Wave-Particle Duality: Electrons Through the Double Slit

The double slit experiment with individual particles is, as Richard Feynman declared, "the only mystery" in quantum mechanics. It's the experiment that forces us to accept that the microscopic world doesn't follow the rules of our everyday experience.

Here's what happens: you set up a double slit experiment (just like Problem 154 with light), but instead of a beam of light, you fire individual electrons, one at a time. Each electron is detected as a single dot on the screen — it arrives as a particle, at one specific location. There's no smearing, no wave pattern. Just a dot.

But if you fire thousands of electrons, one after another, and record where each one lands, something astonishing happens: the dots don't pile up in two blobs behind the slits (which is what you'd expect from particles). Instead, they gradually build up an interference pattern — the same pattern of alternating bright and dark bands that you'd get from waves passing through two slits.

Why this is so strange

Think about what this means:

1. Each electron is detected as a single particle at a single location. It doesn't "spread out." It goes ping at one spot.

2. But if you look at where many electrons land, they form a wave-like interference pattern. Some locations get many hits (constructive interference zones); some get almost none (destructive interference zones).

3. The electron somehow "knows" about both slits — if you close one slit, the interference pattern disappears and you get a single-slit diffraction pattern instead. The electron's behavior at slit A depends on whether slit B is open.

4. You cannot predict where any individual electron will land. You can only predict the probability of it landing at each location. The probability distribution follows the wave interference pattern.

5. If you put a detector at one slit to determine which slit the electron goes through, the interference pattern disappears and you get a two-blob particle pattern. The act of observing which path the electron takes destroys the wave behavior. This is the measurement problem.

The probability amplitude interpretation

Quantum mechanics resolves this by saying that unobserved electrons don't have definite positions. Between the source and the screen, the electron is described by a wave function $\psi(x)$ — a complex-valued probability amplitude that obeys a wave equation (the Schrödinger equation).

The wave function passes through both slits simultaneously and interferes with itself on the other side. The probability of the electron landing at position $x$ is:

$P(x) = |\psi(x)|^{2}$

This probability follows the interference pattern $|\psi_{1} + \psi_{2}|^{2}$, which has the characteristic $\cos^{2}$ fringes. When you fire many electrons, they distribute themselves according to this probability — more where $P$ is high, fewer where $P$ is low. The pattern emerges statistically.

The math: same as light, with de Broglie wavelength

The fringe pattern for electrons is identical to the light pattern from Problem 154, with one substitution: the wavelength is the de Broglie wavelength of the electron:

$\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$

For electrons accelerated through a potential difference $V$:

$\lambda = \dfrac{h}{\sqrt{2m_{e}eV}} = \dfrac{1.226}{\sqrt{V}}\text{ nm}$

Example: electrons at $V = 50$ V have $\lambda = 0.173$ nm — far smaller than visible light. This is why electron microscopes can resolve individual atoms (light can't because its wavelength is too large).

The fringe spacing on the screen:

$\Delta y = \dfrac{\lambda L}{d}$

where $d$ is the slit separation and $L$ is the distance to the screen.

What has actually been done experimentally

This isn't a thought experiment. It's been done:

• Electrons (1961): Claus Jönsson performed the double slit with electrons.
• Single electrons (1989): Tonomura et al. fired electrons one at a time and recorded the gradual buildup of the pattern. This is the definitive demonstration.
• Photons (single-photon regime, 1909): G.I. Taylor dimmed light until only single photons were present. Same result.
• Neutrons (1988): Zeilinger et al. Similar pattern.
• C₆₀ buckyballs (1999): Arndt et al. showed interference with molecules containing 60 carbon atoms — objects 1 nm across!
• Larger molecules (2019): Fein et al. demonstrated interference with molecules of over 2,000 atoms — the largest objects ever shown to exhibit quantum interference.

Interpretations

Physicists disagree about what the double slit experiment means:

• Copenhagen interpretation (Bohr): The electron doesn't have a definite path; it exists as a superposition until measured.
• Many-worlds interpretation (Everett): The electron goes through both slits — in different branches of reality that separate at the measurement.
• Pilot wave theory (de Broglie, Bohm): The electron is a real particle guided by a real wave; it goes through one slit, but the guide wave goes through both.
• Shut up and calculate (pragmatist): The math works; the interpretation is metaphysics.

Common mistakes

• Thinking the electron "splits" and goes through both slits. No — when detected, it's always at one location. The wave function goes through both; the particle appears at one.
• Thinking you need many electrons for interference. Even a single electron interferes with itself. The pattern appears statistically over many trials, but each electron individually carries the wave information.
• Thinking decoherence (which-path detection) is about "disturbing" the electron. Modern experiments show the pattern disappears even with arbitrarily gentle measurements. It's the information about the path, not the physical disturbance, that matters.

Try it in the visualization

Watch individual electrons arrive one at a time as random dots. At first, the distribution looks random. As more accumulate, the interference pattern emerges from the noise — bright bands and dark bands. Toggle "which-path detector" to see the pattern collapse into two classical blobs. Adjust the firing rate to watch the pattern build up faster or slower.