01. Wave–particle duality and the double slit

Wave–particle duality and the double slit

Watch particles arrive one at a time and pile up into interference fringes that nobody drew in advance.

What you will see

The first few dozen dots look random; after a few hundred the fringes surface on their own
Places that received particles with one slit open go empty once the second slit is opened
A which-path detector at the slits kills the fringes instantly; remove it and they return

Assumed background

Superposition and interference of waves (high-school level is enough)
The relation between probability and frequency

Feynman said the double slit “contains the only mystery” of quantum mechanics. That sounds like rhetoric; it is meant literally. Understand this one experiment properly and everything strange that follows is a variation on it.

So do not reach for the formulas yet. Open the scene below, press play, and watch the screen for thirty seconds.

The double slit: how do particles arriving one at a time grow fringes?

Every bright dot on the screen is one independent detection. Watch them land one by one: the fringes emerge on their own, and no single particle ever "knew" where they would be.

Loading 3D scene…

Experiment mode

Open slits

Put a detector at the slits (learn which path)

Parameters

Wavelength λ0.090

Slit separation d0.70

Slit width a0.18

Slit-to-screen distance L9.0

Changing L only stretches the fringes; it does not touch the interference itself

Fringe spacing Δy = λL/d ≈ 1.16

Emission

0 so far

Speed

Display

Wave field behind the slits

Distribution on the screen

The thin line is the predicted intensity I(y); the filled area is the histogram of actual hits. The more particles, the closer they get.

I(y) = |\psi_1 + \psi_2|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\,\mathrm{Re}(\psi_1^*\psi_2)

Δy = λL/d = 1.1570 detectedVisibility High

Field intensity / predicted I(y)
Individual particle hits
Barrier and screen

What to look for

Change nothing at first — just watch the screen for ten seconds. The first few dozen dots look completely random; after a few hundred the fringes surface. That is what the word "probability" actually means here.
Switch to "Left only": the fringes vanish and a broad envelope remains (single-slit diffraction). Now switch back and look at where the dark fringes fall — exactly where particles did arrive with one slit open. Opening a second slit removes particles from places they used to reach; the classical particle picture dies right there.
Switch on the which-path detector: the fringes disappear on the spot and the distribution becomes a plain sum of the two single-slit patterns. Switch it off and they come back. You can have the path information or the interference, never both.
Drag λ and d to check Δy = λL/d: larger λ spreads the fringes, larger d packs them together.
Switch to "Classical wave" and look from above: two point sources spread, overlap and build a radiating fan of bright and dark. The fringes on the screen are just where that fan meets the screen.

✕ "The particle splits, half through each slit"

It does not. Every detection registers one whole particle, never half of one. What interferes is the probability amplitude, not the particle.

✕ "The particles interfere with each other"

Drop the rate to one particle per second so that no two are ever inside the apparatus together: the fringes still appear. The interference is between the two possible paths of a single particle.

✕ "The detector knocks the particle sideways, so the fringes die"

That story holds in some particular set-ups but it is not the reason. Even schemes that extract which-path information without exchanging momentum destroy the fringes. The real cause is that once the environment records the path, the two paths are no longer coherent.

Think it through

Set λ to its smallest and d to its largest: the fringes get too fine to see. A bullet has a de Broglie wavelength of about 10⁻³⁴ m — estimate its fringe spacing and explain why nobody has ever bumped into interference in daily life.
Suppose the path markers behind the slits can be erased afterwards (a quantum eraser). Can the fringes come back? What does that say about *when* a measurement happens?
The envelope width is set by a, the fringe spacing by d. Why do these two lengths control completely different things?

This is an interactive 3D scene (double-slit) and needs JavaScript and WebGL to run.

Three things you just saw

One: every detection is a whole particle. What appears on the screen is always a single dot — never half a dot, never a smear. In that sense it is a particle.

Two: the landing point is unpredictable. Same apparatus, same emission conditions, and there is no way to compute where the next dot goes. Only the probability can be computed.

Three: that probability carries fringes. This is the wave side. But be careful: it is not the particles that interfere, it is the probability amplitudes.

Fringe spacing: a formula you can verify with a slider

In the far field ( $L \gg d$ ) the path difference from the two slits to a point on the screen is $\Delta = d\sin\theta \approx d\,y/L$ . Constructive interference needs $\Delta = m\lambda$ , so neighbouring bright fringes are separated by

\Delta y = \frac{\lambda L}{d}.

The readout shows this number live. Drag $\lambda$ , $d$ and $L$ in turn and measure the spacing by eye each time to confirm it really does follow the formula.

Where the broader envelope comes fromadvanced

The argument above treats each slit as a point source. A real slit has width $a$ , every point inside it radiates, and those contributions interfere too. Integrating across the slit gives the single-slit diffraction factor:

I(y) \propto \underbrace{\cos^2\!\left(\frac{\pi d y}{\lambda L}\right)}_{\text{two-slit interference}} \cdot \underbrace{\mathrm{sinc}^2\!\left(\frac{\pi a y}{\lambda L}\right)}_{\text{single-slit envelope}}.

So there are two length scales on screen: the fine fringes set by the separation $d$ , and the overall envelope set by the width $a$ . Increase $a$ and the envelope narrows, killing the outer fringes first.

Incidentally, this scene does not use that formula. It sums the true optical paths from eleven sub-sources per slit. The benefit is that when you drag $L$ down to something small (the near field), the fringe curvature and envelope distortion appear honestly — the analytic formula has already failed there.

The price of knowing which slit

That one switch in the scene is the heart of this module.

Turn it on: detectors appear at the slits, the fringes vanish, and the distribution collapses to a plain sum of the two single-slit patterns. Turn it off and they come back.

It is not “the detector knocks the particle sideways”

The usual story is that a photon from the detector hits the electron and scrambles its momentum, blurring the fringes. That story works in Heisenberg’s original microscope argument, but it is not the underlying reason.

Later experiments (micromaser which-path markers, for instance) extract path information with essentially no momentum transfer, and the fringes still die. The real reason is that once the environment records the path, particle and detector are in an entangled state

\ket{\Psi} = \tfrac{1}{\sqrt2}\left(\ket{\text{left}}\ket{D_\text{left}} + \ket{\text{right}}\ket{D_\text{right}}\right),

and looking at the particle alone leaves an interference term proportional to $\braket{D_\text{left}}{D_\text{right}} = 0$ . It is gone.

This is the first appearance of decoherence (module 12). It is also why erasing the detector’s record afterwards (a quantum eraser) can bring the fringes back in coincidence counts — the interference was never physically destroyed, only hidden behind path information.

Scale: why daily life shows none of this

De Broglie ties momentum to wavelength:

\lambda = \frac{h}{p}.

An electron at 100 eV: $\lambda \approx 0.12\ \mathrm{nm}$ — a crystal makes a fine “slit”, which is exactly what Davisson and Germer used in 1927.
A 10 g bullet at 500 m/s: $\lambda \approx 1.3\times10^{-34}\ \mathrm{m}$ .

The second number is nineteen orders of magnitude smaller than a nucleus. To get a resolvable 0.1 mm fringe spacing you would need slits closer than $10^{-30}$ m. That is why you have never seen a bullet diffract — not because macroscopic objects are somehow non-quantum, but because no slit can match their wavelength.

Key formulas

De Broglie relation
$\lambda = h/p$
Everything has a wavelength; for macroscopic objects it is unmeasurably small
Interference intensity
$P = |\psi_1 + \psi_2|^2$
Add amplitudes first, then square — the order matters
Fringe spacing
$\Delta y = \lambda L / d$
Measurable directly in the scene
Single-slit envelope
$I \propto \mathrm{sinc}^2(\pi a y/\lambda L)$
The second length scale, set by slit width

Think it through

Drop the emission rate to one particle per minute, so only ever one particle is inside the apparatus. Do the fringes still appear? (They do. The experiment has been done.) What, then, is that particle interfering with?
With only the left slit open, point $y_0$ on the screen receives particles. Open both slits and $y_0$ becomes a dark fringe. Adding a route makes the destination unreachable — say why that is impossible in classical probability theory.
The scene deliberately draws no trajectory from source to screen. What would you be led to believe if it did?

Wave–particle duality and the double slit

What you will see

Assumed background

The double slit: how do particles arriving one at a time grow fringes?

Experiment mode

Parameters

Emission

Display

Distribution on the screen

Three things you just saw

Fringe spacing: a formula you can verify with a slider

The price of knowing which slit

Scale: why daily life shows none of this

Key formulas

Think it through

Go deeper · matching textbook sections

Having finished this module

Experiment mode

Parameters

Emission

Display

Distribution on the screen

Three things you just saw

Fringe spacing: a formula you can verify with a slider

The price of knowing which slit

Scale: why daily life shows none of this

∎Key formulas

Think it through

Go deeper · matching textbook sections

Having finished this module

Key formulas