The Double Slit with Single Particles

The Setup

A particle source fires particles — say, electrons — at a barrier containing two narrow slits, both open. A detector screen sits behind the barrier. The question is: what pattern builds up on the screen after many particles arrive?

Classical intuition says: if a particle is a little bullet, it must go through slit 1 or slit 2. Closing one slit should halve the intensity but keep the shape of the patch. The total pattern with both slits open should be the sum of the two single-slit patches — a double hump with a smooth maximum in the centre.

Experiment delivers something entirely different.

What the Screen Actually Shows

With both slits open, the screen builds up an interference pattern — alternating bright bands (where particles arrive often) and dark fringes (where almost no particle ever lands). The bright-fringe spacing $\Delta y$ satisfies

$\Delta y = \frac{\lambda L}{d},$

where $\lambda$ is the de Broglie wavelength of the particle, $L$ is the distance from the barrier to the screen, and $d$ is the slit separation. This is exactly the result you get from Young's double-slit experiment with light — a wave phenomenon.

One Particle at a Time

The startling version of the experiment fires particles so rarely that only one is in the apparatus at a time — there is no possibility of two particles interfering with each other. Each particle registers as a single localised dot on the screen.

After ten particles the dots look random. After a hundred they hint at a structure. After tens of thousands the full interference pattern has assembled itself, dot by dot.

This result was demonstrated conclusively by Tonomura and collaborators in 1989 for electrons, and has since been reproduced for neutrons, atoms, and even large molecules such as fullerenes (C $_{60}$ ). There is no escape: each individual particle somehow "knows" about both slits.

Why This Demands Quantum Mechanics

The key point is that closing one slit can increase the count at some positions on the screen. In any classical additive model, removing a path can only reduce or leave unchanged the arrival rate at any point. The fact that blocking slit 2 destroys the dark fringes — and lets particles land where they previously could not — proves that the two paths do not simply add as independent probability streams.

The resolution is that we must describe the particle by a probability amplitude $\psi$ , not a probability. The amplitude acquires a contribution from each path:

$\psi = \psi_1 + \psi_2.$

The probability of detecting the particle at a given position is then $|\psi|^2$ , not $|\psi_1|^2 + |\psi_2|^2$ . Expanding the square reveals the interference term:

$|\psi|^2 = |\psi_1|^2 + |\psi_2|^2 + 2\,\operatorname{Re}(\psi_1^* \psi_2).$

The cross-term $2\,\operatorname{Re}(\psi_1^* \psi_2)$ is positive at bright fringes and negative at dark ones. When the slit is blocked, $\psi_2 = 0$ , the cross-term vanishes, and the pattern collapses to a single-slit blob — exactly what is observed.

Which Slit Did It Go Through?

If you install a detector to find out which slit each particle passes through, the interference pattern immediately disappears. The screen shows only the classical double-hump. This is not a matter of the detector physically disturbing the particle; it is a fundamental consequence of quantum measurement. Obtaining "which-path" information collapses the amplitude into a definite path, destroying the coherence between $\psi_1$ and $\psi_2$ that produces fringes.

This complementarity — you can have wave-like interference or particle-like which-path information, but never both simultaneously — is one of the deepest principles of quantum mechanics. It is precisely the principle behind quantum key distribution: any attempt to intercept and re-emit a quantum signal disturbs the state and reveals the eavesdropper.

Summary

| Scenario | Pattern | Reason | |---|---|---| | Both slits open, no which-path info | Interference fringes | Amplitudes from both paths add coherently | | One slit closed | Single-slit diffraction blob | Only one amplitude contributes | | Both slits open, which-path measured | Double-hump (no fringes) | Measurement collapses coherence |

The single-particle build-up experiment makes explicit what the mathematics already implied: interference is a property of the probability amplitude, not of the particle crowd. A single particle interferes with its own possible paths.