Double-Slit Experiment

The double slit stopped being a demonstration and became an instrument. Atom interferometers split and recombine matter waves around macroscopic paths, reading gravity, rotation, and fundamental constants in the phase of the cross term — portable versions now prospect for aquifers. Electron holography, descendant of the biprism experiments, images magnetic fields inside materials by watching them shift fringes. Molecule interferometry keeps raising the mass record — beyond 25,000 u — to ask whether superposition survives at scales where gravity or hypothetical collapse mechanisms might intervene. So far every fringe has shown up on schedule. The same logic runs the quantum technologies: a qubit algorithm is a many-path interferometer whose computational routes are kept indistinguishable so amplitudes can cancel wrong answers and reinforce right ones — and decoherence, the enemy of quantum computers, is exactly the which-path tagging of equation (4), perpetrated by an entire environment. Feynman's "only mystery" turned out to be a design specification.

Run the three classic versions side by side. Bullets through two gaps make two piles — each bullet takes one gap; probabilities add. Water waves make fringes — each wavefront takes both gaps; heights add before you square for intensity. Electrons do both at once: they arrive like bullets and distribute like waves.

The right-hand panel is the heart of quantum mechanics, so look at how strange it is. Every dot is whole — no electron arrives smeared, halved, or dimmed. The wave behaviour lives one level up, in the distribution: dark bands are places where an electron with two available routes cannot land, even though closing one slit — removing an option — would let it land there. What went through both slits is not the electron-as-lump but the amplitude that books its probabilities: the wavefunction, formally introduced one article ahead. Watch the pattern being built, dot by dot, as Tonomura's group filmed it in 1989:

The pattern was in the probabilities from the first electron; the screen just needed samples to render it. That is the sense in which the wavefunction governs single particles, not ensembles — made precise in Born Statistical Interpretation, two articles ahead.

The bookkeeping behind the fringes was set out in Wave–Particle Duality: amplitudes for indistinguishable routes add before squaring, putting the cross term $2\,\mathrm{Re}(\psi_1^*\psi_2)$ on top of the single-route probabilities. That law is the input here — this article turns it into geometry you can measure. Write the detection probability at a screen point as

where: $\psi_1, \psi_2$ — the complex amplitudes for reaching that point via slit 1 or slit 2; the first two terms are what bullets would do; the cross term is the fringes. Everything experimental about the double slit — spacing, brightness, contrast, and its death by observation — is a statement about that last term.

Now install a which-path detector: any device whose state changes according to the slit used — a photon bounced off the electron, a spin flipped near one slit, a stray air molecule scattering differently. The particle-plus-detector system then carries correlated amplitudes, and the screen probability becomes

where: $|d_1\rangle, |d_2\rangle$ — the detector's final states for the two paths, and $\langle d_1|d_2\rangle$ — their overlap, a number between 0 and 1 measuring how badly the detector distinguishes the paths. (The bracket notation is previewed from the Formalism chapter; read it simply as "the overlap of the two records.") The cross term is now held hostage: perfect records ($\langle d_1|d_2\rangle = 0$) kill it entirely; a detector that learned nothing ($\langle d_1|d_2\rangle = 1$) leaves the fringes untouched.

Note what the argument never used: recoil, kicks, or clumsiness. A which-path record sitting in one scattered photon fleeing to infinity erases fringes as effectively as a violent collision. That is why macroscopic objects never show fringes even in principle: the environment is a relentless which-path detector, tagging every dust grain with scattered light and gas molecules. The systematic theory — decoherence, quantum erasers, delayed choice — lives in the Foundations & Entanglement chapter; here it is enough to know the cross term dies by information, not by force.

1 · Jönsson's electrons (confidence builder)

In 1961, Claus Jönsson performed the first true electron double-slit experiment: $50\ \mathrm{kV}$ electrons, slits $d = 2.0\ \mathrm{\mu m}$ apart, screen effectively $L = 35\ \mathrm{cm}$ downstream. Find the fringe spacing, and explain why he needed electron-optical magnification to see it.

Step 1. Wavelength — check relativity first: $50\ \mathrm{keV}$ is 10% of $m_ec^2 = 511\ \mathrm{keV}$, so use the honest formula from de Broglie Matter Waves: $\lambda = hc/\sqrt{K^2 + 2Km_ec^2} = \dfrac{1240\ \mathrm{eV\,nm}}{\sqrt{(5.0\times10^4)^2 + 2(5.0\times10^4)(5.11\times10^5)}\ \mathrm{eV}} = 5.36\ \mathrm{pm}$ (the non-relativistic shortcut gives $5.49\ \mathrm{pm}$, 2.4% high).

Step 2. Fringe spacing from (3): $\Delta y = \dfrac{\lambda L}{d} = \dfrac{(5.36\times10^{-12})(0.35)}{2.0\times10^{-6}} = \boxed{0.94\ \mathrm{\mu m}}$.

Step 3. Interpret: micrometre fringes are far below what film resolves, so Jönsson magnified the pattern electron-optically before recording it. Compare Young: with light ($\lambda \sim 500\ \mathrm{nm}$, $10^5$ times longer), the same geometry gives fringes ~9 cm apart — visible on a wall. Identical physics; only the wavelength budget changed.

2 · Pricing a gentle detector (exam level)

A double slit with equal amplitudes is fitted with a which-path detector whose two records overlap by $g = |\langle d_1|d_2\rangle| = 0.60$. Find the fringe visibility, the ratio of maximum to minimum intensity, and the path distinguishability. Then answer: can a cleverer detector with the same $g$ do better than $D = 0.80$?

Step 1. Visibility from (5) with $a_1 = a_2$: $V = g = \boxed{0.60}$.

Step 2. Intensity ratio: $\dfrac{P_{\max}}{P_{\min}} = \dfrac{1+V}{1-V} = \dfrac{1.6}{0.4} = \boxed{4.0}$ — fringes plainly visible, but washed grey rather than black-bottomed.

Step 3. Distinguishability: $D = \sqrt{1 - g^2} = \sqrt{1 - 0.36} = \boxed{0.80}$ — the detector can identify the path in 80% of the statistically optimal sense.

Step 4. Check the bound: $V^2 + D^2 = 0.36 + 0.64 = 1.00$ — the Englert bound is saturated. No detector design with the same record overlap can beat it: $D = \sqrt{1-g^2}$ is the ceiling, reached only when the detector wastes none of its correlation. Partial knowledge has an exact, non-negotiable price in fringes.

Texts. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., Ch. 1 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., Ch. 1 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 4 · Cohen-Tannoudji, Diu & Laloë, Vol. 1, Complement to Ch. 1 · Townsend, A Modern Approach to Quantum Mechanics, Ch. 1 · Feynman, Leighton & Sands, The Feynman Lectures on Physics, Vol. III, Ch. 1. Key identities used throughout: the intensity expansion $|\psi_1+\psi_2|^2=|\psi_1|^2+|\psi_2|^2+2\mathrm{Re}(\psi_1^*\psi_2)$; the maximum condition $d\sin\theta=m\lambda$ and fringe spacing $\Delta y=\lambda L/d$ from elementary geometry; the $\cos^2$ intensity profile via $1+\cos\delta=2\cos^2(\delta/2)$; the which-path probability $P=|\psi_1|^2+|\psi_2|^2+2\mathrm{Re}[\psi_1^*\psi_2\langle d_1|d_2\rangle]$ and the complementarity bound $\mathcal{V}^2+\mathcal{D}^2\le1$ (Englert, PRL 1996). The Planck constant value used is $h=6.626\times10^{-34}\,\mathrm{J\,s}$ (CODATA). Single-electron buildup is demonstrated in Tonomura et al., Am. J. Phys. 57, 117 (1989).

Prerequisites. Wave–particle duality · de Broglie matter waves · Breakdown of classical physics. Leads to. The wavefunction · Born statistical interpretation · Probability density and normalization · Time-dependent Schrödinger equation.