10.5 Double-slit experiment

Wave-particle duality

Wave-particle duality describes how light and matter can behave as waves or as particles depending on the experiment you run. For a Physics II course, the double-slit experiment is the central demonstration: it shows wave-like interference from light (and even from individual particles), connecting classical wave optics to the quantum ideas you'll encounter later.

Light as waves vs. particles

Light behaves as a wave when it diffracts through narrow openings or produces interference patterns. But it behaves as a particle (a photon) in phenomena like the photoelectric effect, where light ejects electrons from a metal surface only above a certain frequency, regardless of intensity. Einstein's explanation of the photoelectric effect in 1905 established that light carries energy in discrete packets of $E = hf$, cementing its particle nature alongside its wave properties.

Quantum mechanics reconciles these two pictures: light is neither purely a wave nor purely a particle, but something that shows one behavior or the other depending on how you measure it.

Matter waves

In 1924, de Broglie proposed that all matter has wave-like properties, with a wavelength given by:

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

where $h$ is Planck's constant and $p$ is the particle's momentum. For everyday objects, this wavelength is unimaginably small and undetectable. But for electrons and other subatomic particles, the wavelength is large enough to observe. Electron diffraction experiments in the late 1920s confirmed de Broglie's hypothesis directly.

Experimental setup

The double-slit experiment sends light (or particles) through two narrow openings and records the resulting pattern on a screen. The setup has three main components.

Light source

You need a coherent, monochromatic source so that the waves arriving at the two slits maintain a constant phase relationship. A laser works well for this. If you reduce the intensity enough, you can send photons (or electrons) through one at a time, which is how the single-particle version of the experiment works.

Double-slit apparatus

An opaque barrier has two narrow, parallel slits cut into it. The key dimensions are:

• Slit width $a$: narrow enough to produce significant diffraction (typically on the order of micrometers for visible light)
• Slit separation $d$: the center-to-center distance between the slits, which controls the spacing of the interference fringes

Detection screen

A screen (or electronic detector like a CCD) is placed a distance $L$ from the slits. It records where the light lands. The farther the screen is from the slits, the more spread out the fringes become, making them easier to measure.

Interference pattern

The pattern on the screen consists of alternating bright fringes (constructive interference) and dark fringes (destructive interference). This is the signature of wave behavior.

Bright and dark fringes

• Bright fringes appear where waves from the two slits arrive in phase and reinforce each other.
• Dark fringes appear where waves arrive out of phase and cancel each other.
• The central maximum (zeroth-order fringe, $n = 0$) is the brightest because the path lengths from both slits are equal there.
• Fringe intensity gradually decreases as you move away from the center, modulated by the single-slit diffraction envelope.

Fringe spacing

The distance between adjacent bright (or dark) fringes depends on three things: the wavelength $\lambda$, the slit separation $d$, and the slit-to-screen distance $L$. Larger wavelength or larger $L$ spreads the fringes apart; larger $d$ pushes them closer together.

Mathematical analysis

Path difference

When waves travel from each slit to the same point on the screen, they cover slightly different distances. That difference is the path difference:

$\Delta r = d \sin\theta$

where $d$ is the slit separation and $\theta$ is the angle measured from the central axis. This path difference determines whether the waves add up or cancel at that point.

Constructive vs. destructive interference

Constructive interference (bright fringe) occurs when the path difference equals a whole number of wavelengths:

$d \sin\theta = n\lambda \quad (n = 0, \pm1, \pm2, \ldots)$

Destructive interference (dark fringe) occurs when the path difference equals a half-integer number of wavelengths:

$d \sin\theta = \left(n + \frac{1}{2}\right)\lambda \quad (n = 0, \pm1, \pm2, \ldots)$

Here $n$ is the order of the fringe. The central bright fringe is $n = 0$, the next bright fringes out are $n = \pm1$, and so on.

Fringe spacing equation

For small angles (which applies when $L \gg d$), $\sin\theta \approx \tan\theta = y/L$, and the spacing between adjacent bright fringes simplifies to:

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

• $\Delta y$: distance between adjacent bright fringes on the screen
• $\lambda$: wavelength of the light
• $L$: distance from the slits to the screen
• $d$: slit separation

This equation is one of the most useful in wave optics. You can rearrange it to solve for any unknown if you know the other three quantities.

Single-particle behavior

Here's where the experiment gets strange and connects to quantum mechanics.

Probability distribution

When you send particles through one at a time, each one lands at a single, definite spot on the detector. At first the hits look random. But after thousands of particles, the accumulated pattern of hits matches the wave interference pattern exactly. The wave function $\psi$ doesn't tell you where a specific particle will land; instead, $|\psi|^2$ gives the probability density for finding the particle at each position. High-probability regions correspond to bright fringes, and low-probability regions correspond to dark fringes.

Collapse of the wave function

Before detection, the particle is described by a wave function that passes through both slits simultaneously (a superposition). When the detector registers the particle, the wave function "collapses" to a single location. If you add a detector at the slits to determine which slit the particle passes through, the interference pattern disappears and you get two bands instead. The act of determining the path destroys the superposition that produces interference.

Quantum implications

Complementarity principle

Bohr's complementarity principle states that wave-like and particle-like behaviors are mutually exclusive in any single experiment. If your setup reveals which path the particle took (particle behavior), you lose the interference pattern (wave behavior). You can observe one aspect or the other, but never both at once.

Measurement and observation effects

Observing which slit a particle passes through counts as a measurement that disturbs the quantum state. This isn't about clumsy instruments; it's a fundamental feature of quantum mechanics. Even the gentlest possible measurement that gains "which-path" information is enough to destroy interference. This sensitivity of quantum systems to measurement leads to concepts like quantum decoherence, where interactions with the environment effectively act as measurements and suppress quantum behavior.

Historical significance

Young's original experiment

Thomas Young performed the first double-slit experiment in 1801 using sunlight passed through a small hole and then through two closely spaced slits. The interference fringes he observed on a screen provided strong evidence that light is a wave, directly challenging Newton's corpuscular (particle) theory of light. Young's work helped pave the way for the wave theory that culminated in Maxwell's electromagnetic equations decades later.

Feynman's interpretation

Richard Feynman called the double-slit experiment the demonstration that contains "the only mystery" of quantum mechanics. In his path integral formulation, a particle doesn't take one path or the other; it takes all possible paths simultaneously. You calculate the probability amplitude for each path and then add those amplitudes together. Because amplitudes can be positive or negative (or complex), they interfere, producing the pattern. Probabilities themselves don't interfere, but probability amplitudes do.

Modern applications

Electron diffraction

Electrons, with their short de Broglie wavelengths, produce diffraction patterns that reveal atomic-scale structure. Transmission electron microscopes (TEMs) exploit this to image materials at resolutions far beyond what visible light allows. This is widely used in materials science, nanotechnology, and biology.

Quantum computing implications

Quantum computers rely on superposition, the same principle at work in the double-slit experiment. A qubit exists in a superposition of $|0\rangle$ and $|1\rangle$ states, and quantum algorithms use interference of probability amplitudes to amplify correct answers and suppress wrong ones. Maintaining this superposition (coherence) is the central engineering challenge, much like preserving the interference pattern in a double-slit setup requires isolating the system from disturbances.

Variations and extensions

Multiple-slit experiments

Adding more slits (three, four, or hundreds) creates a diffraction grating. The more slits you have, the sharper and narrower the bright fringes become, which is why diffraction gratings are used in spectroscopy to separate wavelengths with high precision. The same interference principles apply; you just have more sources contributing.

Delayed-choice experiments

In John Wheeler's delayed-choice thought experiment (later performed in the lab), the decision of whether to measure "which path" or "interference" is made after the particle has already passed through the slits. The result still matches whichever measurement you chose. This rules out any simple picture where the particle "decides" at the slits whether to be a wave or a particle. It reinforces that quantum properties aren't determined until measurement actually occurs.

Philosophical interpretations

Copenhagen interpretation

Developed by Bohr and Heisenberg, this is the most widely taught interpretation. It holds that the wave function is a complete description of a quantum system, and measurement causes it to collapse into a definite outcome. Before measurement, it's meaningless to ask what the particle is "really doing." Physical properties only become definite when observed.

Many-worlds interpretation

Proposed by Hugh Everett III in 1957, this interpretation says there is no collapse at all. Instead, every possible measurement outcome is realized in a separate branch of a continuously splitting universe. What looks like collapse to you is just your branch seeing one result. This preserves determinism at the cost of an enormous (possibly infinite) number of parallel realities. Both interpretations make identical experimental predictions, so the choice between them remains a matter of philosophy rather than physics.