The Physics of a Double-Slit Experiment: Testing Internal Consistency
A companion to “Did Attention Really Collapse the Photon Wavefunction?”
Companion note: This post is meant to be read alongside my main article, “Did Attention Really Collapse the Photon Wavefunction?” While the main article focuses on direct readings from published figures, this post applies diffraction theory to test their internal consistency.
In the main post, On the Question of Wavefunction Collapse in a Double-Slit Experiment [1], I began with direct readings from the 2012 paper’s figures—the experimental setup diagram, the parameter table, the 3,000-pixel interference pattern, and the FFT power spectrum. Here, I apply standard double-slit diffraction to those reported values and test their internal consistency.
Inputs. Table I (in the main post) reproduces the parameters as published in 2012. Using these inputs, I derive the expected relations among slit separation d, fringe spacing δ, slit-camera distance D, and the main FFT peak frequency, K, and compare those predictions with values obtained from the figures.
Conventions. All equations are labeled A1–A22 and cited by their “A” numbers. I write K for wavenumber (w/n) and P(K) for spectral power (a.u.). The key measure R and fringe visibility V, are defined in this companion post as follows: R via (A14) (with R=V2 in (A15)), V via (A18), while the frequency-unit conversion (px^-1 to w/n) via (A9).
The slit-camera distance D
In a Fraunhofer double-slit diffraction, the slits-to-camera distance, D, is [2]
The pixel width, p, is introduced in the relation for the conversion of units from δ (px) to D(m). Using the reported values (Table I, δ ≈ 69 px, p = 7 × 10^-6 m/px, d = 2 × 10^-4 m: and: λ = 6.328 × 10^-7 m
Summary. The calculated distance based on reported values (15.3 cm) is incompatible with both the 10.4 cm originally reported and the later claim of 14.0 cm.
Interference pattern: fringe spacing δ
The published interference pattern is shown as a smooth intensity curve—without raw camera points or a fitted model. Notable issues:
Axes. The x-axis is the full camera span (1–3000 px); the y-axis is digitized intensity.
Envelope. The central-diffraction band envelope is asymmetric, with intensity falling too rapidly at both ends.
Fringes at edges. Only faint ripples appear near the edges, rather than the full decaying fringe train expected from diffraction.
Likely causes. A slight slit-plate tilt and/or edge contamination on the camera-window rim (fingerprints from handling).
The 28 independent measurements of δ between adjacent fringes across the central diffraction band (Table 3 [5], no pair measured twice) yielded the graphically estimated average [3-5]:
Although the difference from the reported ∼69 px (Table I) seems small, it significantly affects derived quantities—including the main FFT peak frequency K (Table I), as shown below.
Substituting (A3) into (A1) using values listed in table I:
Uncertainty propagation from δ gives:
Final result:
Summary: A measured average fringe spacing of 67.8 px implies slit-camera distance D=15 cm, contradicting the reported distances 10.4 cm (2012) and the later 14.0 cm (2013), as well as the reported FFT peak K, analyzed in a subsequent section.
The number of fringes inside the central interference band N
For a double slit with slit separation d and slit width a, the central diffraction band envelope extends between the two minima at orders, m, of absolute value ∣m∣=d/a. The number of bright interference fringes N contained in the central band, by substituting the reported values (Table I: d=2×10−4 m, a=1×10−5 m), is
In the published graph, however, only about 33 fringes are visible due to the abnormally rapid intensity fall-off near both ends of the pattern. The degraded edges also prevent a reliable estimate of the envelope breadth L from the graph.
Summary. Theory predicts N=39 fringes (40 spacings) within the central diffraction band for the reported a, and d, but only ∼33 are visible in the published pattern.
The breadth, L, of central interference band and the unit conversion formula
From (A3), the central band spans forty fringe spacings between the first missing fringes (orders m=±d/a). Thus:
By contrast, the central-band width was considered to be the full camera length (L′=3000 px), rather than (2712±16) px indicated by the published graph, evidenced by the introduced calibration of the FFT axis via L′ as:
This choice does not use the theoretically correct breadth for the central diffraction band. It does not by itself, however, change the key ratio R=PK/P1, which depends on spectral power values rather than the peak frequency. (As shown later, it is the shift of the spectrum—not a mere rescaling—that alters P1 and thus R.)
The central diffraction band envelope falls more steeply than predicted by the reported parameters (Table I), suppressing the outer fringes, so its breadth L cannot be read reliably from the published graph. Consequently, an alternative single estimate of D involving the unreliable estimate of L from the graph,
cannot yield a trustworthy value of D. Instead, the previous estimate that involved 28 separate fringe spacing measurements on the reported graph with its associated error, Eq. A6, yielded a reliable estimate of D.
Summary. Because the envelope is truncated, the true central-band breadth L cannot be read reliably, so D cannot be inferred from (A10). Separately, treating L′=3000 px in the pixel→wavenumber calibration (A9) only rescales the FFT x-axis: it changes where peaks are plotted but does not change spectral powers. Since experimental outcome depends on spectral powers, the scale choice alone is irrelevant; what matters is the observed spectral shift, which alters P1 and thus R.
FFT spectrum: wavenumber of the principal peak K
Within the central diffraction band envelope, the intensity displays two frequencies: a slow modulation of breadth L (single-slit diffraction) and a faster modulation of period δ (two-slit diffraction). The latter produces the main FFT peak at wavenumber K.
Using the measured fringe spacing (A3) and the implied units conversion (A9, with L′=3000 px), the peak frequency is
Propagating the uncertainty from δ through the derivative of (A11) gives:
Hence,
The reported in 2012 peak frequency is practically one unit higher. This discrepancy is examined next.
Summary: From the measured interference pattern, the correct FFT peak wavenumber is 44.3±0.3 w/n (practically, 44 w/n), not the reported 45 w/n.
The key measure R
As shown, the reviewed publication estimates the key measure R as (Pallikari, 2012)
where PK is the spectral power at the main interference peak and P1 is the power at frequency 1 w/n. (The paper’s original definition is algebraically equivalent; this is the practical form). The visibility relates to R through [1]
Let us estimate the value of R in the following distinct cases: (i) on the reported and (ii) on the reinstated FFT spectrum.
Reported (shifted) spectrum:
As explained in Fig. 2B of On the Question of Wavefunction Collapse in a Double-Slit Experiment (2012), the FFT spectrum starts at K=1; the main peak is at K=45 (w/n) with power P(45)=107.4 (a.u.), while the power at 1 w/n is 109 (a.u.). Thus
which through Eq. (A15) gives fringe visibility V≈0.16.
Reinstated spectrum:
Shifting the reported spectrum left by 1 w/n in order to restore the spectral power at 0 frequency and the missing powers between frequencies 0 and 1 w/n, does not change the peak power, P(44)=P(45)=107.4 (a.u.). Because the spectrum is steep between 0 and 1 w/n, reinstating the spectrum makes the power at 1 w/n to drop from 109 to 107.84(a.u.) increasing the key measure R. Hence
The associated fringe visibility V through Eq. (A15) will be V≈0.60. (The logP1 values 7.4 and 9 are read from the power axis of the reported graph, so the actual spectral powers are 107.4 and 107.84, respectively.)
The spectrum in its reported shifted position yields a reduced key measure R, which appears as evidence for the tested hypothesis, even though it results entirely from the spectral shift; not from any physical effect affecting the spectral powers.
Summary: A (positive) 1 w/n spectral shift lowers R from ~0.36 to ~0.025. It produces a fringe visibility drop from ~0.60 to ~0.16, via (A15), showing an apparent effect without any change on the diffracted light. Even smaller spectral shifts, Δf, (between 0 and 1) are sufficient to reproduce the R-reduction effect reported in the 2012 double-slit study.
The unnaturally shifted spectrum - Additional evidence
Question. Are there further indications that reinstating the FFT spectrum to its proper position is consistent with theory and with the other experimental parameters?
(i) Frequency of the main FFT peak.
From the fringe spacing measured on the reported interference pattern, δ, equations (A3) and (A13) predict that the main FFT peak should lie at 44 w/n, not at 45 w/n as initially reported. When the spectrum is reinstated to its true position, the main peak appears at 44 w/n, as expected from both the theoretical relation and the independently measured parameters.
(ii) The fringe visibility test.
Fringe visibility is defined on the interference pattern by:
where I(max) and I(min) represent the intensities of diffracted light at the center of a bright fringe and the center of a dark fringe in the central interference band, respectively. Using the reported intensities in the central band (Fig. 2C), for a representative bright/dark pair:
Averaging across the strongest central fringes yields the same result, V≈0.60.
From the FFT spectrum, using (A15)–(A17) at 44 w/n:
The agreement between fringe visibility, V, estimated on the reported interference pattern and the reinstated FFT power spectrum, reinforces the correctness of reinstating the spectrum to its correct position so that the main peak is brought to 44 w/n.
Summary: A +1 w/n spectral shift reduces R from about 0.36 to 0.025, which via R=V2 appears as a drop in visibility from ∼0.60 to ∼0.16, creating an apparent effect without any physical modification of the diffracted light. Moreover, as shown below, even smaller shifts (Δf between 0 and 1) are sufficient to reproduce the reported effect.
The role of the key metric R under spectral shifts 0<Δf≤1
What the shift does. From (A16)–(A17), the +1 w/n spectral shift identified in the reported FFT spectrum, lowers the key metric from R(0) = 0.363 to R(1) ≈ 0.025. Under the tested hypothesis this decrease was interpreted as a consciousness effect in selected “attention-towards” sessions, but it is the direct consequence of the spectral shift.
How the spectral shift is established.
(A) Fourier constraints. In a valid FFT spectrum, powers at zero and between 0 and 1 frequency units cannot be missing. The gap in the reported spectrum indicates a shift.
(B) Peak location. The reported main FFT peak at 45 w/n contradicts the measured fringe spacing δ=67.8±0.4 px (from the reported interference pattern), which via (A3) and (A13) predicts a main peak at 44 w/n. Reinstating the spectrum by −1 w/n restores the expected 0–1 w/n spectral powers and places the peak at expected theoretical value, 44 w/n.
(C) Visibility cross-check. The fringe visibility measured on the interference pattern, (A19), V≈0.60, matches the value estimated from the reinstated-spectrum V(44)≈0.60. This independent agreement supports the reinstated FFT position.
Empirical relation. In the 0 − 1 w/n interval, Fig. 3C shows linear dependence of logP1 with Δf
which, combined with (A14), yields the exponential dependence, Fig. 3A
Thus R decreases monotonically and predictably with Δf in the reported FFT spectrum.
Mechanism. Small shifts raise the spectral power at 1 unit of frequency while leaving the main peak unchanged (Fig. 2B), inflating the denominator in (A14) and thereby lowering R. Any Δf∈(0,1] reduces R; +1 w/n shift is simply the largest in that range. In practice, a mixture of smaller shifts Δf (in range 0 − 1) likewise lowers R and, together with pooled z-scoring and the observed timing imbalance, reproduces the published standardized means (Fig. 4).
Implication. Random, invisibly small shifts within 0<Δf≤1 can produce large reductions in R (up to ∼93%) by inflating P1, while PK remains fixed; no consciousness-related influence is required.
Summary. A +1 w/n shift yields the maximal drop in R (from ∼0.36 to ∼0.025); crucially, smaller shifts also suffice to simulate the reported effect, in line with R(Δf) in (A22).
These findings show that the (2012) reported results arise from spectral shifts introduced during data processing, rather than from any physical influence on the diffracted light.
References
1. Pallikari, F. (2012). On the question of wavefunction collapse in a double-slit diffraction experiment. arXiv:1210.0432 [quant-ph].
2. Hecht, E. (2002). Optics (4th ed.). Addison-Wesley.
3. Pallikari, F. (2022). The double-slit experiment and the conscious observer [Video under construction]. YouTube.
4. Pallikari, F. (2024a). Evaluating a double-slit diffraction and psychokinesis experiment [Video under construction]. YouTube.
5. Pallikari, F. (2025). Did Attention Really collapse the photon wavefunction? Substack essay.