Why does the wave packet spread even though there's no force acting on it?

Spreading is not caused by a force but by dispersion, a wave phenomenon. The packet is a superposition of plane waves with different momenta. For a free particle, the phase velocity depends on momentum (v_phase = p/2m). These components move apart over time, deconstructively interfering in the tails and broadening the packet's position distribution. This is intrinsic to the wave description of matter.

What does it mean that the simulator sets ħ = m = 1?

This is a common simplification in computational physics known as using 'natural units.' It removes physical constants from equations, making the math cleaner and allowing us to focus on the functional relationships. All results are in these scaled units. To connect to a real system, you would reintroduce the actual values of ħ and the particle's mass.

How is the initial momentum (k₀) related to the particle's motion?

The parameter k₀ (wave number) is related to the average momentum of the packet: ⟨p⟩ = ħk₀. It determines the group velocity, v_group = ħk₀/m, which is the speed at which the packet's peak moves. A larger k₀ means a faster-moving packet, but it does not stop the spreading, which is governed by the momentum *uncertainty* (the width of the packet in momentum space).

Can a wave packet ever stop spreading?

A free Gaussian wave packet never stops spreading; its width σ(t) increases indefinitely. However, spreading can be halted or reversed by an external potential, such as a harmonic oscillator potential, which provides a restoring force that can produce stable, non-dispersing wave packets (coherent states). This simulator models free space, where no such potential exists.

Gaussian Wave Packet

A Gaussian wave packet provides a fundamental model for a localized quantum particle in free space. This simulator visualizes the time evolution of such a packet, governed by the time-dependent Schrödinger equation. The initial state is a minimum-uncertainty Gaussian, defined by its initial width σ₀ and central momentum k₀. The core physics demonstrated is wave packet dispersion: even without a potential, the packet spreads in position space over time. This is a direct consequence of the wave-like nature of quantum particles and the dispersion relation for free particles, E = p²/2m, which links energy to momentum quadratically. Different Fourier components (momentum states) within the packet travel at different phase velocities, causing the packet to broaden. The simulator calculates the time-dependent width as σ(t) = σ₀ √(1 + (ħt / (2mσ₀²))²), using the simplification ħ = m = 1. Students can manipulate the initial width and momentum to observe their profound impact on the spreading rate and group velocity. Key learnings include the Heisenberg uncertainty principle (Δx Δp ≥ ħ/2), the distinction between phase and group velocity, and how a pure momentum state (a plane wave) has infinite position uncertainty and does not spread, while a localized packet must have momentum uncertainty and therefore inevitably disperses.

Who it's for: Undergraduate students in introductory quantum mechanics or modern physics courses, and educators seeking to visualize wave packet dynamics and quantum uncertainty.

Key terms

Wave Packet
Schrödinger Equation
Uncertainty Principle
Wave Packet Spreading
Dispersion Relation
Group Velocity
Fourier Transform
Gaussian Function

How it works

Tighter initial localization implies broader momentum spread, so the wavefunction fans out. Green dashed lines mark ±σ as a rough width guide.

Frequently asked questions

Why does the wave packet spread even though there's no force acting on it?: Spreading is not caused by a force but by dispersion, a wave phenomenon. The packet is a superposition of plane waves with different momenta. For a free particle, the phase velocity depends on momentum (v_phase = p/2m). These components move apart over time, deconstructively interfering in the tails and broadening the packet's position distribution. This is intrinsic to the wave description of matter.
What does it mean that the simulator sets ħ = m = 1?: This is a common simplification in computational physics known as using 'natural units.' It removes physical constants from equations, making the math cleaner and allowing us to focus on the functional relationships. All results are in these scaled units. To connect to a real system, you would reintroduce the actual values of ħ and the particle's mass.
How is the initial momentum (k₀) related to the particle's motion?: The parameter k₀ (wave number) is related to the average momentum of the packet: ⟨p⟩ = ħk₀. It determines the group velocity, v_group = ħk₀/m, which is the speed at which the packet's peak moves. A larger k₀ means a faster-moving packet, but it does not stop the spreading, which is governed by the momentum *uncertainty* (the width of the packet in momentum space).
Can a wave packet ever stop spreading?: A free Gaussian wave packet never stops spreading; its width σ(t) increases indefinitely. However, spreading can be halted or reversed by an external potential, such as a harmonic oscillator potential, which provides a restoring force that can produce stable, non-dispersing wave packets (coherent states). This simulator models free space, where no such potential exists.

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Frequently asked questions

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