Double-Slit Experiment

The geometry

Setup: a coherent monochromatic source (laser, or filtered light through a pinhole) illuminates a barrier with two narrow parallel slits separated by distance d. Light spreads from each slit and reaches a screen at distance L away. The two slits act as two coherent sources; their waves overlap and interfere.

At a point on the screen at height y above the center, the path from one slit is slightly longer than from the other. To a good approximation (L >> y, L >> d):

Path difference  δ ≈ d · sin θ ≈ d · y / L

Constructive interference (bright fringe) when δ = m·λ:

y_m = m · λ · L / d            (bright fringe at order m = 0, ±1, ±2, ...)
y_dark = (m + ½) · λ · L / d   (dark fringe between)
Δy   = λ · L / d               (fringe spacing — adjacent bright fringes)

Worked example — 633 nm laser, 0.5 mm slits, 2 m screen

The textbook setup, with realistic numbers:

λ = 633 nm = 6.33 × 10⁻⁷ m   (He-Ne laser)
d = 0.5 mm = 5 × 10⁻⁴ m       (slit separation)
L = 2 m                       (screen distance)

Δy = λ · L / d
   = (6.33 × 10⁻⁷ × 2) / (5 × 10⁻⁴)
   = 2.532 × 10⁻³ m
   ≈ 2.5 mm

Easily visible bands of light and dark, about 2.5 mm apart, on a screen across the room. Move the screen to L = 4 m and the spacing doubles. Halve the slit gap to d = 0.25 mm and the spacing doubles again.

Worked numbers across sources

Source	λ	d (slits)	L (screen)	Fringe spacing Δy = λL/d
He-Ne laser (red)	633 nm	0.5 mm	2.0 m	2.53 mm
Diode laser (violet)	405 nm	0.5 mm	2.0 m	1.62 mm
Green laser	532 nm	0.25 mm	1.5 m	3.19 mm
Sodium lamp (filtered)	589 nm	0.4 mm	1.0 m	1.47 mm
Electron beam (50 keV)	5.5 pm	2 µm	0.5 m	1.4 µm
Thermal neutrons	0.18 nm	0.1 mm	5 m	9 µm
Cold C₆₀ molecules (Zeilinger 1999)	~2.5 pm	100 nm grating	1.25 m	~30 µm

Single-photon interference — the quantum twist

Reduce the laser intensity until on average only one photon at a time is in the apparatus. The detector records discrete clicks at random positions. Build a histogram of click positions over thousands of photons.

The histogram is the same fringe pattern. Each photon's probability amplitude goes through both slits, the two amplitudes superpose, and |amplitude|² gives the probability of detection at each screen point. The photon is detected as a point; its propagation is wavelike.

The first demonstration: G. I. Taylor, 1909. Modern canonical version: Akira Tonomura's electron build-up animation (1989) shows the pattern emerging dot by dot. Same physics works for atoms (1991), C₆₀ molecules (1999), and recently molecules of ~25,000 amu (2019).

The which-path measurement

Add a detector at one slit that records which photon passed through. The fringe pattern vanishes — you get a smooth single-slit-like spread. The act of obtaining which-path information collapses the superposition.

This isn't an instrumental disturbance issue. Quantum eraser experiments (Scully & Drühl 1982) show you can destroy the which-path information after the photon hits the screen and recover the fringe pattern in correlated coincidences. It's the existence of which-path information in principle that matters, not when you decide to look.

JavaScript — fringe calculations

// Bright fringe positions
function brightFringes(wavelength, slitSeparation, screenDistance, mMax = 5) {
  const out = [];
  for (let m = -mMax; m <= mMax; m++) {
    out.push({ m, y_m: m * wavelength * screenDistance / slitSeparation });
  }
  return out;
}

const lambda = 633e-9;
const d = 0.5e-3;
const L = 2.0;

const fringes = brightFringes(lambda, d, L, 3);
console.log(fringes.map(f => ({ m: f.m, y_mm: (f.y_m * 1000).toFixed(2) })));
// [-7.60, -5.07, -2.53, 0.00, 2.53, 5.07, 7.60]  (mm)

// Fringe spacing
const dy = lambda * L / d;
console.log(`Fringe spacing: ${(dy * 1000).toFixed(2)} mm`);  // 2.53 mm

// Intensity pattern: I(y) = 4·I₀·cos²(π·d·y / (λ·L)) · sinc²(π·a·y/(λ·L))
// (a = single-slit width — modulates the envelope)
function intensity(y, lambda, d, a, L, I_0 = 1) {
  const k = Math.PI / (lambda * L);
  const cos2 = Math.cos(k * d * y) ** 2;
  const sincArg = k * a * y;
  const sinc2 = sincArg === 0 ? 1 : (Math.sin(sincArg) / sincArg) ** 2;
  return 4 * I_0 * cos2 * sinc2;
}

// Sample the pattern
for (let y = -0.005; y <= 0.005; y += 0.0005) {
  const I = intensity(y, lambda, d, 0.1e-3, L);
  console.log(`y = ${(y*1000).toFixed(1)} mm, I/I_0 = ${(I).toFixed(3)}`);
}

// Single-photon Monte Carlo: sample positions from the intensity distribution
function samplePhoton(lambda, d, a, L, halfWidth = 0.01) {
  // Rejection sampling
  let y, I, p;
  do {
    y = (Math.random() - 0.5) * 2 * halfWidth;
    I = intensity(y, lambda, d, a, L);
    p = Math.random() * 4;  // max intensity ~ 4 (when slits add coherently)
  } while (p > I);
  return y;
}

// Simulate the build-up
const hits = [];
for (let i = 0; i < 100000; i++) hits.push(samplePhoton(lambda, d, 0.1e-3, L));

// Bin and show histogram
function histogram(values, bins = 40, half = 0.01) {
  const counts = new Array(bins).fill(0);
  for (const v of values) {
    const idx = Math.floor((v + half) / (2 * half) * bins);
    if (idx >= 0 && idx < bins) counts[idx]++;
  }
  return counts;
}

const hist = histogram(hits, 40, 0.01);
console.log('Histogram:', hist);
// The histogram traces the cos²·sinc² interference pattern, even though
// each photon was a discrete click at a single position.

Where the double slit lives in physics

• Teaching wave optics. The canonical demo that light is a wave (Young 1801) — every introductory optics course replays it.
• Demonstrating quantum mechanics. Single-photon and electron buildup is the most direct evidence of probability amplitudes; "the only mystery of quantum mechanics" per Feynman.
• Matter-wave experiments. Cold atoms, neutrons, fullerenes, large molecules — all show fringe patterns when sent through a grating or slit pair.
• Quantum eraser and delayed-choice experiments. Probe the role of information in quantum mechanics (Wheeler 1978; Scully & Drühl 1982; many modern variants).
• Atom interferometry. Used in inertial sensors, fundamental tests of equivalence principle, and proposed dark-matter detectors — descendants of Young's setup.
• Quantum-information primers. The cleanest classical analogue of qubit superposition; pedagogically inseparable from |0⟩ + |1⟩.

Common mistakes

• Using sin θ for large angles. The simple formula y_m = m·λ·L/d assumes y << L (small-angle). For wide-angle fringes use y_m = L·tan(arcsin(m·λ/d)).
• Ignoring the single-slit envelope. The pattern is cos² (two-slit interference) modulated by sinc² (single-slit diffraction). Beyond a few fringes from center, the envelope dims the bands.
• Forgetting coherence. Pure white light or incoherent sources won't produce visible fringes — you need monochromatic and spatially coherent light (laser, or filtered point source).
• Treating which-path detectors as "disturbance". The fringe pattern vanishes because of the existence of which-path information, not because of mechanical jiggling — delayed-choice experiments prove this.
• Equating "wave or particle" with "either-or". Quantum mechanics says the same object exhibits both behaviors. The fringe is wavelike propagation; the click is particle detection.
• Skipping the factor of m for higher orders. The mth bright fringe is m × λL/d from center. Forgetting this misplaces the higher orders.