Quantum Wave Packet Dispersion Calculator
Calculates how a quantum wave packet's spatial width grows over time due to quantum dispersion. Use it when studying free-particle evolution, matter-wave optics, or ultracold atom experiments.
About this calculator
A free quantum particle is described by a wave packet — a localized superposition of plane waves. Even without any force, the packet spreads over time because different momentum components travel at different phase velocities (dispersion). For a Gaussian wave packet with initial width σ₀, the width at time t is σ(t) = √(σ₀² + (ℏt / (m · σ₀²))²), where ℏ = 1.055 × 10⁻³⁴ J·s and m is the particle mass. The term ℏt/(m·σ₀²) is the 'spread parameter' — it grows when the particle is light, the initial packet is narrow, or the time elapsed is long. At t = 0, σ = σ₀ as expected. The group velocity v_g = ℏk/m (where k is the central wave number) determines where the center of the packet moves, while dispersion controls the width.
How to use
Consider an electron (m = 9.109 × 10⁻³¹ kg) with initial width σ₀ = 1.0 × 10⁻¹⁰ m after t = 1.0 × 10⁻¹⁵ s (1 femtosecond). Step 1: Compute spread term: ℏt/(m·σ₀²) = (1.055×10⁻³⁴ × 1×10⁻¹⁵) / (9.109×10⁻³¹ × (1×10⁻¹⁰)²) = 1.055×10⁻⁴⁹ / 9.109×10⁻⁵¹ ≈ 1.158 m. Step 2: σ(t) = √((1×10⁻¹⁰)² + (1.158)²) ≈ √(1.158²) ≈ 1.158 m. The packet has spread enormously — from atomic scale to over a meter — demonstrating how rapidly a light particle delocalizes.
Frequently asked questions
Why does a quantum wave packet spread even in empty space with no forces?
Spreading occurs because a localized packet must be built from a superposition of momentum eigenstates, each propagating at a different phase velocity v = p/(2m) = ℏk/(2m) for a free non-relativistic particle. The components drift apart over time, broadening the probability distribution. This is purely a wave phenomenon with no classical counterpart for a point particle. Heavier particles spread much more slowly because the velocity differences between momentum components are smaller, which is why macroscopic objects exhibit negligible quantum spreading.
How does initial wave packet width affect how fast it spreads?
Counterintuitively, a narrower initial packet spreads faster. The spread rate scales as ℏ/(m·σ₀²), so halving σ₀ quadruples the rate of spreading. A narrow packet corresponds (by the uncertainty principle) to a large range of momentum components, which diverge rapidly. A broad, well-spread packet has a narrowly peaked momentum distribution, so the components stay close together longer. This trade-off is central to designing matter-wave interferometers and atom traps where coherence must be maintained.
What is the difference between group velocity and phase velocity for a quantum wave packet?
Phase velocity v_ph = ω/k = ℏk/(2m) is the speed at which the peaks of individual plane-wave components move, while group velocity v_g = dω/dk = ℏk/m is the speed at which the envelope (center) of the packet moves. For a free non-relativistic particle, v_g = 2·v_ph, and v_g equals the classical particle velocity p/m. The center of the wave packet travels at v_g, which is what you would measure as the particle's velocity. Dispersion — the fact that v_ph depends on k — is responsible for the spreading of the packet over time.