Vågpaketet

Vågpaketet, or the wave packet, is a localized disturbance formed by superposing waves with a spread of wavelengths. In quantum mechanics, the state of a particle is described by a wave function, and a localized particle state is often modeled as a vågpaket obtained by combining plane waves with amplitudes around a central wave number k0 (corresponding to momentum p0 = ħk0). The spread in wave numbers implies a spread in momentum, in line with the Heisenberg uncertainty principle, which explains why a localized packet cannot have perfectly definite position or momentum.

In a free particle setting, the different components of a vågpaket travel with different phase and group velocities, causing the packet to spread over time. For a nonrelativistic particle, the Schrödinger equation leads to a group velocity v_g = ħk/m, so the packet moves with an average velocity p0/m while gradually broadening. A common mathematical example is a Gaussian vågpaket, which retains a Gaussian shape while its width increases with time. The typical time dependence of the width Δx(t) is Δx(t) = Δx0 sqrt(1 + (ħ t / (2 m Δx0^2))^2), illustrating dispersion-driven spreading.

The center of the vågpaket follows classical-like motion, with its position expectation value Δx advancing according to the average momentum. The spread in position and momentum is constrained by the uncertainty principle, Δx Δp ≥ ħ/2. In specific potentials or relativistic regimes, the evolution and spreading can differ, but the vågpaket remains a fundamental tool for describing localization and dynamics in quantum systems. The concept is also used in optics to describe short pulses and in signal processing as a Fourier-analytic representation of localized signals.