Qm Photons And Radiation Modes — Examples

Adopt the state and composition postulate P1, the observable and probability postulate P2 (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), the radiation extension postulate clause (i) (Radiation extension of the framework: transverse mode and photon assignments) and the declared fixed-mode polarisation model The fixed-mode polarisation model of the framework with polarisation space $V$, orthonormal basis $(e_1,e_2)$ and analyser projections $P_{\theta }$. Assume the Axioms of Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The Axiom of Choice). Fix an angle $\theta \in \mathbb{R}$. This example compares two declared models of the same analyser:

(a) The quantised-model ratio. Consider the pure one-photon polarisation preparation of the ray $[e_1]$ and the declared ideal analyser at angle $\theta$. By The analyser outcome law of the fixed-mode polarisation model: Born probabilities and the Malus ratio the outcome law of the analyser is $p_{+1}=\mathrm{\parallel }{P}_{\theta }{e}_1{\mathrm{\parallel }}^2={\cos }^2\theta ,p_{-1}={\sin }^2\theta ,$ so the transmitted Born probability is ${\cos }^2\theta$ in the unit $1$ of dimensionless probabilities.

(b) The declared classical transverse-amplitude model. Declare the following elementary model, with no claim that it is Maxwell theory: its field data are complex vectors $E\in V$ whose coordinates are dimensionless amplitude ratios relative to one fixed reference amplitude $E_{\mathrm{r}\mathrm{e}\mathrm{f}}$; no absolute field strength is assigned. The analyser at angle $\theta$ acts on the field data by the orthogonal projection $E⟼P_{\theta }E;$ the model's declared dimensionless relative intensity of a field datum is $I(E):=\mathrm{\parallel }E{\mathrm{\parallel }}^2,$ and the transmitted intensity fraction is $I(P_{\theta }E)\mathrm{/}I(E)$ for $E\mathrm{\ne }0$. This is a declared idealisation of the classical description of a polariser as projecting the transverse field amplitude (the Jones-vector reading); it imports no Maxwell equation and makes no detector claim.

The example computes both ratios for the same declared data and compares them.

• What is compared. Only the two declared models on the declared two-dimensional space are compared. No claim is made that the elementary classical model reproduces the photon counting statistics or the non-classical states of the radiation sector. Quantum count laws are computed in Number statistics of a single declared mode: sharp, Poisson and geometric preparations; A two-bin comparison of a declared classical field model and the quantised model: identical normalised laws do not discriminate states the additional classical count law, preparation and detector declarations required for a count-statistics comparison.
• Sources and status. The coincidence of the classical transmitted fraction with the quantum transmitted probability is explicitly noted in the cited source; the classical model here is an elementary declared idealisation constructed for the comparison and is not a derivation from Maxwell theory.

Adopt the state and composition postulate P1, the observable and probability postulate P2 (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), the radiation extension postulate (Radiation extension of the framework: transverse mode and photon assignments), the radiation sector definition The radiation sector of the framework: modes and the number, energy and momentum assignments and the declared ideal counting model The declared ideal counting, energy-window and analyser detection models of the radiation sector. Assume the Axioms of Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The Axiom of Choice). Fix a unit mode vector $\phi \in \mathfrak{h}$ with coordinate representatives in ${\mathrm{m}}^{3\mathrm{/}2}$ under $d^{3}k$ by clause (b) of The radiation sector of the framework: modes and the number, energy and momentum assignments; write $U=U_{\phi }$ for the fixed-mode isometry of The fixed-mode isometry of Fock space and the transfer of number and ladder operators and let the dimensionless label $z\in \mathbb{C}$ and a declared effective mode energy $E_0>0$ in joules (equivalently a declared effective frequency ${\omega }_0=E_0\mathrm{/}\mathrm{\hslash }>0$ in ${\mathrm{s}}^{-1}$) be given parameters of this example. The example compares the count statistics of the following three declared preparations of the mode subspace, all read with the declared ideal total-number detector:

(a) The $n$-quanta mode state. For $n\in \mathbb{N}$, the pure preparation of the ray of $U{e}_n={\phi }^{\otimes n}$.

(b) The mode-coherent state. The pure preparation of the ray of $U{\psi }_z$, with dimensionless coherent-state label $z$ and ${\psi }_z=e^{-\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{/}2}{\sum }_m\frac{z^m}{\sqrt{m!}}{e}_m$.

(c) The declared effective-energy mode preparation. For a parameter $\beta >0$ in ${\mathrm{J}}^{-1}$, the density operator ${\rho }_{\beta ,E_0}:=(1-q){\sum }_{n\in \mathbb{N}}{q}^n{T}_{U{e}_n,U{e}_n},q:=e^{-\beta E_0}\in (0,1),$ the transfer to the mode subspace of the single-mode weights of Single-mode thermal trace with its parameter $\omega$ set equal to the declared effective energy $E_0$. This is a declared conditional preparation of the model, not a derivation from any radiation-matter interaction and not a statement that the mode is in thermal equilibrium.

The example computes the count law, mean and variance of the three preparations. It compares (b) and (c) at equal positive mean, and includes (a) in that comparison when the common mean is a positive integer and $n$ is chosen equal to it.

• Independent effective-energy assignment and Gibbs compression. The example independently declares $E_0$; a normalisable mode is not an eigenmode of the free energy, so the free Hamiltonian does not restrict to $E_0{N}^{\mathrm{\prime }}$ on this mode subspace. Nevertheless a compression of its bounded Gibbs operator is possible. Put $a_{\beta }:=⟨e^{-\beta \epsilon }\phi ,\phi ⟩$. Since $0<e^{-\beta \epsilon (k)}<1$ off the null set $\{k=0\}$ and $\mathrm{\parallel }\phi \mathrm{\parallel }=1$, $0<a_{\beta }<1$. The sector multiplier representation of Transverse continuum free Fock Hamiltonian gives $e^{-\beta T_n}=M_{{\prod }_i{e}^{-\beta \epsilon (k_i)}}$ on the symmetric transverse sector. Its pairing on ${\phi }^{\otimes n}$ is $a_{\beta }^n$ by the finite-product inner-product integral of Euclidean L² product unitary: L²(R^a)⊗̂L²(R^b) ≅ L²(R^{a+b}), and different sectors have zero cross pairing. Hence $U^{\ast }{e}^{-\beta H_f}U$ is the diagonal operator with entries $a_{\beta }^n$; its trace is $(1-a_{\beta }{)}^{-1}$ and its normalization gives the geometric weights $(1-a_{\beta })a_{\beta }^n$. At each fixed $\beta$ these correspond to the effective scale ${\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}(\beta )=-{\beta }^{-1}\log a_{\beta }>0$, with the real logarithm of The natural logarithm as the inverse of the exponential function; positivity follows because the exponential is strictly increasing with $\exp (0)=1$ (The exponential function is strictly increasing). The scale is generally dependent on $\beta$. This is compression and normalization of a positive operator, not a normalized Gibbs state of the full continuum or an invariant free-energy mode. The full Gibbs operator remains non-trace-class by clause (e) of Number and energy statistics of a declared mode: number states, coherent states and the continuum thermal limitation; no thermalisation or temperature identification follows.
• Counts are model quanta. All three laws are laws of the declared ideal total-number detector; a laboratory photon-counting comparison would require a declared counter response, efficiency and sampling model, and reported data. None is supplied or assumed here.

Adopt the state and composition postulate P1, the observable and probability postulate P2 and the preparation and apparatus discipline P7 (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, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), the radiation extension postulate clause (i) (Radiation extension of the framework: transverse mode and photon assignments) and the declared fixed-mode polarisation model The fixed-mode polarisation model of the framework with polarisation space $V$ and orthonormal basis $(e_1,e_2)$. Assume the Axioms of Countable Choice, Dependent Choice and Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain, The Axiom of Choice). Declare the following two-bin comparison:

1. an orthonormal pair $(u_{+},u_{-})$ of $V$, read as the pair of orthogonal fringe bin directions of a declared two-bin detector, with projections $Q_{\pm }x:=⟨x,u_{\pm }⟩u_{\pm }$;
2. a declared amplitude datum $a\in V\setminus \{0\}$ whose complex coordinates are dimensionless ratios relative to one fixed reference amplitude $A_{\mathrm{r}\mathrm{e}\mathrm{f}}$; $a$ need not have unit norm;
3. the declared classical field model, whose state datum is the amplitude coordinate $a$ relative to $A_{\mathrm{r}\mathrm{e}\mathrm{f}}$, whose dimensionless relative bin intensities are $I_{\pm }(a):=\mathrm{\parallel }{Q}_{\pm }a{\mathrm{\parallel }}^2$, and whose normalised two-bin law is $I_{\pm }(a)\mathrm{/}(I_{+}(a)+I_{-}(a))$;
4. the quantised model reading of the same datum: the pure preparation of the ray $[a]$ of the polarisation space, measured by the declared ideal sharp measurement with outcome projections $(Q_{+},Q_{-})$ declared directly in the sense of Declared ideal measurements and instruments of the framework clause (a), with the Born law of P2. This arbitrary-bin sharp measurement need not belong to the real-angle analyser family of the original frame.

Then:

(a) $Q_{+}+Q_{-}=I$ and the normalised stated laws of the two declared models coincide for every datum $a\mathrm{\ne }0$: $\frac{I_{\pm }(a)}{I_{+}(a)+I_{-}(a)}=p_{\pm }([a])=\frac{\mathrm{\parallel }{Q}_{\pm }a{\mathrm{\parallel }}^2}{\mathrm{\parallel }a{\mathrm{\parallel }}^2}=\frac{\mathrm{\mid }⟨u_{\pm },a⟩{\mathrm{\mid }}^2}{\mathrm{\parallel }a{\mathrm{\parallel }}^2};$

(b) writing $a=a_1+a_2$ for any declared decomposition of the datum into two path components, both declared laws contain the interference cross-term $\frac{2\mathrm{Re}(⟨u_{\pm },a_1⟩\overline{⟨u_{\pm },a_2⟩})}{\mathrm{\parallel }a{\mathrm{\parallel }}^2},$ so the presence of a fringe pattern of this two-bin form does not distinguish the two declared models;

(c) the two declared models are nevertheless different physical models: in the classical model the datum is a field amplitude with a deterministic intensity, while in the quantised model it is a state ray with a Born law; a discriminator requires a prediction on which two fully specified models differ, together with a declared detector response, preparation and sampling model in the sense of P7. Count statistics on a declared number-state preparation are a candidate for a further comparison only after a classical field-and-detector count law and the corresponding preparations in both models are declared. The deterministic classical intensities do not supply that count law, so no count-statistics discriminator follows from the present declarations;

(d) this item constructs no such discriminating comparison and reports no data; it asserts no measurement, no count, no frequency and no empirical conclusion, and it does not claim that the classical model agrees with the quantised model beyond the declared two-bin law.

• Classical waves interfere. The cross-term of clause (b) is present in the declared classical intensity as well as in the Born law, so this two-bin interference prediction alone does not distinguish the two declared models or refute the declared classical model; conversely, the declared quantised model predicts the pattern too, so the absence of a visible pattern in finitely many detections would not by itself falsify it. Statistical comparison requires the sampling model of P7 and reported data.
• What "classical" means here. The classical model of this item is the elementary declared transverse-amplitude model with intensity equal to squared amplitude norm; it is not Maxwell's theory, and the item imports no Maxwell theorem. A test of classical electromagnetism would require that framework's own postulates and field solutions.

What the three items establish. Malus and Jones ratios of the declared analyser: the same cos^2 from the Born law and from a classical transverse amplitude shows that the Born probability ${\cos }^2\theta$ of the declared ideal analyser on the preparation along $e_1$ and the transmitted intensity fraction ${\cos }^2\theta$ of the declared elementary classical transverse-amplitude model coincide for every declared angle, so that this ratio alone does not establish field quantisation. Number statistics of a single declared mode: sharp, Poisson and geometric preparations computes the count laws and their means and variances for three declared preparations of one mode -- sharp $n$, Poisson for dimensionless coherent-state label $z$ with mean and variance $\mathrm{\mid }z{\mathrm{\mid }}^2$, and the geometric law of a declared effective-energy preparation with ${\epsilon }_0>0$ joules and $\beta >0$ in ${\mathrm{J}}^{-1}$, whose mean is $1\mathrm{/}(e^{\beta {\epsilon }_0}-1)$ -- and shows different fluctuations for the coherent and geometric preparations at equal positive mean. A number state joins that equal-mean comparison when the common mean is a positive integer and $n$ is chosen equal to it. A two-bin comparison of a declared classical field model and the quantised model: identical normalised laws do not discriminate proves that the declared classical and quantised models assign the same normalised two-bin law, with the same interference cross-term, for every declared amplitude datum, and concludes that a two-bin fringe law of that form is not a discriminator between the two declared models.

What they rest on. All three items are conditional on the state and composition postulate P1, the observable and probability postulate P2, the adopted radiation extension and the declared fixed-mode or radiation-sector vocabulary of the main page; the counting results use only the declared ideal total-number detector of The declared ideal counting, energy-window and analyser detection models of the radiation sector. The classical models appearing in the first and third items are explicit elementary declared models -- dimensionless amplitude coordinates relative to a fixed reference amplitude, with relative intensity equal to squared amplitude norm -- and are not Maxwell's theory; the effective mode energy ${\epsilon }_0$ of the second item is a declared parameter, not a sharp energy of a normalisable mode.

Different objects. A one-photon preparation is a unit ray of the transverse one-particle space read in the one-photon sector; a mode-coherent preparation is a unit ray of the radiation Fock space with no sharp number when $z\mathrm{\ne }0$, while $z=0$ is the vacuum with sharp number zero; and the declared classical intensity datum is a complex transverse amplitude with its declared intensity. The three are distinct mathematical objects of the declared models, and no identification of one with another is made.

What is not claimed. The page records no experiment, no apparatus, no calibration, no measured count, no frequency, no efficiency and no uncertainty budget, and none is invented. It does not claim that any laboratory polariser, counter or interferometer realizes the declared ideal models; that claim is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). It does not assert thermal equilibrium or a temperature for the geometric preparation, and it does not assert that the coincidence of the classical and quantum ratios is evidence for field quantisation. Single-photon polarisation and counting experiments remain research candidates for a separate reported-experiment item; a finite sample of detections is a random object under any declared sampling model, visible fringes can fail to appear with nonzero probability under a correct coherent model, and a $p$-value or confidence level is not the probability that a model is false. Comparing any of these conditional predictions with data would test the combined model and apparatus assumptions, not prove the model or the framework.

Mathematical hypotheses. The conditional deductions summarized here retain the choice hypotheses of their mathematical suppliers; in particular, the transverse free-field Fock supplier assumes Countable Choice, Dependent Choice and Choice. The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain and The Axiom of Choice record these assumptions.