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Probability Cloud Hoodie

PROBABLY HERE | Pixelated Physics

\[\psi_{nlm}(r,\theta,\phi) = R_{nl}(r)\,Y_l^m(\theta,\phi)\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

This is not a picture of where the electron goes. It is a map of where you would find it if you looked. Hydrogen’s 5f orbital, |ψ₅₃₀|² sliced through the xz-plane: one radial node at 20 a₀, three angular. The electron is probably here. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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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

The electron in hydrogen does not follow an orbit. The square of its wavefunction gives the probability of finding it in each small volume, and for the 5f state (n=5, l=3, m=0) that probability forms the lobed, layered cloud in the art.

This is not a picture of where the electron goes. It is a map of where you would find it if you looked.

The state is hydrogen 5f: n = 5, l = 3, m = 0. Its wavefunction splits cleanly into a radial part and an angular part, ψ = R(r)Y(θ,φ). Square it and you get a probability density; Max Born proposed that reading in 1926, the same year Schrödinger wrote down the equation.

The empty bands are nodes, surfaces where |ψ|² is exactly zero. A state has n − 1 of them, four here. One is radial: a sphere at r = 20 a₀, where the radial factor changes sign. Three are angular: the equatorial plane and two cones at about 39° and 141° from the axis, where P₃(cos θ) vanishes.

The art is a slice through the xz-plane. Colour gives the sign of ψ, brightness gives |ψ|², and the dither is the texture of chance.

Probably here. Never in the gaps.

Equations

\[\psi_{nlm}(r,\theta,\phi) = R_{nl}(r)\,Y_l^m(\theta,\phi)\]
\[dP = |\psi|^2\,d^3r\]
\[n=5,\; l=3,\; m=0\]
SymbolMeaningUnit
\(\psi_{nlm}\) hydrogen wavefunction with quantum numbers n, l, m \(\mathrm{m^{-3/2}}\)
\(R_{nl}(r)\) radial part of the wavefunction \(\mathrm{m^{-3/2}}\)
\(Y_l^m(\theta,\phi)\) spherical harmonic, the angular part \(\mathrm{1}\)
\(r, \theta, \phi\) distance from the nucleus, polar angle, azimuthal angle \(\mathrm{m, rad, rad}\)
\(dP\) probability of finding the electron in the volume element d^3r \(\mathrm{1}\)
\(d^3r\) volume element \(\mathrm{m^3}\)
\(n, l, m\) principal, orbital and magnetic quantum numbers (here 5, 3, 0) \(\mathrm{1}\)

Sources

  1. Schrödinger (1926) Quantisierung als Eigenwertproblem (Erste Mitteilung), Ann. Phys. 79, 361
  2. Born (1926) Zur Quantenmechanik der Stoßvorgänge, Z. Phys. 37, 863
  3. Griffiths, Introduction to Quantum Mechanics, Ch. 4: The Hydrogen Atom
  4. Feynman Lectures on Physics Vol. III Ch. 19: The Hydrogen Atom and the Periodic Table

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

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