• Δ-X Heisenberg Uncertainty pixel-art physics T-shirt, white tee, front view

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Δ-X

The Heisenberg Runner / Uncertainty Principle Physics

\[\sigma_x\,\sigma_k \ge \tfrac{1}{2}, \qquad p = \hbar k \;\Rightarrow\; \sigma_x\sigma_p \ge \tfrac{\hbar}{2}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

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. The physics panel below has the equation, every symbol and the sources.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Para ter uma escala: o tamanho M tem 20 in (50.80 cm) de largura, cerca de 3.143 × 1034 comprimentos de Planck (lP = 1.616255 × 10−35 m). Uma fita métrica é mais rápida.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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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.

Δ-X is never in one place long enough to have one speed. That is not a character flaw. It is a theorem about waves.

Any wave confined to a short stretch of space has to be built by adding many wavelengths together. A single pure wavelength goes on forever, the same everywhere. To make a short packet you add neighbouring wavelengths so that they cancel away from the centre, and the shorter the packet, the wider the range you need. The first relation on this page is the exact form, for spreads measured as standard deviations: the width in position times the width in wavenumber is at least one half. That is mathematics, not physics. It holds for sound, radio, water waves, anything.

You can hear it. A tone burst with a spread in time of one millisecond cannot have its frequency pinned down to better than about 80 hertz. Make a click shorter and it loses its pitch altogether.

Quantum mechanics adds one sentence. Louis de Broglie's relation says a particle's momentum is ħ times its wavenumber. Multiply through, and the wave fact becomes the Heisenberg relation: position spread times momentum spread is at least ħ/2. Earle Kennard proved it in this exact form in 1927, and showed that packets with a Gaussian, bell-curve shape reach the bound exactly. Every other shape does worse.

The consequence is not small. Confine an electron to the width of an atom, about a tenth of a nanometre, and its momentum must spread so much that its velocity spread is at least roughly 600 kilometres per second. Nothing pushed it. Squeezing the wave did that. Electrons in atoms move fast because atoms are small.

Δ-X keeps running, because standing still in a small space was never an option.

Equations

\[\sigma_x\,\sigma_k \ge \tfrac{1}{2}, \qquad p = \hbar k \;\Rightarrow\; \sigma_x\sigma_p \ge \tfrac{\hbar}{2}\]
SymbolMeaningUnit
\(\sigma_x\) position spread \(\mathrm{m}\)
\(\sigma_k\) wavenumber spread \(\mathrm{m⁻¹}\)
\(k\) wavenumber \(\mathrm{m⁻¹}\)
\(p\) momentum \(\mathrm{kg·m/s}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)

Sources

  1. Kennard (1927) Zur Quantenmechanik einfacher Bewegungstypen, Z. Phys. 44, 326
  2. Feynman Lectures on Physics Vol. I Ch. 38: The Relation of Wave and Particle Viewpoints
  3. Feynman Lectures on Physics Vol. III Ch. 2: The Relation of Wave and Particle Viewpoints
  4. Encyclopaedia Britannica — Uncertainty principle

Source Kennard, E. H. (1927). Zur Quantenmechanik einfacher Bewegungstypen. Zeitschrift für Physik 44, 326–352.doi.org/10.1007/BF01391200

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