vågpacket

A vågpacket, or wave packet, is a localized disturbance described as a superposition of waves with a range of wavelengths or momenta. In physics, it often serves as a bridge between wave and particle pictures, capturing how a particle-like excitation can be localized in space while possessing a spread in momentum.

A wave packet is typically formed by combining plane waves with amplitudes A(k) centered around a central wave number k0. The packet can be written as psi(x,t) = ∫ A(k) e^{i(kx − ω(k)t)} dk, where ω(k) is the dispersion relation. If A(k) is chosen as a Gaussian, the resulting wave packet is approximately Gaussian in position, centered at x ≈ v_g t, with group velocity v_g = dω/dk|_{k0}. The uncertainty principle implies a trade-off between spatial width σ_x and momentum width σ_k, with σ_x σ_k ≈ 1/2 in suitable units.

Spreading occurs when the medium is dispersive, i.e., when ∂^2 ω/∂k^2 ≠ 0. In such cases, the packet broadens over time. For a free nonrelativistic quantum particle, ω(k) = ħ k^2/(2m), so the center moves at v_g = ħ k0/m, and the width evolves as σ_x(t) = σ_x0 √[1 + (ħ t/(m σ_x0^2))^2]. This spreading reflects the coexistence of position and momentum uncertainties and the wave-like nature of the quantum state.

Vågpacket concepts appear in quantum mechanics, where particles are described by wave packets, and in classical wave contexts such as optics and water waves, where localized pulses are analyzed as packets of Fourier components.