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Quantum Oscillator Hoodie

NEVER QUITE STILL | Pixelated Physics

\[\hat H = \frac{\hat p^2}{2m} + \frac12 m\omega^2\hat x^2\]
$54.99 USD
$54.99 USD
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Put a quantum particle in a parabolic well, a mass on an ideal spring, and it can only have certain energies. The first six states of the quantum harmonic oscillator, |ψₙ|² drawn on their levels E = (n + ½)ħω. The lowest is not zero. 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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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 quantum spring can only hold energy in equal steps of ħω, and its lowest level still carries half a step. That zero-point energy means the oscillator is never perfectly still, and each level has its own Hermite-function wave shape.

Put a quantum particle in a parabolic well, a mass on an ideal spring, and it can only have certain energies.

They form an evenly spaced ladder, E = (n + ½)ħω. Werner Heisenberg found it in 1925 with his new matrix mechanics, and Schrödinger found it again from his wave equation a year later.

Each rung in the art is the probability density |ψₙ|² for n = 0 to 5, a Hermite polynomial times a Gaussian, drawn on its own energy level inside the parabola. Level n has n zeros where the particle is never found.

The lowest rung is not at zero. Confining the particle forces some momentum on it, by the uncertainty principle, so even the ground state keeps ½ħω.

Never quite still.

Equations

\[\hat H = \frac{\hat p^2}{2m} + \frac12 m\omega^2\hat x^2\]
\[E_n = \left(n+\tfrac12\right)\hbar\omega\]
\[\psi_n(\xi) = \frac{1}{\sqrt{2^nn!\sqrt\pi}}\,H_n(\xi)\,e^{-\xi^2/2},\quad \xi = x\sqrt{m\omega/\hbar}\]
SymbolMeaningUnit
\(\hat H\) Hamiltonian, the energy operator \(\mathrm{J}\)
\(\hat p, \hat x\) momentum and position operators \(\mathrm{kg·m/s, m}\)
\(m\) particle mass \(\mathrm{kg}\)
\(\omega\) angular frequency of the oscillator \(\mathrm{rad/s}\)
\(E_n\) energy of level n \(\mathrm{J}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(H_n\) Hermite polynomial of degree n \(\mathrm{1}\)
\(\xi\) dimensionless position \(\mathrm{1}\)

Sources

  1. Heisenberg (1925) Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, Z. Phys. 33, 879
  2. Schrödinger (1926) Quantisierung als Eigenwertproblem (Zweite Mitteilung), Ann. Phys. 79, 489
  3. Griffiths, Introduction to Quantum Mechanics, Sec. 2.3

Source Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik 33, 879–893.doi.org/10.1007/BF01328377

Field notes

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