Double Slit Experiment — Measurement

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Detection Pattern vs Theoretical Intensity

Detected Hits (Blue)

Quantum |ψ₁+ψ₂|²

P₁′ (slit 1)

P₂′ (slit 2)

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Simulation Context

Guided Inquiry

1 · The apparatus

Here’s the setup. A point source emits monochromatic radiation at wavelength λ, and a screen with two slits is placed in between the source and a detector screen.

2 · The classical prediction

Let’s warm up with just one slit. Classically, if one slit is closed, what form do you think the distribution on the detector screen takes?

3 · Young’s result

Now both slits are open. Young’s double slit experiment shows an interference pattern, i.e. crests and troughs — watch it build up on the screen.

4 · What it establishes

So what does that pattern really prove? What conclusion do you think can be drawn from Young’s double slit experiment?

5 · Detectors at the slits

Now for the twist. Particle detectors are about to be placed in front of the two slits. What pattern do you think forms on the detector screen?

Listen

Parameters

Source & Slits

Slit Separation (d)

1600 nm

400 nm1600 nm

Slit Width (a)

500 nm

100 nm500 nm

Wavelength (λ)

480 nm

■ 380 nm700 nm ■

Screen Distance (L)

4000 nm

4000 nm8000 nm

Animation Speed

1.00x

0.25x2x

Detectors OFF — Quantum interference active

Unmeasured: I ∝ |ψ₁ + ψ₂|² = cos²(πd sinθ/λ) × sinc²(πa sinθ/λ)
Interference fringes visible — no “which path” information

The Physics

The double slit with which-path detectors · what this sim shows, the physics behind it, and every control

At a glance

This sim sends particles one at a time through two slits and builds the arrival pattern on a screen, hit by hit. You can change the geometry and the wavelength, close either slit, and place a which-path detector at each slit. It is the standard test of what a measurement does to a quantum pattern.

What you'll learn

How the pattern on the screen depends on the slit separation d, the slit width a, the wavelength λ and the screen distance L.
How single arrivals add up to a probability pattern, and how to read the chart that compares them.
How the intensity law is built from the amplitudes of the two routes.
How a which-path detector enters that calculation.

Plan view, not to scale. The openings are d apart and a wide; the screen is L behind them. The route to P from the lower opening is longer by d sin θ.

What's on screen

S1 (pink) and S2 (violet): the left and right slit, d apart.
Detectors (amber): the which-path sensors at the slits, shown only when enabled.
Screen: each hit leaves one dot, across x = −4000 to +4000 nm.
Blue bars (chart): hits in 80 bins across the screen, tallest bin scaled to 1.
Green curve (chart): the probability pattern the sim computes, peak scaled to 1. Detector mode adds orange curves (section 06).

See itRun the sim with both slits open. Watch for: single dots arriving one at a time, and the blue bars slowly following the green curve.

The controls

Press ⌖ on a row to see where that control sits in the sim. Moving any of the four geometry sliders restarts the screen.

Control	What it changes	Range
Source & Slits	Turns the source on or off, and opens or closes each slit (Slit L is S1, Slit R is S2). The chart shows what the open slits give.	3 toggles
Slit separation d	The centre-to-centre distance of the two openings.	400–1600 nm opens at 1600
Slit width a	The width of each opening (both are equal).	100–500 nm Reset sets 500
Wavelength λ	The wavelength of the source. It also sets the colour of the waves.	380–700 nm opens at 480
Screen distance L	How far the screen sits behind the barrier.	4000–8000 nm opens at 4000
Animation speed	How fast the wavefronts travel and the hits arrive. The pattern itself does not change.	0.25×–2× Reset sets 1×
Enable Detectors	Places a which-path detector at each slit. It first sets d = 1600, a = 500, λ = 480 and L = 4000 nm and clears the screen; turning it off restores your earlier values.	top bar opens off
Detector status	States whether the detectors are off or on.	badge
Chart caption	Writes the intensity law for the current mode.	panel
Chart	The hit histogram against the computed curve (section 01).	panel
Play / Pause	Runs the sim or freezes it.	top bar
Reset	Detectors off, d = 1600, a = 500, λ = 480, L = 4000 nm, speed 1×, and a clear screen. It leaves the source and slit toggles as they are.	top bar

Fixed: the screen always spans x = −4000 to +4000 nm, the two openings are always equal, and every hit is drawn as one dot.

Two routes, one amplitude

A particle that can reach the screen through either opening has a wave function that is the sum of the two routes, ψ = ψ₁ + ψ₂. The probability density P for where it lands is the square of that sum.

The first two terms are what each route gives alone. The third is the interference.

The cross term changes sign from place to place. Where it is positive the density is more than the two routes add up to; where it is negative it is less, down to zero. Each particle still lands at one point. P is the density of many such points, so the histogram follows it only after many arrivals.

Key idea. Amplitudes add first and are squared afterwards. That is why a point can receive fewer arrivals with both slits open than with one.

See itBoth slits open, nothing recording the path. Watch for: the dark fringes staying nearly empty while the dots pile up at the bright ones.

Spacing and heights have separate causes

The green curve is one formula, which the sim evaluates at 80 points across the screen:

I₀ is the central value. The sim scales its curve so that this is 1.

Fringes. The cos²φ factor sets where the bright and dark fringes fall. It depends on d and λ.
Envelope. The (sin β/β)² factor exists because each opening has a width. It multiplies the fringes and sets how tall they are away from the centre.
Narrow slits, flat envelope. At λ = 480 nm and L = 4000 nm the envelope at the screen edge is 0.93 for a = 100 nm and 0.10 for a = 500 nm.
Only its falling side. The envelope's first zero needs a sin θ = λ. For every slider setting the screen ends before that, so the zero never appears.

See itRun a = 500 nm, then a = 100 nm, with d, λ and L unchanged. Watch for: the outer fringes of the green curve rising from 0.24 to 0.95 of the central one.

Where the fringes land

Two routes arrive in step when their path difference is a whole number of wavelengths. At angle θ that difference is d sin θ (figure in section 01), so the cos² factor peaks and vanishes where

bright: d sin θ = mλ dark: d sin θ = (m + ½)λ m = 0, ±1, ±2, …

The sim uses sin θ = sin(arctan(x/L)) exactly, with no small-angle step.

On the screen x = L tan θ, so the m-th bright fringe sits at x_m = Lmλ/√(d² − m²λ²), which is close to mλL/d when mλ ≪ d. Larger λ or L spreads the fringes; larger d crowds them. An order exists only if mλ/d ≤ 1, and it is visible only if x_m ≤ 4000 nm.

d a λ L

These sliders move the sim too.

Separation d, at λ = 480 nm, L = 4000 nm	first bright fringe	bright fringes on screen
1600 nm	x₁ = 1258 nm	5
800 nm	x₁ = 3000 nm	3
400 nm	none (λ/d > 1)	1

See itRun d = 1600 nm, then d = 800 nm, with a = 500 nm. Watch for: the fringes moving apart, from five bright fringes to three.

A path record removes the cross term

A which-path detector is a second quantum system. It ends in state |D₁⟩ if the particle used slit 1 and |D₂⟩ if it used slit 2, so the full state carries the detector too:

ψ₁ and ψ₂ are the same functions as before.

The cross term is multiplied by the overlap ⟨D₁|D₂⟩. The sim's detector is ideal, so the two states are completely distinguishable and ⟨D₁|D₂⟩ = 0. What is left is |ψ₁|² + |ψ₂|²: the two one-slit densities added, with no fringes. The calculation depends on the record existing, not on anyone reading it.

What the sim draws: with detectors on, each particle is shown passing through one slit, and the chart shows two bumps P₁′ and P₂′ centred on the slits, with σ = 0.025·Lλ/a (96 nm at d = 1600, a = 500, λ = 480, L = 4000 nm). That is the sim's display model, not a computed diffraction pattern. What it shares with the calculation is the point: no fringes.

See itThe same geometry with the detectors on. Watch for: the green fringe curve replaced by orange bumps at the slits, with the dots piling into them and no dark fringes.

Where it matters

One electron at a time. Tonomura and colleagues (1989) sent electrons singly through an electron biprism, the electron version of two slits. Each arrived as one dot, and fringes showed only after thousands.
Large molecules. Arndt and colleagues (1999) saw interference from C₆₀ molecules passing through a grating with a 100 nm period. Larger mass means a shorter wavelength and finer slits.
Which-path records. Dürr, Nonn and Rempe (1998) stored the path of atoms in their internal states. The fringes disappeared, with momentum kicks too small to account for it.

Estimates from λ = h/p. The sim's own wavelengths (380–700 nm) are those of light.

Go deeper: partial path knowledge

V is the contrast of the fringes; D is how well the path can be told.

The inequality is the general statement: partial path knowledge leaves partial fringes.
The sim shows only the two ends: D = 0 with fringes at full contrast, and D = 1 with none.

Summary

In one line. With the path unrecorded, amplitudes add and fringes appear; once the path is recorded, probabilities add and the fringes go.

Key points

The pattern is the density of many single arrivals.
Bright fringes sit where d sin θ = mλ. The width a sets an envelope that multiplies them.
A path record sets the cross term to zero and leaves ψ₁ and ψ₂ unchanged.
The sim's detector is ideal; its two bumps are a display model.

Common mix-ups

"Each particle splits and lands as a smear." Each lands at one point. The pattern is only visible in many arrivals.
"The detector spoils the fringes by pushing the particles." In the calculation the cross term goes because ⟨D₁|D₂⟩ = 0, not because of a force. The sim contains no push.
"Closing a slit only dims the same pattern." It changes the pattern: with one slit closed the fringes are gone.

Things to try on your own

At λ = 480 nm and L = 4000 nm, find the largest d (10 nm steps) that still shows only three bright fringes.
Enable the detectors, then set a = 100 nm. Work out σ first, then compare it with d/2.
Close one slit with the detectors off and compare the pattern with the two-slit one at the same settings.