Problem — Wave-particle complementarity

Problem

In a double-slit experiment with electrons, if “which slit it passes through” is detected, the interference pattern disappears. Explain in terms of wave packets and position-momentum uncertainty.

Solution

Interference requires uncertainty about the path. The fringe pattern arises because the electron’s wave passes through both slits and overlaps with itself. This requires that it is not known which slit it passed through: the two “paths” remain indistinguishable and their amplitudes add coherently.

Revealing the path destroys the coherence. To determine the slit one must interact with the electron (for example by illuminating it with a photon). This position measurement (small $\Delta x$, we know where it passed) imposes, by the uncertainty relation

$\Delta x\,\Delta p \gtrsim \frac{\hbar}{2}$

an uncontrollable disturbance of the momentum (large $\Delta p$). This random “kick” randomly dephases the two components of the wave: the fringes blur until they disappear, leaving the simple sum of two blobs (particle-like behaviour).

Language of wave packets. A packet well localised in position (narrow in $x$) is, by Fourier, wide in momentum: it contains many different wavelengths and does not produce sharp fringes. An extended packet (wide in $x$, narrow in $p$) has a well-defined wavelength and interferes well, but does not allow one to say “where it is”. One cannot have both: this is complementarity.

Summary. Path information (particle-like aspect) and fringe visibility (wave-like aspect) are complementary quantities: gaining one causes the loss of the other. This is not a technological limitation of the apparatus, but a fundamental property imposed by the uncertainty principle.

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Topics: Fisica quantistica Concepts: Dualismo onda-particella · Principio di indeterminazione · Interferenza