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With two coherent paths, the probability amplitude at a screen is ${\psi }_1(x)+e^{i\varphi }{\psi }_2(x)$. The Born rule givesThe final term shifts when the relative phase changes, producing interference fringes. For slit separation $d$ in the far field, the phase difference is approximately $2\pi d\sin \theta /\lambda$; constructive interference occurs at $d\sin \theta =n\lambda$. Finite slit widths add a diffraction envelope, so an ideal equally bright, infinitely narrow slit model does not predict the full observed envelope.

single-electron interference pattern buildup starts with isolated localized detector events at sufficiently low source intensity. Repeated events accumulate the interference probability distribution. Thus detection is discrete while propagation is described by coherent probability amplitudes. The effect cannot be explained by interference between different particles if the apparatus is occupied by at most one particle at a time. A motivating realization used a field-emission electron source, an electron biprism to recombine two paths, electron lenses and a position-sensitive electron counter; it recorded individual arrivals building up the fringes. This was a biprism realization of two-path interference, not a plate with two literal slits.^[ref]

The crucial comparison is between both coherent paths open and a preparation that supplies which-path information. If the paths become correlated with detector states $|D_1⟩,|D_2⟩$, the joint state is ${\psi }_1|D_1⟩+e^{i\varphi }{\psi }_2|D_2⟩$. Ignoring that detector multiplies the interference term by $⟨D_1|D_2⟩$. Orthogonal detector states remove the term completely, while identical detector states preserve it. This is a consequence of entanglement and the partial trace; a human need not read the detector.

In a real test, check the single-path distributions, detector background, source coherence and stability of the relative phase. Classical phase drift can wash out fringes without supplying which-path information. Conversely, localized detections alone do not establish coherent two-path propagation; the controlled phase dependence of their accumulated distribution provides the discriminating evidence.

A sequence of individual localized electron detections can accumulate a two-path interference distribution when the apparatus maintains coherence. The 1989 experiment led by Akira Tonomura used a field-emission source, an electron biprism, electron lenses and a position-sensitive electron counter. It realized two coherent paths without literal machined double slits. Low occupation of the instrument excludes interactions between different electrons as the origin of the fringes.^[ref]

Wave-particle duality names the coexistence of localized detection events and wave-like interference in quantum mechanics. A photon or electron is detected as an individual event, yet the distribution of many events depends on probability amplitudes from alternative paths. If the alternatives remain coherent, adding the probability amplitudes before applying the Born rule gives an interference term; simply adding the separate path probabilities does not.

In the single-electron experiment of Akira Tonomura and collaborators, an electrostatic biprism divided and overlapped the electron wave, and a position-sensitive detector recorded individual arrivals. Successive accumulated records developed interference fringes. This makes the apparent tension concrete: localization describes the detector events, while matter-wave interference describes their statistical distribution.^[ref]