Δ-X: Heisenberg Runner / Uncertainty Principle
Quantum Foundations arc, page 2 of 193. Back to The Pixelated Physics Universe
Who they are
Squeeze a quantum particle’s position and its momentum spreads out. Δ-X runs on the uncertainty relation: the tighter a quantum state’s position spread Δx, the wider its momentum spread Δp. Δ-X wears it in pixel art on a unisex tee.
Δ-X is a character of the Quantum Foundations arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
A wave packet squeezed into a small region has to be made of many wavelengths, so its momentum is spread out. The product of the spread in position and the spread in momentum can never be smaller than ħ/2; this is a property of waves, not a fault of our instruments.
$$ \begin{gathered} \sigma_x\,\sigma_k \ge \tfrac{1}{2}, \qquad p = \hbar k \;\Rightarrow\; \sigma_x\sigma_p \ge \tfrac{\hbar}{2} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\sigma_x$ | position spread | m |
| $\sigma_k$ | wavenumber spread | m⁻¹ |
| $k$ | wavenumber | m⁻¹ |
| $p$ | momentum | kg·m/s |
| $\hbar$ | reduced Planck constant | J·s |
Sources
- Kennard (1927) Zur Quantenmechanik einfacher Bewegungstypen, Z. Phys. 44, 326
- Feynman Lectures on Physics Vol. I Ch. 38: The Relation of Wave and Particle Viewpoints
- Feynman Lectures on Physics Vol. III Ch. 2: The Relation of Wave and Particle Viewpoints
- Encyclopaedia Britannica — Uncertainty principle
Crossovers
Δ-X meets Nickela (Electrons Diffract 1927). Nickela fires electrons at her crystal and gets a sharp diffraction beam, because each electron has a definite de Broglie wavelength λ = h/p. Δ-X points out the cost: a momentum that sharp means the electron's position is spread across many atomic planes. That spread is exactly why the planes can act together as a grating.
Δ-X meets Broadling (Natural Linewidth). Δ-X trades position for momentum. Broadling makes the same kind of trade with time and energy: a state that lives only for a time τ has an energy width Γ = ħ/τ. Neither of them can get a sharp value of both quantities at once.
Δ-X meets Molassa (Doppler Cooling Limit). Molassa slows atoms with laser light, but every photon she scatters gives the atom a random kick. That random recoil heating, balanced against the cooling, sets her Doppler limit. Δ-X nods: even light carries momentum ħk, and nobody gets the kicks for free.
Wear the lore
Concept keywords: Heisenberg Runner / Uncertainty Principle, Δ-X, Quantum Foundations, physics lore, Pixelated Physics