Complementarity

The double-slit experiment sits at the center of quantum mechanics. Fire electrons (or photons, or even molecules) one at a time through two narrow slits, and over many trials a striped interference pattern builds up on the detector screen — the hallmark of wave behavior. Now ask: which slit did that electron go through? The moment you obtain a definitive answer, the interference fringes vanish. This is Bohr's principle of complementarity: wave behavior and particle (which-path) behavior are mutually exclusive aspects of the same object. You can reveal one fully only at the cost of the other.

The double-slit setup

Place a barrier with slits at $x = 0$ separated by distance $d$ , and a detector screen a distance $L$ away. When no which-path information is available, the probability of landing at height $y$ on the screen is proportional to the intensity of two overlapping waves. The path-length difference between the wave from the upper slit and the wave from the lower slit is approximately

$\Delta = d \sin\theta \approx \frac{d \, y}{L}$

for small angles $\theta$ (where $\tan\theta \approx \sin\theta \approx y/L$ ). Constructive interference (a bright fringe) occurs when $\Delta = n\lambda$ for integer $n$ , giving fringe positions

$y_n = \frac{n \lambda L}{d}.$

This is the interference fringe spacing $\Delta y = \lambda L / d$ , entirely analogous to Young's experiment with light.

Which-path information destroys interference

Suppose you add a detector just behind one slit — a weak laser beam, a microscope, anything that can distinguish slit A from slit B. To resolve which slit the electron passed through, the detector must transfer at least one photon's worth of momentum to the electron. By the Heisenberg uncertainty principle, a transverse momentum kick

$\Delta p_y \gtrsim \frac{h}{d}$

shifts the electron's transverse position by an uncertain amount comparable to the fringe spacing. Averaging over this random kick washes the fringes out completely.

This is not merely a failure of cleverness: any interaction sufficient to record which-path information necessarily disturbs the electron's transverse momentum enough to erase the interference pattern. The two pieces of information — which path was taken and whether fringes exist — are complementary in Bohr's precise sense: both cannot be simultaneously sharp.

Why the wavefunction picture is consistent

In the formalism of quantum mechanics the electron is described by a wavefunction $\psi(\mathbf{r})$ . When no measurement is made at the slits, $\psi$ passes through both openings simultaneously; the two partial waves $\psi_A$ and $\psi_B$ add, and the probability density on the screen is

$|\psi_A + \psi_B|^2 = |\psi_A|^2 + |\psi_B|^2 + 2\operatorname{Re}(\psi_A^* \psi_B).$

The last term is the interference term; it encodes the phase relationship between the two paths. When a which-path measurement collapses the electron onto a definite slit, $\psi$ is no longer a coherent superposition of $\psi_A$ and $\psi_B$ — the off-diagonal interference term vanishes, and the screen pattern reduces to the incoherent sum $|\psi_A|^2 + |\psi_B|^2$ , which has no fringes.

The key insight is that obtaining information about a quantum system always has a physical cost: the measurement apparatus becomes entangled with the system, and the resulting entanglement destroys the phase coherence responsible for interference. Complementarity is therefore not a philosophical puzzle but a precise consequence of the quantum formalism.