superposition, no ensemble or random-phase average, detector model, finite-sample frequency or visibility estimate is introduced, and nothing in the proof uses any property distinguishing classical from quantum descriptions of light beyond the adopted Maxwell equations and the definitions; in particular classical electromagnetic waves interfere, so these formulas do not by themselves discriminate between descriptions. The identities are conditional on the adopted postulate and make no empirical claim.

Source notes

The cited treatments state the superposition principle for linear Maxwell theory and the monochromatic amplitude representation on which the two-beam sum is written; the interference cross term and its averaged form are proved here from the framework's own definitions and period-average lemma. The scope qualification (d) records explicitly that this is a deterministic field calculation, not a detection or sampling analysis.

Superposition intensity and the coherent interference cross term

Statement

Facts & Assumptions

Proof

Source notes

Depends on

Dependency tree · 2 levels

Sources