What does a narrow wave packet cost in momentum?
What does a narrow wave packet cost in momentum?
- Compare broad and narrow initial packets.
- Record Δk for each Δx.
- Increase spreading rate and follow the envelope.
What to measure
Units and parameter definitions are shown beside every laboratory control and live reading.
The idea underneath
A localised wave packet requires a superposition of wave numbers. A Gaussian packet is a minimum-uncertainty state with ΔxΔk=1/2, so narrowing position broadens its wave-number distribution. Free evolution spreads the envelope because its component waves have different group velocities. The display uses an analytic Gaussian spreading envelope with an illustrative phase clock. It wraps the packet centre at the display edge for continuous inspection; the wrap is not a physical periodic boundary calculation.
Δx Δk=½ · σ(t)=σ₀√[1+(αt/σ₀²)²]Units and parameter definitions are shown beside every laboratory control and live reading.
Build the argument
- State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
- Normalise the wavefunction so the integral of |ψ|² is one.
- Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.
Worked example 1
Δx=0.5 nm gives Δk=1 nm⁻¹.
Worked example 2
Δx=1 nm gives Δk=0.5 nm⁻¹.
Applications
Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.
Common misconception
A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.
A further investigation
Halving Δx changes the minimum Δk to…
Check your understanding
Halving Δx changes the minimum Δk to…
- Twice its value
- Half its value
- Zero
Explanation
Their product remains 1/2 for a Gaussian minimum-uncertainty packet.
Model limits
Analytic Gaussian envelope; the phase/playback rates are illustrative and are not a calibrated electron propagation time.