Qm Interference And Local Detection — Examples

Adopt the declared two-path preparation and detection model The declared two-path preparation model and its ideal detection bins in its spin-zero reading $r=1$, and the coherent and which-path detection laws Coherent and which-path detection laws of the declared two-path model. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), as required by the preparation and detection-law suppliers. Fix lengths $0<d<L<\mathrm{\infty }$, measured in metres, and declare the two intervals $I_1:=[-\frac{d}{2}-\frac{L}{2},-\frac{d}{2}+\frac{L}{2}],I_2:=[\frac{d}{2}-\frac{L}{2},\frac{d}{2}+\frac{L}{2}]$ on the first spatial coordinate, measured in metres, whose overlap region $O:=I_1\cap I_2=[\frac{d}{2}-\frac{L}{2},-\frac{d}{2}+\frac{L}{2}]$ has length $L-d>0$; let $g\in L^2({\mathbb{R}}^2;\mathbb{C})$ satisfy $\mathrm{\parallel }g{\mathrm{\parallel }}_2=1$ and choose a Borel representative, which exists by completion of Borel Lebesgue measure ($\mathcal{L}({\mathbb{R}}^n)$ is exactly the completion of the restriction of ${\lambda }_n$ to the Borel sets). Its amplitude unit is ${\mathrm{m}}^{-1}$ in the two-dimensional position measure; declare the flat path amplitudes $f_1:=L^{-1\mathrm{/}2}{\mathbf{1}}_{I_1},f_2:=L^{-1\mathrm{/}2}{\mathbf{1}}_{I_2}\mathrm{.}$ Define the declared path components by ${\psi }_j(x_1,x_2,x_3):=f_j(x_1)g(x_2,x_3)(j=1,2)\mathrm{.}$ In these SI conventions each $f_j$ has unit ${\mathrm{m}}^{-1\mathrm{/}2}$ and each ${\psi }_j$ has unit ${\mathrm{m}}^{-3\mathrm{/}2}$, as required for a normalized position wave function on ${\mathbb{R}}^3$; the probabilities below are dimensionless. A slab bin is a set $B_1\times {\mathbb{R}}^2$ with $B_1\subseteq \mathbb{R}$ Borel. Then:

(a) ${\psi }_1,{\psi }_2\in H_0=L^2({\mathbb{R}}^3;\mathbb{C})$ with $\mathrm{\parallel }{\psi }_1\mathrm{\parallel }=\mathrm{\parallel }{\psi }_2\mathrm{\parallel }=1$, so the declared data are admissible, and the model is admissible at every phase: on $I_1\setminus I_2$ the combination $f_1+e^{i\phi }{f}_2$ equals $L^{-1\mathrm{/}2}$ independently of $\phi$, and that region has length $d>0$. Since $\mathrm{\parallel }g\mathrm{\parallel }=1$, $\mathrm{\parallel }{\psi }_1+e^{i\phi }{\psi }_2{\mathrm{\parallel }}^2=\mathrm{\parallel }{f}_1+e^{i\phi }{f}_2{\mathrm{\parallel }}^2\ge d\mathrm{/}L>0$.

(b) For every slab bin, with $N_{\phi }:=\mathrm{\parallel }{f}_1+e^{i\phi }{f}_2{\mathrm{\parallel }}^2=2+\frac{2(L-d)}{L}\cos \phi$, ${\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G(B_1\times {\mathbb{R}}^2)=\frac{1}{N_{\phi }}{\int }_{B_1}\mathrm{\mid }{f}_1+e^{i\phi }{f}_2{\mathrm{\mid }}^{2}dx,{\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w)}^G(B_1\times {\mathbb{R}}^2)=\frac{w\mathrm{\mid }{B}_1\cap I_1\mathrm{\mid }+(1-w)\mathrm{\mid }{B}_1\cap I_2\mathrm{\mid }}{L}\mathrm{.}$

(c) The coherent density $\mathrm{\mid }{f}_1+e^{i\phi }{f}_2{\mathrm{\mid }}^2$ equals $\frac{2(1+\cos \phi )}{L}$ on $O$, equals $\frac{1}{L}$ on the symmetric difference $I_1\mathrm{△}{I}_2$, and vanishes outside $I_1\cup I_2$; consequently ${\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G(O\times {\mathbb{R}}^2)=\frac{(L-d)(1+\cos \phi )}{L+(L-d)\cos \phi },{\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G((I_j\setminus I_{3-j})\times {\mathbb{R}}^2)=\frac{d}{2L+2(L-d)\cos \phi }$ for $j=1,2$, and for the balanced which-path preparation $w=\frac{1}{2}$ ${\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(1\mathrm{/}2)}^G(O\times {\mathbb{R}}^2)=\frac{L-d}{L},{\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(1\mathrm{/}2)}^G((I_j\setminus I_{3-j})\times {\mathbb{R}}^2)=\frac{d}{2L}\mathrm{.}$

(d) At $\phi =\pi$ the coherent law assigns probability zero to the whole overlap region $O$ of positive length, while the balanced which-path law assigns it probability $(L-d)\mathrm{/}L>0$; every slab bin contained in $O$ has coherent probability zero, every slab bin meeting $I_1\setminus I_2$ or $I_2\setminus I_1$ in a set of positive length has positive coherent probability, and every slab bin meeting $O$ in a set of positive length has positive which-path probability.

The computation is exact in the declared model; it asserts no detector response, no sampling model, no measured count and no laboratory realization.

• The dark region is a feature of the declared flat model. The vanishing of the coherent density on all of $O$ at $\phi =\pi$ uses the flat amplitudes and the exact cancellation $f_1+e^{i\phi }{f}_2=0$ there; a model with unequal weights or non-flat amplitudes need not vanish on a region. At $\phi =\pi$, a slab bin $B_1\times {\mathbb{R}}^2$ has positive coherent probability exactly when $\mathrm{\mid }{B}_1\cap (I_1\mathrm{△}{I}_2)\mathrm{\mid }>0$; it has zero coherent probability exactly when this intersection has length zero. Adding points in $O$ or outside $I_1\cup I_2$ does not change that probability. These are ideal-bin probabilities; recorded-event probabilities require a declared response model.
• No data are used. The example computes probabilities in the declared model; it does not report counts, and the flat amplitudes are a declared idealization, not a measured profile.

Adopt the declared two-path preparation and detection model The declared two-path preparation model and its ideal detection bins with path components ${\psi }_1,{\psi }_2\mathrm{\ne }0$ satisfying $⟨{\psi }_1,{\psi }_2⟩=0$ (orthogonal components), and the coherent and which-path detection laws Coherent and which-path detection laws of the declared two-path model. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), as required by the preparation and detection-law suppliers. Fix an integer $n\ge 2$ and declare the finite phase-averaging model: the relative phase takes the $n$ values ${\phi }_k:=\frac{2\pi k}{n}(k=0,1,\dots ,n-1)$ with equal statistical weight $1\mathrm{/}n$, so that the averaged bin law of a Borel set $B\subseteq {\mathbb{R}}^3$ is $\bar{p}(B):=\frac{1}{n}{\sum }_{k=0}^{n-1}{\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}({\phi }_k)}^G(B)\mathrm{.}$ Put $N:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2$ and $w_{\ast }:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2\mathrm{/}N$. Then:

(a) $\mathrm{\parallel }{\psi }_1+e^{i{\phi }_k}{\psi }_2{\mathrm{\parallel }}^2=N$ for every $k$, so every coherent preparation of the declared model has the same normalization $N$;

(b) the averaged cross term vanishes bin by bin: $\frac{1}{n}{\sum }_{k=0}^{n-1}2\mathrm{Re}(e^{-i{\phi }_k}{\int }_B⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}dx)=0;$

(c) consequently $\bar{p}(B)={\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w_{\ast })}^G(B)$ for every Borel set $B$: the declared finite phase-averaging model reproduces the which-path law with the norm weights, while each single phase shows the cross term of Coherent and which-path detection laws of the declared two-path model.

The averaging over the phase is a declared statistical model of the preparation ensemble in the sense of clause (ii) of the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions; the computation asserts no decoherence mechanism, no detector response, no counts and no empirical result, and it does not cover the continuous uniform phase distribution, which would require its own integral model.

• Source scope. Tong supplies the two-slit cross term and discusses which-path monitoring; Binney and Skinner discuss alternative-route amplitudes and spatial averaging over unresolved fringes. Neither cited passage specifies this finite random-phase ensemble. Its hypotheses are declared locally, and its averaged law is proved here using the detection laws and the sum of roots of unity.
• The averaging is not a dynamics. Nothing here evolves the state or removes coherence dynamically; the computation concerns a declared ensemble of preparations whose relative phase is randomized over a finite set. A continuous uniform phase distribution is a different declared model and is not computed here.
• Orthogonality is used. The identification of the averaged law with the which-path law uses $⟨{\psi }_1,{\psi }_2⟩=0$, which makes the normalization independent of the phase. For non-orthogonal components the averaged law need not coincide with the which-path law, and no claim is made about that case.

The source is the primary preprint of Bach, Pope, Liou and Batelaan (arXiv:1210.6243v1, 22 October 2012), reporting a controlled double-slit electron diffraction experiment. The reported apparatus consists of a thermionic tungsten filament with electrostatic lenses producing an electron beam of energy 600 eV; a collimation aperture of 2 micrometre width and 10 micrometre height placed 16.5 cm from the source; a double-slit 30.5 cm beyond the collimation aperture, milled with a 30 keV Ga+ focused-ion-beam system into a 100 nm silicon-nitride membrane coated with about 2 nm of gold, with two slits of 50 nm width, 280 nm centre-to-centre separation and 4 micrometre height, each with a 150 nm support midway; a movable mask 5 micrometre wide, driven by a piezoelectric actuator and placed 230 micrometre downstream of the double-slit; an electrostatic quadrupole lens magnifying the pattern; and a two-dimensional microchannel plate and phosphor screen imaged by a CCD camera. The body of the paper gives the mask height as 10 micrometre while the caption of figure 1 gives 20 micrometre; the inconsistency is recorded and not resolved here. The comparison is a reported experiment of the framework in the sense of the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions; the identification of the reported device with the quantities of any model's preparation or detection is an apparatus identification carrying the source's limitations.

Two procedures are reported. First, for the slit-control measurements the mask is moved across the double-slit so that predominantly one slit, then both slits, then the other slit transmit; the recorded image frames for each mask position are summed and the accumulated image normalized, producing the distributions labelled P1 (slit 1), P12 (both slits) and P2 (slit 2). Second, for the build-up measurement the mask is centred, the source intensity is reduced until the detection rate in the pattern is about 1 Hz, the quadrupole lens is set to magnify the central five diffraction orders, and localized detections are recorded with a blob-localization scheme; the reported final accumulation took about 2 hours. The paper reports an average distance between consecutive electrons of 2.3e6 m at this rate and energy, and states that this ensures at most one electron is present in the 1 m apparatus at any time. No arrival-process law, occupancy bound, event-selection rule, dead-time model or independence test is reported, so that statement remains a reported apparatus-behaviour inference; no finite-sampling analysis under a declared independent-trials model is performed here or in the source.

The paper reports, and the retained figures show, slit-selective probability distributions: a single-slit central diffraction order in the P1 and P2 configurations and a double-slit distribution in the P12 configuration, with weak double-slit contributions visible in the first diffraction order of the nominal single-slit configurations, which the paper attributes to the finite separation between the mask and the double-slit. For the mask-centred build-up the figure captions report accumulated localized-event counts of 2, 7, 209, 1004 and 6235 in five successive images of the central five orders, the paper describing the earliest images as appearing random and a pattern becoming discernible at the largest counts, with a clearly visible pattern in the final image. Each reported detection is a localized blob whose size represents the localization error of the detection scheme; the accumulated distribution, not any single event, carries the reported pattern.

The report supplies no numerical uncertainty budget for its observations. In detail: no calibrated detector efficiency or background map is given; no numerical value of the localization error is given although the blob size is said to represent it; no uncertainty is given for the beam energy, the slit dimensions or the mask position; no statistical independence of the recorded events is tested; no event-selection or image-processing error rate is quantified; and no significance level, confidence interval or fitted comparison with a competing distribution is reported. The mask-height inconsistency between the body text and the figure caption is unresolved. The reported claim that at most one electron is present at a time rests on the average-separation estimate and not on a bound on the probability of two electrons in the apparatus. The approximate rate of about 1 Hz and duration of about 2 hours are not converted here into a trial count, efficiency or calibrated rate, and no uncertainty is manufactured for them.

The source interprets the slit-selective patterns and the build-up of a diffraction pattern from localized detections as a realization of Feynman's two-slit thought experiment, illustrating interference and wave-particle duality for electrons. Read as a report, the accumulated distributions and their dependence on which slits transmit are consistent with a coherent electron-amplitude model under the reported preparation and apparatus conditions; the report's own comparison is qualitative and no competing incoherent or classical model is specified, fitted or tested, so the report does not by itself discriminate between such models. Classical wave models also exhibit interference, so a reported pattern does not establish the quantum framework or exclude every alternative; conversely, nothing here falsifies a model, and each recorded event remains a localized detection whose relation to any theoretical amplitude is a model identification. A controlled test of the declared two-path law Coherent and which-path detection laws of the declared two-path model would require a declared preparation model, a declared detector response including efficiency, resolution and background, a declared competing law, and a declared sampling model with its statistical method; none of these is supplied by this report, and no such analysis is performed here. Agreement of the reported distributions with a model would be evidence about the combined model, apparatus and analysis assumptions, not a proof of the model.

The two computations. An ideal flat two-path distribution: exact coherent and which-path bin laws computes, in the declared two-path model with flat product-form amplitudes, the exact coherent and which-path probabilities of the overlap and outer regions of the declared detection coordinate: at relative phase $\pi$ the coherent law assigns probability zero to the whole overlap region while the balanced which-path law assigns it positive probability, and a slab bin $B_1\times {\mathbb{R}}^2$ has positive coherent probability at that phase exactly when $\mathrm{\mid }{B}_1\cap (I_1\mathrm{△}{I}_2)\mathrm{\mid }>0$, where $I_1,I_2$ are the declared intervals and $I_1\mathrm{△}{I}_2$ is their symmetric difference (the two outer regions). Mere nonempty intersection does not suffice. A declared finite phase-averaging model reproduces the which-path law for orthogonal components shows that for orthogonal path components a declared ensemble whose relative phase is uniformly distributed over the finitely many $n$-th roots of unity has the which-path law with the norm weights as its averaged law. Both are conditional computations in the declared model under all the stated supplier hypotheses, including Countable Choice and Choice, and the pair's detection laws (Coherent and which-path detection laws of the declared two-path model); neither is a dynamics, an environmental decoherence model, a detector response or a statement about an apparatus.

The reported experiment. Reported controlled double-slit electron diffraction with single-electron detection (Bach, Pope, Liou and Batelaan) reports the primary preprint of Bach, Pope, Liou and Batelaan: a 600 eV electron beam through a nanofabricated double-slit with slit-selective distributions and a single-electron event build-up over 2, 7, 209, 1004 and 6235 accumulated localized events. The report supplies no calibrated efficiency or background, no numerical localization error, no uncertainty for its apparatus parameters, no occupancy bound and no statistical comparison with a specified competing model; the mask-height inconsistency between its body and its figure caption is recorded as unresolved. The item therefore supports a qualitative operational account only, and its agreement with any model is evidence about the combined model and apparatus assumptions, not a proof of the framework, not a demonstration that every event is independent, and not a demonstration that only one electron is in the apparatus at a time.

What the page does not address. No photon experiment is reported here; a single-photon polarization or counting experiment belongs to the photon and radiation page, and the classical wave comparison for light is a separate declared model. No experiment testing the which-path or marker variants of the arrangement is reported. No claim is made that the finite sample of the reported experiment would distinguish the coherent from the which-path recorded law; A finite sample can fail a criterion that rejects a positive-probability sequence shows that both can fail or pass a finite-sample criterion and that coincident recorded laws are indistinguishable by any criterion.

Evidence status. The page combines exact conditional computations with one faithfully reported experiment. It contains no simulation presented as data, no invented count, precision or significance, and no conclusion about nature; the empirical qualifications of the report are carried in its Observations and Uncertainty sections and are not strengthened here.