🌊 Quantum Wave Packet
A Gaussian wave packet is a superposition of plane waves ψ(x) = exp(ikx) weighted by a Gaussian in momentum space, giving ψ(x,0) = exp(ik₀x)·exp(−x²/(4σ₀²)). In a free-particle dispersion relation ω(k) = ħk²/(2m), different frequency components travel at different phase velocities, causing the packet to spread over time. The packet's centre moves at the group velocity v_g = ħk₀/m while each wave crest moves at the phase velocity v_φ = ħk₀/(2m). Heisenberg uncertainty: Δx·Δp ≥ ħ/2 — a narrow packet spreads faster. 🇺🇦 Українська
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How dispersion works
The exact free-particle solution is computed analytically: the Gaussian spreads as σ(t) = σ₀·√(1 + (ħt/(2mσ₀²))²) while its centre moves as ⟨x⟩(t) = x₀ + (ħk₀/m)t. The probability density |ψ|² remains a Gaussian at all times. The minimum uncertainty product Δx·Δp = ħ/2 holds only at t=0; it increases as the packet spreads. This spreading is a purely quantum-mechanical effect with no classical counterpart — it reflects the wave-like nature of matter described by the de Broglie relation λ = h/p.