Quantum Wave Packet: a free particle that spins and spreads

ψ (the helix)
Re ψ, shadow on back wall
Im ψ, shadow on floor
envelope |ψ|
peak marker
crest marker

Time t 0.00

Wave number k₀ 4.0 Starting width σ₀ 1.8

View

Time t: 0.00
Envelope speed v_g: 4.00
Crest speed v_p: 2.00
Width σ(t): 1.800
Peak at x: 0.00
Total probability: 1.0000
Wavelength λ: 1.571
Δx · Δp: 0.500

The big idea

A free quantum particle is a traveling bump of probability. The bump moves at the particle's speed, the crests inside it move at half that speed, and the bump slowly spreads, because a narrow bump has to be built from a spread of speeds.

Memory trick: “Bump at full, crests at half; squeeze it tight and it spreads out fast.”

A bit of history: Erwin Schrödinger published his wave equation in 1926. Later that year Max Born proposed that |ψ|² gives the probability of finding the particle. In 1927 Werner Heisenberg described the uncertainty relation, and Earle Kennard proved the exact form Δx · Δp ≥ ℏ/2 the same year. A Gaussian packet like this one is the shape that reaches that limit exactly.

To think about: “For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.” (1 Corinthians 13:12, KJV)