A Thought Experiment on Double-Slit Interference from a Source with Positional Fluctuation: Push-forward of the Source-Position Distribution to the Fringe-Shift Distribution (Shape Preservation)
Thought experiment (model calculation), not a measurement. For a fixed geometry (L=10, W=5, lambda0=1) a stationary point source with positional fluctuation P(y)=cos^2(pi y/lambda) is passed through a double slit. Each trial shifts an identically shaped far-field fringe by the exact geometric path difference u(y), a nearly linear map of the source position. Over repeated trials the fringe-shift distribution is the push-forward rho(u)=P(y(u))|dy/du| of P, preserving the cos^2 shape because the map is paraxially linear; the ~3% departure is the non-paraxial nonlinearity (1/2 tan^2 theta). This is a change-of-variables (push-forward), not a derivation of a probability law. Distinguished from intensity-accumulation visibility loss (spatial coherence / van Cittert-Zernike).
Thought experiment (model calculation), not a measurement. For a fixed geometry (L=10, W=5, lambda0=1) a stationary point source with positional fluctuation P(y)=cos^2(pi y/lambda) is passed through a double slit. Each trial shifts an identically shaped far-field fringe by the exact geometric path difference u(y), a nearly linear map of the source position. Over repeated trials the fringe-shift distribution is the push-forward rho(u)=P(y(u))|dy/du| of P, preserving the cos^2 shape because the map is paraxially linear; the ~3% departure is the non-paraxial nonlinearity (1/2 tan^2 theta). This is a change-of-variables (push-forward), not a derivation of a probability law. Distinguished from intensity-accumulation visibility loss (spatial coherence / van Cittert-Zernike).
Thought experiment (model calculation), not a measurement. Extends the position-readout push-forward (companion paper, isSupplementTo) to a localized source: localization is represented by a constant-amplitude odd-harmonic sum (isolated peak wave S_N). Although S_N is a sum of odd harmonics, it localizes in the far field WITHOUT changing shape only when the source-slit diagonal r_k=sqrt(L^2+(W/2)^2) is an integer/half-integer multiple of the fundamental wavelength: sqrt(L^2+(W/2)^2)=(m/2) lambda0. Alignment is a band of width ~1/(2N) (narrower for larger N); a ~3% change in lambda0 decides interference vs scattering. The central-alignment tolerance is W/L-independent; larger W gives greater off-axis fragility. An aligned localized source inherits the cos^2 fringe-shift readout under positional fluctuation but its peaks fall below cos^2 off-axis (extra scattering loss). The single wavelength N=1 has none of this fragility and reproduces the companion paper to machine precision. Net result is negative: localization adds an alignment constraint and off-axis fragility; shape preservation is conditional and fragile, with N=1 the robust special case. Distinguished from spatial coherence / van Cittert-Zernike visibility loss.
A numerical experiment that simulates and observes the de Broglie relation lambda=h/p inside a single-source double-slit classical wave model. An electron is modeled as a spatially localized wave built from an equal-amplitude sum of odd harmonics (N>1). The central result is NOT the value of h (which is a near-tautological self-consistency, since lambda0=h/p is the input and the resonance lattice is dense ~1e-19 m): it is (i) that a highly localized odd-harmonic classical wave can survive two-slit interference as a sharp single peak only under an inter-harmonic alignment condition, and (ii) that the observed central fringe spacing DeltaX = lambda' D/W is set by the fundamental wavelength and is independent of the harmonic count N (N controls only the localization sharpness ~1/(N+1)). Scope is stated explicitly: this is neither a first-principles derivation of p*lambda=h nor a derivation of a Delta_x*Delta_p ~ h uncertainty relation. Each electron is given independent Monte Carlo draws of source position (cos^2, +/-lambda0/2) and initial wavelength (cos^2, +/-1%), with interference, alignment and exclusion recomputed exactly per trial. Bilingual (Japanese + English) drafts with source programs.
3.1 Interference of a Single Photon The kinds of study originally conducted by Thomas Young - in particular, the celebrated "double slit" experiment that we briefly examined in Chapter 1 (see Fig. 1) - provide a capacity to reveal far more than the wave nature of light, important as that is. True enough, the observed characteristic fringe pattern, when monochromatic light passes through two closely spaced slits onto a screen, appears to be simply explicable on the basis of waves from each slit undergoing constructive and destructive interference at different positions on the screen. Yet, astonishingly, these patterns of interference are still exhibited when individual photons of a low-intensity input beam reach the screen one at a time, showing that it cannot in fact be a matter of different photons having to interfere with each other. Indeed, Dirac enunciated the general principle that photons can never directly interact with each other. The double slit and other such experiments indicate that each photon interacts in a delocalized, wave-like manner with both slits at the same time. Successive experiments build interference patterns that conform to an exact mathematical distribution formula, but the position at which an individual photon is detected in any particular instance cannot be predicted; this is a quintessential illustration of quantum uncertainty. Using the path integral approach of quantum theory, the phenomenon is more accurately represented by introducing a summation over all possible routes between the source and the screen on which the pattern appears, signifying an entire "lifetime" for each photon involved in the experiment. This relates to the fact that the actual trajectory is unknown without an observation, and thus we only obtain the probabilities of the photon striking the screen at different positions and times; all trajectories (potentialities) must be considered in a superposition. Each route is considered equally probable, and the variation in phase produces quantum interference. The differences in phase of each trajectory, at varying positions on the screen, generates the fringe pattern. An intriguing variation on this experiment involves "marking" the photons so that the actual route of each photon to the screen is known. Remarkably, in such a case, the fringe patterns do not appear because quantum interference cannot occur. A consequence of the observation of the photon route is an effect that, in other quantum mechanical connections, is known as a collapse of the wavefunction. However, it is possible for the fringe patterns to reappear when information on the route of the photons is lost; this is known as a quantum eraser.
Abstract The double slit experiment was the first demonstrative proof of the wave nature of light. It was expounded by the English physician-physicist Thomas Young in 1801 and it soon helped lay to rest the then raging Newton–Huygens debate on whether light consisted of a fast-moving stream of particles or a train of progressive waves in the ether medium. In the experiment, light is made to pass through two very narrow slits spaced closely apart. A screen placed on the other side captures a pattern of alternating bright and dark bands called fringes which are formed as a result of the phenomenon of interference. In prior work by the same author, it was shown that the conventional analysis of Young’s experiment that is used in many introductory physics textbooks, suffers from a number of limitations in regards to its ability to accurately predict the positions of these fringes on the distant screen. This was owing to the adoption of some needless and paradoxical assumptions to help simplify the geometry of the slit barrier-screen arrangement. In the new analysis however, all such approximations were discarded and a hyperbola theorem was forwarded which was then suitably applied to determine the exact fringe positions on screens of varied shapes (linear, semi-circular, semi-elliptical). This paper further builds on that work by laying down the mathematical framework necessary for counting fringes and then comparing their distributions on differently shaped screens, using MATLAB software package for numerical–graphical simulation. In addition, a pair of equivalent laws of proportionality are predicted that govern the distribution of fringes independent of the shape of the detection screen employed.
Electron double-slit experiments, the essence of quantum mechanical “wave/particle duality”, have been extensively and continuously studied because their unrealistic interpretations have attracted many researchers. Here, we discuss recent sophisticated V-shaped double-slit experiments to elucidate electron-path information, i.e., which slit electrons pass through, which path they pursue, and where they arrive and interfere on the image plane. In a transmission electron microscope, an optically zero-propagation distance condition was realized, where the double-slit position was imaged just on the detector plane. Interference fringes composed of dot images were controlled by using two electron biprisms. Using a V-shaped double slit, we confirmed that it is possible to observe interference fringes only when the path information of individual electrons is not available. Furthermore, we studied how the interference fringes would shift their detected positions when electron wave phases were modulated before illumination to the double slit. Phase shifts were generated by tilting electron beams. We confirmed that each single electron as a wave changed its phase even when it traveled alone inside the microscope.
Electron double-slit experiments, the essence of quantum mechanical “wave/particle duality,” have been extensively and continuously studied because their unrealistic interpretations have attracted many researchers. Here, we discuss recent sophisticated V-shaped double-slit experiments to elucidate electron-path information, i.e., which slit electrons pass through, which path they pursue, and where they arrive and interfere on the image plane. In a transmission electron microscope, an optically zero-propagation distance condition was realized, where the double-slit position was imaged just on the detector plane. Interference fringes composed of dot images were controlled by using two electron biprisms. Using a V-shaped double slit, we confirmed that it is possible to observe interference fringes only when the path information of individual electrons is not available. Furthermore, we studied how the interference fringes would shift their detected positions when electron wave phases were modulated before illumination to the double slit. Phase shifts were generated by tilting electron beams. We confirmed that each single electron as a wave changed its phase even when it traveled alone inside the microscope.