# Çift Yarık Deneyi (`cift-yarik-deneyi`): verification report

Rebuilt November 2026 from `sablon/sablon.html`. The core CSS and the `Q4` script are byte-identical to the current template, including the two coordinator fixes: the canvas clear in `cv.redraw` and `details > *{box-sizing:border-box}`. File: `simulasyonlar/cift-yarik-deneyi.html`, about 133 KB before packaging.

## 1. Audit of the original (Gemini 2025, `kaynak-eski/cift-yarik.js`, last ~5 KB)

| # | Problem | Evidence in the original | Correct physics |
|---|---|---|---|
| 1 | **The screen shows intensity squared.** | The screen loop computes `x=max(x,Math.pow(t,2))` and then `r=Math.pow(t,2)/x`, where `t=v(e)` is already cos²·sinc², the intensity. Each pixel's colour is scaled by `r`. | Intensity on a screen or photographic plate is proportional to I = \|ψ\|², not I². Squaring narrows every fringe and steepens the envelope. Halfway between a bright and a dark fringe, I/I₀ = cos²(π/4) = 0.5, but the old screen showed 0.25. The first side lobe of the single-slit envelope (4.7 % of the peak) dropped to 0.2 %. Test 24 checks the correct value. |
| 2 | **Arbitrary units.** | The sliders are "Dalga boyu 10–60 birim", "Yarık Genişliği 1–15 birim" and "Yarıklar Arası Mesafe 10–80 birim", with distances in canvas pixels (screen at x = 630, barrier 30–570 px). | There is no link to any real setup. The new version uses SI units and presets from real experiments (Section 2). |
| 3 | **Small-angle formula used everywhere, giving unphysical fringes.** | `u=y−300` and `c=π·d·u/(λ·L)` with L = 630 − barrier, which can be as small as 60 px while \|u\| reaches 300 px. So sinθ is replaced by u/L, which can reach 5. | sinθ = y/√(y²+L²) ≤ 1. With d = 10 and λ = 60 (d < λ), physics allows only the zeroth order, yet the original drew many fringes at "sinθ" > 1. Near the barrier the Fraunhofer formula is also invalid (L < d²/λ). The new model uses Fresnel integrals or the full Kirchhoff path sum. |
| 4 | **The wavefronts are decorative.** | Plane-wave lines are drawn every λ/2 with crests and troughs in the same colour. They are offset by `p·λ/(2π) mod λ` while the wavelets advance by `p mod (λ/2)`, so the two regions move at different speeds. Lines start at `e+n` with `n` up to λ and leak up to λ past the barrier. The semicircles from the slits are fixed arcs, not a computed field. | The new wave view is a computed field: per-pixel superposition of the slit contributions. It shows the instantaneous field at true scale when λ can be resolved (ripple-tank preset), and the time-averaged \|ψ\|² otherwise. |
| 5 | **No particle behaviour.** | None in the code. | One-at-a-time detection sampled from \|ψ\|² is the quantum point of the experiment (Feynman III §1; Tonomura 1989). It is now the centre of the lab. |
| 6 | **No contrasts.** | None. | Added: a classical prediction P₁ + P₂, closing either slit, a which-path detector, and (İleri) partial distinguishability with V = √(1 − D²). |
| 7 | **UX issues.** | Animation never stops (no pause, ignores reduced motion). Colour is an ad hoc rainbow mapped from the unitless "wavelength". The slit width can be 1 px, so the slit is invisible. There is no intensity plot or readout. | The new version uses `Q4.loop`, `Q4.reduced`, `Q4.wl2rgb` (CIE-based) and labelled plots with units. |

**Kept:** all four original controls: wavelength, slit width, slit separation, and barrier position (now the slit–screen distance L). They now use physical units and log scales. The animated wave view and the coloured screen strip are kept as well.

## 2. What the new version models

**Geometry.** N parallel, infinitely long slits (default N = 2), each of width a, with centres spaced d apart. The screen is at distance L. The problem is two-dimensional and scalar. Slits are numbered from the top; "Yalnız üst/alt" leaves only the top or bottom slit open.

**Amplitude of slit j at (y, z).** All three methods are normalised so that |ψ| → 1 in the geometrically lit region.
- *Fraunhofer:* ψ = a/√(λz) · sinc(πa sinθ/λ) · e^{−ik s_j sinθ}, with sinθ = y/√(y²+z²) (non-paraxial angle). Used only as a comparison curve at İleri and in analytic tests.
- *Fresnel (paraxial):* ψ = e^{−iπ/4}/√2 · [C(w) + iS(w)] evaluated between w₁ and w₂, where w = (s − y)√(2/λz). The Fresnel integrals use the Numerical Recipes `frenel` algorithm (power series for |x| ≤ 1.5, complex continued fraction above, accuracy about 1e−10). This covers near field and far field in one formula. It is used whenever all angles are below 0.12 rad and λ < 0.05a, which covers every light, electron and C₆₀ preset.
- *Kirchhoff (exact path):* each slit is split into M point sources, with M chosen adaptively so the phase step between neighbours is below π/6. Each source uses the 2D Green function √(1/iλr)·e^{ikr} and the obliquity factor (1 + cosθ)/2. The phase is computed as k·Δy²/(z + r) to avoid cancellation at large kr. Used for wide angles (ripple tank).

**Born rule with path information.** I = (1 − V)Σ|ψ_j|² + V|Σψ_j|², summed over open slits.
- V = 1: quantum, no path information.
- V = 0: classical particles, or a which-path detector with D = 1.
- V = √(1 − D²): partial which-path information (İleri), from Englert 1996, equality for pure states.

**Temporal coherence (İleri, and the C₆₀ preset).** The pattern is averaged over a Gaussian λ distribution with a given FWHM Δλ/λ, using 9-point Gauss–Hermite quadrature.

**de Broglie wavelengths.**
- Electron: λ = h/√(2m_e eU(1 + eU/2m_ec²)), relativistic.
- C₆₀: λ = h/(mv) with m = 720 u.
- Constants are CODATA 2018 from `Q4.C`; u = 1.66053906660e−27 kg is defined locally.

**Particles.** Each detection is sampled from the tabulated screen distribution: 1201 points over ±Y, trapezoidal CDF, inverse-CDF sampling. When the path is known (classical, or D = 1), a slit is chosen first with weight ∫|ψ_j|², then y is sampled from that slit's own distribution. This shows which slit "fired" without changing the statistics. The coordinate along the slit is uniform.

Outputs:
- 2D detector front view in the style of Tonomura's frames.
- Histogram (120 bins) against the theoretical curve. At Lisans and above, the classical P₁ + P₂ curve is drawn dashed at equal particle number. At İleri, the Fraunhofer curve is drawn dotted.
- Markers for envelope zeros, Δy, and missing orders when d/a is close to an integer.

**Wave view (side view).** A computed map of |ψ|² over 0…X_view (adjustable at Lisans and above). Columns are normalised individually; when fringes are finer than one cell, the cell is supersampled and averaged so no moiré appears. The vertical axis is stretched (the caption gives the factor). When the view is isotropic and λ ≥ 4 px, the instantaneous field Re[ψe^{i(kx−ωt)}] is animated instead. Markers show a²/λ and d²/λ (the near-field/far-field boundaries).

**Presets.**

| Preset | λ | a | d | L | Notes |
|---|---|---|---|---|---|
| Red laser | 632.8 nm | 40 µm | 250 µm | 1 m | PASCO OS-8453 double slit 0.04/0.25 mm |
| Green laser | 532 nm | 40 µm | 250 µm | 1 m | |
| Ripple tank | 1.5 cm | 1 cm | 4 cm | 30 cm | wide angles, Kirchhoff, crests at true scale; particle view disabled (classical wave) |
| Jönsson 1961 | 50 kV → 5.36 pm | 0.5 µm | 2.0 µm | 0.35 m | AJP 1974 translation, body text; L < d²/λ = 0.75 m, so the Fresnel and Fraunhofer curves visibly differ |
| Tonomura 1989 | 50 kV | 0.8 nm (equiv.) | 5.74 nm (equiv.) | 1.5 m | biprism modelled as two equivalent virtual sources, chosen to reproduce 1.4 mm fringes on the detector (0.7 µm × 2000 magnification). The values are an equivalent model, not physical slits, and the page says so. |
| Arndt 1999 C₆₀ | 220 m/s → 2.52 pm | 50 nm | 100 nm | 1.25 m | the real experiment used a grating; shown as a double slit with the same a and d (d = 2a removes even orders); Δv/v = 60 % FWHM. At İleri, N up to 12 approaches the grating. |
| Missing order | 532 nm | 80 µm | 240 µm | 1 m | d = 3a |

## 3. Verification table (Doğrulama tab: 24/24 pass)

| # | Test | Expected | Model | Source / basis | Result |
|---|---|---|---|---|---|
| 1 | Fringe spacing from numerically located maxima (He-Ne, d = 0.25 mm, a = 5 µm, L = 1 m) | λL/d = 2.5312 mm | 2.5309 mm | Young relation. With a = 40 µm the envelope slope pulls the maxima inward by about 1 %, which is real physics and is noted on the page. | pass (0.01 %) |
| 2 | Spacing of dark fringes, real slide (a = 0.04 mm) | 2.5312 mm | 2.5312 mm | d sinθ = (m + ½)λ; zeros are not shifted by the envelope | pass |
| 3 | First envelope zero, single slit | λL/a = 15.820 mm | 15.820 mm | a sinθ = λ (Hecht §10.2) | pass |
| 4 | Missing order, d = 3a | I₃/I₀ = 0 | 1.5e−33 (I₁ = 0.684, I₂ = 0.171) | analytic | pass |
| 5 | Bright fringes in the central envelope, d = 3a | 2d/a − 1 = 5 | 5 | analytic; two remnants at 0.4 % of the peak excluded by a 1 % threshold | pass |
| 6 | Sampler KS test, 10⁵ samples (seeded) | √n·D < 1.63 | 0.769 | α = 0.01 | pass |
| 7 | Sampler χ² test, 80 bins | below the 0.1 % limit (~115) | 68.6 (75 dof) | | pass |
| 8 | Electron λ at 50 kV, relativistic | 5.3551 pm | 5.3553 pm | λ[nm] = 1.22639/√(U(1 + 0.97848e−6 U)) (Williams & Carter); Tonomura 1989: 0.054 Å | pass |
| 9 | Electron λ at 50 kV, non-relativistic | 5.4848 pm | 5.4847 pm | 1.22643/√U nm | pass |
| 10 | C₆₀ λ at 220 m/s | 2.5 pm | 2.519 pm | Arndt et al. 1999 (rounded value) | pass (0.8 %) |
| 11 | Davisson–Germer, 54 eV | 1.67 Å | 1.669 Å | de Broglie prediction quoted by DG 1927 (measured 1.65 Å) | pass |
| 12 | Electron speed at 50 kV | 0.413 c | 0.413 c | relativistic kinematics; Hitachi: "about 40 %" | pass |
| 13 | C₆₀ grating (N = 40), first order at 1.2 m | 34 µm | 33.6 µm | Nairz, Arndt & Zeilinger, AJP 71, 319 (2003): λ = 2.8 pm, d = 100 nm. A pure double slit with a = 55 nm would put the peak at 30 µm, which is noted. | pass (1.2 %) |
| 14 | Which-path pattern = P₁ + P₂ | 0 | 8.6e−17 relative | Feynman III §1-6 | pass |
| 15 | Englert limits V(0) = 1, V(1) = 0, D² + V² = 1 | | | Englert, PRL 77, 2154 (1996) | pass |
| 16 | Visibility measured from the model pattern, D = 0.6 | 0.8 | 0.8000 | (I_max − I_min)/(I_max + I_min) | pass |
| 17 | Intensity ≥ 0 over 1500 random configurations (N 2–5, random V, open slits, spread) | ≥ 0 | min 0 | (1 − V)Σ\|ψ\|² + V\|Σψ\|² ≥ 0 | pass |
| 18 | Fresnel integral C(1) (S(1) shown in the note) | 0.7798934 | 0.7798934 | Abramowitz & Stegun Table 7.7 | pass |
| 19 | Far field: Fresnel = Fraunhofer at L = 100 d²/λ | max diff / peak < 1 % | 3.8e−4 | analytic limit | pass |
| 20 | Kirchhoff path sum vs Fresnel integral, near field (L = 5 cm < d²/λ) | < 1 % | 3.6e−3 | two independent methods (Fraunhofer deviates by about 8.6 % here) | pass |
| 21 | Flux conservation ∫P₁₂ = ∫(P₁ + P₂) | 1 | 1.0000 | interference redistributes and does not create | pass |
| 22 | Single-slit flux = a | 40 µm | 39.75 µm | normalisation; the 0.6 % deficit is the tails beyond ±25 cm | pass |
| 23 | Near field, N_F = 1000: fraction of light inside the geometric shadow | 1 | 0.993 | geometric optics limit (edge-diffraction tail falls off slowly) | pass |
| 24 | I/I₀ halfway between a bright and a dark fringe | cos²(π/4) = 0.5 | 0.5000 | analytic; the original showed 0.25 | pass |

I also cross-checked the Fresnel integrals against brute-force quadrature at x = 0.5, 1.4, 1.6, 2.5, 3.7 and −2 (agreement to 1e−10) during development (node).

## 4. Historical claims with sources

All of these were verified by web research. Primary sources were used where available, with OCR from archive scans.

- **Young, Bakerian Lecture, read 12 Nov 1801**, *Phil. Trans.* 92, 12 (1802). **Second Bakerian Lecture, read 24 Nov 1803**, *Phil. Trans.* 94, 1–16 (1804).
  - The "slip of card, about one-thirtieth of an inch in breadth" in a sunbeam admitted through a needle hole in a window shutter.
  - Blocking one edge with a small screen made "all the fringes … immediately disappear".
  - Verified from the JSTOR scan via Zenodo.
- **The myth question.** The only two-slit text in Young is in *A Course of Lectures on Natural Philosophy* (1807), vol. 1, Lecture XXXIX, pp. 464–465: "the simplest case appears to be, when a beam of homogeneous light falls on a screen in which there are two very small holes or slits…". The optical figure is Plate XXX Fig. 442; Plate XX Fig. 267 shows water waves. The passage gives no apparatus, date or measurements. Historians treat it as unclear whether Young performed it, citing Kipnis 1991, Robinson 2006 and Rothman 2003; I only had these second-hand. The page presents it as an open question. Physics World 2002 does say "1801 double-slit", which the primary record does not support.
- **Fresnel.** Memoir deposited 29 Jul 1818, prize 15 Mar 1819; Poisson's objection and Arago's observation of the spot. **Arago & Fresnel**, *Ann. Chim. Phys.* 10, 288 (1819): orthogonally polarised rays do not interfere. Calling this a "path marker" is our framing, and the card attributes it to us.
- **Taylor**, *Proc. Camb. Phil. Soc.* 15, 114 (1909). Shadow of a needle, gaslight through a slit, smoked glass. Longest exposure 2000 hours ("about 3 months"). Light equal to "a standard candle burning at a distance slightly exceeding a mile". No loss of sharpness. Not a double slit, and the paper does not use the word "photon"; the page says both.
- **de Broglie** thesis defended 25 Nov 1924 (secondary source); Nobel 1929. **Davisson & Germer**, *Nature* 119, 558 (Apr 1927) and *Phys. Rev.* 30, 705 (Dec 1927): 54 eV, peak at 50°, 1.65 Å measured vs 1.67 Å predicted (secondary source). **Thomson & Reid**, *Nature* 119, 890 (1927). Nobel 1937.
- **Jönsson**, *Z. Phys.* 161, 454 (1961); English translation *AJP* 42, 4 (1974).
  - 50 kV electrons, copper foil, up to five slits.
  - The body text gives 0.5 µm width and 2.0 µm spacing; the abstract gives 0.3 µm and 1 µm. We used the body values.
  - Observation plane 350 mm beyond the slits; electron-optical magnification up to 100×.
- **Merli, Missiroli & Pozzi**, *AJP* 44, 306 (1976), received 1974.
  - Biprism, TV image intensifier, "one or a few electrons … in 0.04 sec".
  - Film "Interferenza di elettroni", awarded in Brussels in 1976.
  - The 2003 priority dispute in Physics World letters is from secondary sources (Physics World, Rosa 2012); the letters themselves were not read.
- **Tonomura, Endo, Matsuda, Kawasaki & Ezawa**, *AJP* 57, 117 (1989).
  - 50 kV, about 10³ e/s, mean spacing between electrons 150 km.
  - Frames of 10, 100, 3000, 20 000 and 70 000 electrons; fringes form in about 20 min.
  - Biprism filament < 1 µm at +10 V; 0.7 µm fringes in the image plane, magnified 2000× to 1.4 mm.
  - Hitachi's web page says "10 electrons per second", contradicting the paper. We use the paper.
- **Arndt et al.**, *Nature* 401, 680 (1999).
  - v ≈ 220 m/s, λ = 2.5 pm, SiN grating with period 100 nm and slit width 50 nm, laser-ionisation detection.
  - Δv/v ≈ 60 % FWHM, from the group's web page.
  - The 1.25 m flight path comes from the group's website (search snippet only). The test uses the AJP 2003 numbers instead (1.2 m, 34 µm).
- **Englert**, *PRL* 77, 2154 (1996): D² + V² ≤ 1, with equality for pure states.
- **Physics World poll**, R. P. Crease, Sept 2002: #1 Young's double slit applied to single electrons; #5 Young's light interference.
- **Feynman**, *Lectures* Vol. III §1-1. Verbatim: "…which has in it the heart of quantum mechanics. In reality, it contains the *only* mystery." The first two sentences were verified; caltech.edu was blocked, so a PDF copy of the chapter was used.
- **Dirac**: "Each photon then interferes only with itself. Interference between two different photons never occurs." This is the 2nd edition (1935) onward, §3. The 1930 edition has "can never occur" in §5. The page cites the 2nd edition and notes that HOM two-photon interference qualifies the second sentence.
- **Bach, Pope, Liou & Batelaan**, *New J. Phys.* 15, 033018 (2013): 600 eV, slits 62 nm wide and 272 nm apart, movable mask, 6235 electrons in about 2 h.
- **Grangier, Roger & Aspect**, *Europhys. Lett.* 1, 173 (1986): single-photon interference in a Mach–Zehnder interferometer, not a double slit; the page says so.

## 5. Known limitations

- **Geometry and wave model.** Scalar, two-dimensional (infinitely long slits). No polarisation, spin or charge effects, and no slit-edge interactions (image charge for electrons, van der Waals forces for C₆₀, which narrow the effective slit in the real experiment).
- **Source and detector.** Point source with full spatial coherence. Only temporal coherence is optional (Δλ/λ). The detector is ideal.
- **Tonomura preset.** An equivalent two-source model: the biprism's overlap window is replaced by the sinc envelope of a very narrow "virtual slit", and the electron-lens magnification is folded into L. The side-view near-field structure in that preset therefore has no physical meaning; the caption and the Fizik tab say it is an equivalent model.
- **C₆₀ preset.** Shown as a double slit; only N ≤ 12 is available at İleri, while the real grating had about 100 illuminated slits. Collimation and detector-width broadening are not modelled.
- **Which-path detector.** Modelled through the overlap of detector states only. There is no recoil or decoherence dynamics, and no quantum eraser.
- **Side view.** Wave crests are drawn only when λ ≥ 4 px on an isotropic scale, which happens only for the ripple tank. Otherwise the view is time-averaged with a stretched vertical axis, and the caption says so. In the ripple-tank preset the side view is isotropic (about ±8 cm) while the screen strip and plots cover ±33 cm. Each is labelled with its own scale.
- **Classical prediction.** Defined operationally as P₁ + P₂, each measured with one slit open (Feynman's argument). It is not a ballistic "bullet" model. For realistic far-field parameters it gives one broad peak, not two bands; two bands appear when the screen is in the near field (N_F ≳ 1). The page explains this.

## 6. Core-template changes

None in my copy. The two template fixes the coordinator announced mid-task (`ctx.clearRect` in `cv.redraw`, and `details > *{box-sizing:border-box}`) are present, because my build splices from the current `sablon/sablon.html`. I verified that the core CSS and the Q4 script are byte-identical to the template.

One suggestion for the template: `Q4.ready()` only parses `#tab-level`. I added simulation-side deep links (`#lab-1&p=tonomura&v=particle&n=6000&wp=1&D=0.6&o=top&ph=classical&xv=…`) for screenshots and teacher links. A generic hook for this in the core might be useful.