Double-slit experiment with single electrons

What just happened?

The electrons travel one at a time — never two at once. At small $N$ the hits look like random dots. But once you collect millions of hits, they arrange into regular fringes: bright and dark bands. This is interference — a phenomenon characteristic of waves, not of classical balls.

Wave-particle duality:

Every particle has an associated de Broglie matter wave of wavelength:

where $h$ is the Planck constant and $p$ is the electron momentum. It is this wave $\psi$ that passes through both slits and interferes with itself. The probability of hitting a given spot on the screen is described by $|\psi|^2$:

where $d$ is the slit separation, $L$ is the distance to the screen, and $x$ is the position on the screen. It is this formula that generates the fringes seen in the simulation.

The role of measurement — why does the detector spoil the pattern?

When you turn on the which-slit detector, you measure which way the electron went. The wave function collapses — there is no longer a superposition of the two paths, so the interference term vanishes. What remains is the plain sum of two single slits, that is two peaks:

The very act of observation changes the result — one of the deepest features of quantum mechanics. The invariant holds: Σ hits = N — every fired electron reaches the screen; only the distribution changes.

Why you need this for the exam:

This extends the topic of interference and diffraction and the de Broglie wave. The trick: fringes = wave (interference), two peaks = a particle "caught" by measurement. Problems combine $\lambda = h/p$ with the bright-fringe condition $d\sin\theta = n\lambda$ — write down what is given ($d$, $L$, $\lambda$), and compute the positions of the maxima.

Remember: a single electron interferes with itself — $|\psi|^2$ gives fringes. The which-slit measurement destroys interference and two peaks remain. Matter is simultaneously a wave and a particle.