Free Particle

The “Simplest” Particle — Not So Simple

A free particle (zero potential, V = 0) is the first system in every quantum mechanics textbook. Classically it is trivial: a particle moves at constant velocity with a perfectly sharp position and momentum at every instant. Quantum-mechanically, even this minimal case reveals dramatic departures from that picture.

Wave Packet: Localisation and Spreading

A Gaussian packet localises the particle at the cost of a spread in momentum (Heisenberg relation Δx·Δp ≥ ℏ/2). The packet spreads as it evolves:

A classical free particle never spreads — this broadening is a purely quantum effect. Different momentum components travel at different speeds, pulling the packet apart.

The Detector

In this pedagogical setup, the detector is modeled phenomenologically; this is no longer a strictly free-particle Hamiltonian once detection is included. We therefore treat this as a particle with a phenomenological detector and ask: how does a detection event happen? The three views displayed here give different answers:

Important distinction: the lower |ψ|² strip is a position distribution at fixed time, while detector clicks are modeled as time-of-arrival events in a finite detector window.

• Collapse view: before the detector fires there is no particle position. The wavefunction encodes probabilities. When a detection occurs, the wavefunction collapses — it disappears and the particle is localised at the detector.
• Pilot-wave view (de Broglie–Bohm): the particle has a definite position at all times, guided by the velocity field v(x,t) derived from the wavefunction. The detector fires when the particle’s actual trajectory enters the detector window. In this setup the displayed single-particle (conditional/effective) wave description is updated after detection, while the ontology remains unambiguous (a definite trajectory throughout).
• Many-worlds view (Everett): the wavefunction never collapses. Instead the universe “branches”: one branch contains a detector that fired, another does not. Both are equally real.

Wavefunction (exact analytic solution)

Gaussian wave packet (free-particle propagator):

where σ(t) = σ₀√(1+τ²), τ = ℏt/(mσ₀²), x̅(t) = x₀ + v_gt, and the phase φ includes a carrier wave, a chirp term, and a Gouy-like phase.

Pilot-wave guidance equation

The particle velocity in the pilot-wave view is:

Wave packet: different parts of the packet carry different local phase gradients. Particles near the front feel a larger phase gradient (higher “local k”) and are pushed forward; particles at the rear are slowed. This velocity differential is what spreads the packet. The pilot-wave velocity v_B at the particle’s current position is shown in the readout panel.

Compared to the classical prediction v = ℏk/m, a Bohmian particle born near the leading edge of the packet starts slightly faster; one born near the trailing edge starts slower. In an ensemble all particles spread exactly with |ψ|², reproducing the Born-rule statistics.

Phase velocity vs. group velocity

The group velocity is the speed of the envelope (and the pilot-wave particle near the packet centre). The phase velocity is the speed of the wavefronts, which are not directly observable.

Effective absorbing detector

In a rigorous treatment the detector would enter the Hamiltonian as an imaginary (absorbing) potential −i W(x), so that the Schrödinger equation becomes

This causes local amplitude decay at rate W/ℏ: the probability density “leaks” out of the detector region, representing absorption. In this simulation the analytic free-particle wavefunction is multiplied by a smooth exponential damping factor inside the detector window as a pedagogical approximation:

where f(x) is a smooth profile that equals 1 inside the detector and tapers to 0 outside over a narrow edge, and γ = 0.02 fs⁻¹ is a fixed absorption rate (corresponding to an imaginary-potential strength W ≈ γℏ ≈ 13 meV). This is not a full numerical PDE solve but correctly captures the qualitative physics: the wave amplitude is reduced in the detector region as the packet passes, providing backreaction on the guidance field in the pilot-wave view.

Internal units

Lengths in nm · times in fs · energies in eV · mass = electron mass.