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Wave Packet Hoodie

PRECISE NOW, VAGUE LATER | Pixelated Physics

\[i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Pin down where a free particle is, and its wavefunction starts to spread. A free Gaussian wave packet at seven moments: it travels, its envelope widens as σ(t)² = σ₀² + (ħt/2mσ₀)², and its peak drops. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

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.

Full refund & replacement policy

The physics

A free quantum particle described by a narrow wave packet does not stay narrow. The narrower it starts, the wider its spread of momenta, so the faster it spreads out over time.

Pin down where a free particle is, and its wavefunction starts to spread.

The packet is a Gaussian envelope around a carrier wave of wavenumber k₀. It is built from a range of wavenumbers, and in free space each travels at a different speed, so the packet widens. The width grows as σ(t)² = σ₀² + (ħt/2mσ₀)².

The narrower the start, the faster the spread, because a narrow packet needs a wide range of momenta. That is the uncertainty principle at work.

The art shows the exact solution at seven times from top to bottom. The outline is the envelope ±|ψ|, the inner line is the real part of ψ, and the falling height is real: total probability stays 1 while it spreads.

Precise now, vague later.

Equations

\[i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}\]
\[\sigma(t)^2 = \sigma_0^2 + \left(\frac{\hbar t}{2m\sigma_0}\right)^2\]
\[v_g = \frac{\hbar k_0}{m}\]
SymbolMeaningUnit
\(\psi\) wavefunction \(\mathrm{m^{-1/2}}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(m\) particle mass \(\mathrm{kg}\)
\(x, t\) position and time \(\mathrm{m, s}\)
\(\sigma(t)\) width of the probability density \(\mathrm{\psi}\)
\(\sigma_0\) initial width \(\mathrm{m}\)
\(k_0\) central wavenumber \(\mathrm{m^{-1}}\)
\(v_g\) group velocity \(\mathrm{m/s}\)

Sources

  1. Schrödinger (1926) Quantisierung als Eigenwertproblem, Ann. Phys. 79, 361
  2. Griffiths, Introduction to Quantum Mechanics, Sec. 2.4 and Problem 2.22
  3. Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. 1, Complement G_I

Source Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. Annalen der Physik 384, 361–376.doi.org/10.1002/andp.19263840404

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