3.1.1 Single-particle detections accumulating into fringes
Direct a monoenergetic beam of electrons or photons through two narrow parallel slits of separation \(d\) toward a position-sensitive detector screen at distance \(L\). When the source flux is attenuated so that at most one particle traverses the apparatus at any instant, each detection event registers at a single localized pixel \(x\) rather than smearing across the screen. Yet as thousands of independent single-particle arrivals accumulate, the spatial histogram builds up alternating bright and dark interference fringes.
3.1.2 Aperture geometry and open versus monitored slits
The physical boundary conditions define the experiment: a collimating aperture prepares transverse spatial coherence across both slits, free propagation maps path-length differences into phase differences across the screen coordinate \(x\), and the detector array bins individual arrivals. Closing one slit changes the physical preparation to a single-aperture diffraction experiment; inserting a path-recording sensor couples the particle to an extra degree of freedom. Comparing single-slit, unmonitored double-slit, and monitored double-slit runs compares three physically distinct apparatuses.
3.1.3 Diffraction envelopes and schematic trial markers
In the interactive laboratory below, individual arrivals are sampled from a normalized probability distribution across \(64\) spatial screen bins. The lines drawn between the slits and the screen represent the two interfering amplitude alternatives \(\psi_L(x)\) and \(\psi_R(x)\) contributing to each screen position \(x\), not observed classical trajectories, and a smooth Gaussian single-slit envelope modulates the two-slit interference fringes.
Visual animation · Section 3.1
Localized arrivals and two-aperture geometry
Interactive visual model for 3.1 Localized arrivals and two-aperture geometry (Single-particle detections accumulating into fringes · Aperture geometry and open versus monitored slits · Diffraction envelopes and schematic trial markers). Use Animate or Step to follow the physical state evolution and quantitative predictions.
3.2 Superposing amplitudes and de Broglie wavelength
3.2.1 Cross terms from coherent amplitude addition
When a particle can reach screen position \(x\) through two indistinguishable slit alternatives without leaving a path record elsewhere, quantum mechanics adds the complex probability amplitudes \(\psi_L(x)\) and \(\psi_R(x)\) before squaring:
Adding classical probabilities \(|\psi_L(x)|^2+|\psi_R(x)|^2\) omits the interference cross term \(2\operatorname{Re}[\psi_L^*(x)\psi_R(x)]\) and predicts a featureless sum of two single-slit profiles.
3.2.2 Far-field path difference and fringe period
Where the two slit envelopes have equal magnitude \(|\psi_L(x)|=|\psi_R(x)|\) and relative phase difference \(\delta(x)\), the detection probability is proportional to \(1+\cos\delta(x)\), vanishing at destructive phases \(\delta=(2n+1)\pi\) and doubling the single-path sum at constructive phases \(\delta=2n\pi\). In the far-field regime \(L\gg d\), the geometric path-length difference \(d\sin\theta\approx dx/L\) produces the phase difference
where \(\lambda\) is the wavelength and \(\Delta x\) is the spatial period between adjacent interference maxima.
3.2.3 Momentum scaling and the visibility parameter
For matter waves of momentum \(p\), the de Broglie relation \(\lambda=h/p\) links higher particle momentum to a shorter wavelength \(\lambda\) and therefore a tighter fringe spacing \(\Delta x=hL/(pd)\) at fixed geometry. When partial distinguishability or phase noise reduces coherence between the two slit alternatives by a visibility factor \(V\in[0,1]\), the screen distribution takes the form \(P(x)\propto E(x)[1+V\cos\delta(x)]\), where \(E(x)\) is the diffraction envelope: setting \(V=0\) eliminates the two-slit fringes while leaving the broad envelope \(E(x)\) intact.
Visual animation · Section 3.2
Superposing amplitudes and de Broglie wavelength
Interactive visual model for 3.2 Superposing amplitudes and de Broglie wavelength (Cross terms from coherent amplitude addition · Far-field path difference and fringe period · Momentum scaling and the visibility parameter). Use Animate or Step to follow the physical state evolution and quantitative predictions.
3.3 Disentangling path records from phase drift
3.3.1 Why fringes rule out independent slit probabilities
Recording discrete clicks rules out models in which a single particle spreads into a continuous classical charge cloud across the detector face, while the appearance of zeroes in the two-slit pattern rules out models in which each particle passes through a single slit with a probability independent of whether the other slit is open. Note, however, that interference fringes alone do not exclude nonlocal hidden-variable dynamics (such as Bohmian mechanics) in which a guiding wave depends on both open apertures.
3.3.2 Mechanical stability, transverse coherence, and contrast loss
In a laboratory matter-wave interferometer, verifying two-slit interference requires comparing the two-slit histogram against the sum of the two single-slit histograms under matched beam current, while controlling mechanical vibration, stray electromagnetic fields, and finite source size. Crucially, observing reduced visibility \(V<1\) does not by itself prove that a which-way detector measured the particle’s path: classical beam-pointing jitter or slow thermal phase drift averages \(\cos(\delta+\phi)\) over random phases \(\phi\) and washes out the fringes in the exact same way.
3.3.3 Finite-sample histograms versus expected intensity
At small event counts \(N\), the empirical bin fractions fluctuate strongly around the smooth theoretical curve \(P(x)\), and no single localized detection ever exhibits a fringe spacing by itself. Chapter 8 and Chapter 10 examine how coupling the particle to a controllable quantum path marker suppresses and conditionally restores the cross term \(2\operatorname{Re}[\psi_L^*\psi_R]\).
Visual animation · Section 3.3
Disentangling path records from phase drift
Interactive visual model for 3.3 Disentangling path records from phase drift (Why fringes rule out independent slit probabilities · Mechanical stability, transverse coherence, and contrast loss · Finite-sample histograms versus expected intensity). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the experiment
Set the visibility slider to \(V=1\), step through individual arrivals, and then press Play to watch the \(64\)-bin histogram converge toward the interference curve. Reset with \(V=0\) to compare the unmodulated diffraction envelope.
Interactive model · Synthetic observations
Build an interference pattern
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Read the values as a table
How can separate detection events form an interference pattern? The text and worked examples remain available without JavaScript. Animation starts only when requested; Step gives a motion-free way to follow the model.
A worked example
Electrons with de Broglie wavelength \(\lambda=0.05\text{ nm}\) (\(5\times 10^{-11}\text{ m}\)) pass through two slits separated by \(d=1.0\text{ }\mu\mathrm{m}\) (\(10^{-6}\text{ m}\)) and strike a detector screen at distance \(L=1.0\text{ m}\). Estimate the fringe spacing \(\Delta x\).
Using the far-field relation \(\Delta x=\lambda L/d\), we find \(\Delta x=(5\times 10^{-11}\text{ m})(1.0\text{ m})/(10^{-6}\text{ m})=5\times 10^{-5}\text{ m}=50\text{ }\mu\mathrm{m}\). Resolving this \(50\text{ }\mu\mathrm{m}\) period requires transverse source coherence across the \(1.0\text{ }\mu\mathrm{m}\) slit pair and a detector spatial resolution finer than \(\Delta x\) (or magnifying electron optics).
Summary
Electrons, neutrons, atoms, and molecules accumulate double-slit interference fringes with de Broglie wavelength \(\lambda = h/p\) even when particles traverse the apparatus one at a time.
Each detection event is localized at a single pixel, while the probability density \(P(x) = |\psi_1(x) + \psi_2(x)|^2\) governs the long-run spatial envelope and fringe visibility \(\mathcal{V}\).
Transverse source size, velocity dispersion, mechanical vibration, and residual gas scattering reduce fringe contrast without turning individual clicks into smeared waves.
Try it yourself
Problem 3.1
At a dark fringe position \(x_0\), the two slit amplitudes are \(\psi_L(x_0)=1/2\) and \(\psi_R(x_0)=-1/2\) in normalized units. What is the detection probability \(P(x_0)\) when both slits are open, and what would it be if probabilities were added without interference?
Hint
Compare \(|\psi_L+\psi_R|^2\) with \(|\psi_L|^2+|\psi_R|^2\).
Worked solution
Adding the amplitudes first gives \(\psi_L(x_0)+\psi_R(x_0)=1/2-1/2=0\), so \(P(x_0)=|0|^2=0\). Adding the individual single-slit probabilities would instead give \(|1/2|^2+|-1/2|^2=1/4+1/4=1/2\).
Problem 3.2
How does the matter-wave fringe spacing \(\Delta x\) change if the particle momentum \(p\) is doubled while \(d\) and \(L\) remain fixed?
Hint
Combine \(\lambda=h/p\) with \(\Delta x=\lambda L/d\).
Worked solution
Doubling the momentum \(p\) halves the de Broglie wavelength \(\lambda=h/p\), which halves the fringe spacing \(\Delta x=\lambda L/d\).
Problem 3.3
Does observing a smooth, fringe-free screen distribution (\(V=0\)) prove that a path detector recorded which slit each particle traversed?
Hint
Consider the effect of random phase fluctuations \(\delta\to\delta+\phi(t)\) between the two arms.
Worked solution
No. Mechanical vibration, transverse beam incoherence, or random thermal phase drift averages \(\cos(\delta+\phi)\) to zero over the run and produces \(V=0\) without any accessible path record.
What to carry forward
Single-particle detections are localized events whose ensemble probability distribution \(P(x)=|\psi_L(x)+\psi_R(x)|^2\) reflects interference between indistinguishable alternatives. Never mistake schematic amplitude lines for measured particle trajectories.
Sources and further exploration
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.